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\section{Introduction}
The theory of belief functions (also called Evidential Theory or The Dempster-Shafer Theory) is a well known framework for reasoning under uncertainty \cite{Dempster}. It is widely used to model imperfect information. We can distinguish the imprecision (lack of accuracy), the uncertainty (lack of compliance comparing to the real world, due to the source of information) and even the inconstancy where is characterized by a high level of conflict.
The discrete case of belief functions, knowning a large succes, have been applied to several research fields, medical, military, risk management, information fusion \cite{Martin08,Martin07,Denoeux}, etc.\\
Otherwise, Smets in \cite{BFRN}, defined the basic notions of continuous belief functions to describe them in the extended set of reals.
Recently Attiaoui {\em et al.} \cite{Attiaoui} proposed a similarity measure for continuous belief functions based on Smets' formalism using the distance of Jousselme~\cite{J1}.
Our work in this paper will consider the notion of inclusion and how two continuous belief functions can be included in each other. This operation will help us later to take into account this specific characteristic during the information fusion and the measurement of the conflit.
Thus, we will define two forms: the strict inclusion and the partial one. To do so, we will present a measure called a degree of inclusion of an interval (the focal element of a continuous belief function) in the second one.
This work presents a new way to determine the relation of inclusion by considering a new vision for the continuous case within the theory of belief functions.
\section{Theory of belief functions: an overview}
This section recalls the necessary background related to the evidential theory. It has been developed by Dempster in his work on upper and lower probabilities \cite{Dempster}. Based on that, he was able to represent more precisely the observed data.\\
Later, in his book "A mathematical Theory of evidence'' \cite{shafer1976mathematical}, Shafer, presented that, any information defined by an expert characterized by basic belief assignments has two functions: a credibility and a plausibility function corresponding respectively to the lower and upper probabilities of Dempster.
\\
The theory was further developed by Smets in \cite{TBM,Smets89} who proposed the Transferable Belief Model (TBM). This model presents a pignistic probability induced by a belief function which is built by defining a uniform probability from each positive mass. Moreover, in terms of upper and lower probabilities, it can be considered as the center of gravity of the set of probabilities dominating the belief functions. He also introduced new tools for information fusion and decision making according to \cite{DuboisPS01}.
\\
The objective of the evidential theory is to represent the information which is transmitted by a source concerning an event.
A belief function must take in consideration all the possible events on which a source can describe a belief. Based on that, we can define the frame of discernment.
\\
The frame of discernment is a finite set of disjoint elements noted $\Omega$ where $\Omega= \{\omega_{1},...,\omega_{n}\}$. This theory allows us to affect a mass on a set of hypotheses not only a singleton like in the probabilistic theory. Thus, we are able to represent ignorance, imprecision and uncertainty.
\begin{eqnarray}
m: 2^{\Omega} \mapsto [0,1].
\end{eqnarray}
\begin{eqnarray}
\sum_{X\subseteq \Omega} m(X) = 1.
\end{eqnarray}
The principle functions in the belief functions theory are:
\textbf{The credibility function (\emph{bel})}: this function measures the strength of the evidence in favor of a set of propositions for all $X \in 2^{\Omega}$:
\begin{eqnarray}
bel(X)= \sum_{Y\subseteq X, Y\neq \emptyset} m(Y).
\end{eqnarray}
The credibility is interpreted as a degree of justified support given to proposition X by the available evidence.
\textbf{The plausibility function (\emph{pl})}: expresses the maximum amount of specific support that could be given to a proposition $X \in 2^{\Omega}$. $pl(X)$ is then obtained by summing the $bba's$ given to the subsets $Y$ such that $Y \cap X \neq \emptyset$ :
\begin{eqnarray}
pl(X)= \sum_{Y\in 2^{\Omega}, Y\cap X \neq \emptyset} m(Y).
\end{eqnarray}
\\
It measures the degree of belief committed to the propositions compatible with X. \\
\textbf{The commonality function (\emph{q}):} this function measures the set of $bbas$ affected to the focal elements included in the studied set, for all $X \in 2^{\Omega}$:
\begin{eqnarray}
q(X)= \sum_{Y \supseteq X} m(Y).
\end{eqnarray}
\subsection{Inclusion as a conflict measure for discrete belief functions}
Recently Martin in \cite{Martin12} defined a degree of inclusion as involved in the measurement made in order to determine the conflict during the combination of two discrete belief functions.\\
The author presented an index of inclusion having binary values where $Inc(X_{1},Y_{2})=1$ if $ X_{1} \subseteq Y_{2}$ and $0$ otherwise with $X_{1}$, $Y_{2}$ being respectively the focal elements of $m_{1}$ and $m_{2}$.
This index is then used to measure the degree of inclusion of the two mass functions and defined as:
\begin{eqnarray}
d_{inc}= \frac{1}{|F_{1}| |F_{2}|} \sum_{X_{1}\in F_{1}} \sum_{Y_{2}\in F_{1}} Inc(X_{1},Y_{2})
\end{eqnarray}
\begin{eqnarray}
\sigma_{inc}(m_{1},m_{2})= max(d_{inc}(m_{1},m_{2}),d_{inc}(m_{2},m_{1}))
\end{eqnarray}
Where $d_{inc}$ is the degree of inclusion of $m_{1}$ in $m_{2}$ and inversely.
\\
This inclusion is used as a conflict measure for two mass functions, using it like presented:
\begin{eqnarray}
Conf(m_{1},m_{2})=1-\sigma_{inc}(m_{1},m_{2})d(m_{1},m_{2})
\end{eqnarray}
where $d(m_{1},m_{2})$, is the distance of Jousselme:
\begin{eqnarray}
d(m_{1},m_{2})=\sqrt{\frac{1}{2}(m_{1}-m_{2})^{T} \underline{\underline{D}} (m_{1}-m_{2})};
\end{eqnarray}
where $\underline{\underline{D}}$ is a matrix based on the dissimilarity of Jaccard expressed by $D(A,B)= 1$ if $A=B=\emptyset$ otherwise, $D(A,B)= \frac{\mid A \cap B \mid }{\mid A \cup B \mid }$ if $\forall A,B \in 2^{\Omega}$
\section{Continuous belief functions}
In the previous sections, we have presented different specifications of discrete belief functions. Unfortunately, these functions do not allow us to manipulate continuous data that can be provided by sensors in different areas like: search and rescue problems \cite{Dore2010}, classification issues, information fusion, etc.
\\
Some researches were interested in representing belief functions in continuous frame of discernment like Strat in \cite{Strat}, and Smets in \cite{BFRN}.
\\
In following sections, we will present the several proposals that allows us to describe continuous belief functions. First, we explain how to extend these functions on the real numbers. To do so, we will focus on Smets' approach to represent continuous belief functions by using probability densities. Later, we will remember the other approaches for the continuous case of the theory of belief functions.
\subsection{Continuous belief functions on $\overline{\mbox{I\hspace{-.15em}R}}$ }
Smets based on the TBM's background, used the same representation than Strat, and proposed the belief functions in the extended set of reals noted $\overline{\mbox{I\hspace{-.15em}R}} = \mbox{I\hspace{-.15em}R}\cup \{-\infty, +\infty\}$.\\
However, using the belief function framework to model information in a continuous frame is not an easy task mainly to the complex nature of the focal elements.
Comparing to the discrete domain, on real numbers, in (Smets 2005) $bba$ becomes \emph{basic belief densities} $(bbd)$ defined on an interval $[a,b]$ of $\mbox{I\hspace{-.15em}R}$.\\
\subsubsection{Basic belief densities}
A generalization of the classical $bba$ into a basic belief density ($bbd$) denoted $m^{\emph{\scriptsize I}}$ on the interval \emph{I}. He defined the $bbd$ where all focal elements are closed intervals or $\emptyset$.\\
Given a normalized $bbd$ $m^{\emph{\scriptsize I}}$, Smets defined another function $f$ on $\mbox{I\hspace{-.15em}R}^{2}$, where:
\begin{eqnarray}
\left\{
\begin{array}{ll}
f(a,b)= m^{ I} ([a,b]), & a\leq b, \\
f(a,b)=0, & a>b.
\end{array}
\right.
\end{eqnarray}
$f$ is called a probability density function ($pdf$) on $\mbox{I\hspace{-.15em}R}^{2}$.
\subsection{Continuous belief functions induced by probability density functions}
Let's consider several belief functions characterizing a unique source of information (the source is subjective and evidential). Smets proposed a pignistic transformation of the belief functions (representing the knowledge of the source) in order to obtain probabilities. The probabilities are used to ease the decision making. Pignistic probabilities are noted BetP having densities also noted $betf$.
For each probability density, we have a set of belief functions with which they are compatible. This set is called an \emph{Isopignitic}. The main issue is to choose one belief function from this set. To do so, we consider the principle of least commitment proposed in \cite{DPr,Hsia}. This principle supports the idea of choosing the belief function that involves the least an expert.
It can be considered as a natural approach to select the less committed \emph{bba} from a subset.
A particular type of belief functions describes the best this principle which are the consonant belief functions where focal elements are nested \cite{DPr2}.
\subsection{Continuous belief functions: other representations}
Some other approaches have been proposed in order to describe continuous belief functions.
\cite {Nguyen08} introduced in the notions of a source constituted by a probability space and a multivalued mapping which is able to define the lower probability defined by a $\Gamma$ function.
This function can hold at the same time two notions: on one hand, it defines both of the lower and upper probability, on the other hand, it considers random sets.
We can say that $\Gamma$ as a multivalued mapping is measurable with respect to the spaces that it characterizes.
Moreover, he supposes that $\Gamma$ is a measurable mapping, then it is a random set by specifying its probability distribution.
Thus the probability distribution of a random set $\Gamma$ is precisely the basic probability assignment.
We say that there is a correspondence established between belief functions on a source $S$ and the probability distribution of random sets. This relation can be expressed by its density on $P(S)$.
\\
Dor\'e {\em et al.} in \cite {Dore2011a}, proposed a similar approach founded on an index function that can be assumed as $\Gamma$. This function can describe the set of focal elements of a continuous belief function. In this case, every index has its own probability measure where there is an allocated weight to a set of focal elements using a credal measure.
Every set is described according to its index and its probability density.
The formalism of Smets takes into consideration only to closed intervals, in \cite {Dore2010}, the author extended classical continuous belief functions by proposing belief functions where focal elements are not represented by intervals. He uses $\alpha cuts$ to measure to area of the portions of multimodal distributions.
\subsection{Similarity measure within continuous belief functions}
Attiaoui et al. in \cite{Attiaoui} proposed a similarity measure based on the distance of Jousselme using Smets' formalism. This distance uses a scalar product as a scalar product is defined on $\overline{\mbox{I\hspace{-.15em}R}}$ by:
\begin{eqnarray}
\langle f,g \rangle = \displaystyle{\int_{x=-\infty}^{+\infty} \int_{y=-\infty}^{+\infty} f([x,y])g([x,y])dx dy}
\end{eqnarray}
Here, the authors presented a new method to measure the similarity founded on the properties of belief functions on real numbers, they were able to define a distance
between two densities in an interval $\emph{I}$.
\begin{eqnarray}
\label{CSP}
\langle f_{1},f_{2}\rangle = & &
\int_{\tiny -\infty}^{\tiny +\infty} \!\! \int_{\tiny y_{i}=x_{i}}^{\tiny +\infty} \!\! \int_{ -\infty}^{\tiny +\infty} \int_{\tiny y_{j}=x_{j}}^{\tiny y_{j}= \tiny +\infty} \\
&& \!\!\! f_{1}(x_{i},y_{i}) f_{2}(x_{j},y_{j}) \delta(x_{i},x_{j},y_{i},y_{j}) dy_{j} dx_{j} dy_{i} dx_{i} \nonumber
\end{eqnarray}
The scalar product of the two basic belief densities is noted: $\langle f_{1},f_{2}\rangle$ with a function $\delta$ defined as $\delta: \mbox{I\hspace{-.15em}R} \longrightarrow [0,1] $
\begin{eqnarray}
\label{IntLenEq}
\delta(x_{i},x_{j},y_{i},y_{j})= \frac{ \lambda (\llbracket max(x_{i},x_{j}), min(y_{i},y_{j})\rrbracket)}{ \lambda(\llbracket max(y_{i},y_{j}), min (x_{i},x_{j}) \rrbracket )}
\end{eqnarray}
where $\lambda$ represents the Lebesgue measure used for the interval's length and $\delta (x_{i},x_{j},y_{i},y_{j})$ is an extension of the dissimilarity of Jaccard applied for the intervals in the case of continuous belief functions.
\begin{eqnarray}
\llbracket a,b\rrbracket= \left\{
\begin{array}{ll}
\emptyset, & \mbox{ if } a > b \\
$[a,b]$, & \mbox{ otherwise. }
\end{array}
\right.
\end{eqnarray}
Therefore, the distance between two basic belief densities is defined by the following equation:
\begin{eqnarray}
\label{Dist}
d(f_{1},f_{2})= \sqrt{\frac{1}{2} (\|f_{1}\|^{2} + \|f_{2}\|^{2}-2 \langle f_{1}, f_{2}\rangle)}
\end{eqnarray}
We noticed that the standard deviation is influencing this measure. As long as the difference between the distributions grows, the more the distance is rising.
This representation proposed a natural approach that allows us to manipulate and also study the behavior of continuous belief functions induced by normal and exponential distributions.
\section{Inclusion within continuous belief functions}
Within this section, we will consider two forms of inclusion: a strict and a partial inclusion.
We will present their mathematical expressions, and explain how do we build them. But first, we enumerate several properties that must be satisfied by the relation of inclusion.
\subsection{Properties of the inclusion}
The inclusion defined between two intervals $[x_{i},y_{i}]$ and $[x_{j},y_{j}]$ in a set \textsl{I} satisfies the following requirements:
\textbf{Property 1: Non-negativity}\\
\begin{eqnarray}
\begin{array}{l}
Inc(f_{i},f_{j})\geq 0.\nonumber
\end{array}
\end{eqnarray}
Namely, the inclusion of the first interval in the second one must never be negative.
\textbf{Property 2: asymmetry}
\\
\begin{eqnarray}
\begin{array}{l}
Inc(f_{i},f_{j}) \neq Inc(f_{j},f_{i}), \forall f_{i}\neq f_{j}. \nonumber
\end{array}
\end{eqnarray}
No need for the relation of inclusion to be symmetric.
\\
\textbf{Property 3: Upper bound}\\
\begin{eqnarray}
\begin{array}{l}
Inc(f_{i},f_{j})=1. \nonumber
\end{array}
\end{eqnarray}
This property implies a total inclusion of the first interval in the second one other.
\\
\textbf{Property 4: Lower bound}
\\
\begin{eqnarray}
\begin{array}{l}
Inc(f_{i},f_{j})=0. \nonumber
\end{array}
\end{eqnarray}
This property implies the absence of any intersection or inclusion of the first interval in the second one.
\subsection{Strict inclusion}
Here, we will define the strict inclusion between two continuous belief functions represented by two basic belief densities $bbd$.
\\
First, we use these distributions to deduce a degree of inclusion between the $bbds$ and then we can be able to measure the value inclusion between our continuous belief functions.
\\
Let's consider two continuous $pdfs$: $f_{1}$ and $f_{2}$.
If one distributions is included in the second one, then the strict inclusion is expressed by the following equation:
\begin{eqnarray}
\label{IStr}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! IncStr(f_{1},f_{2})=\int_{-\infty}^{+\infty}\!\! \int_{y_{i}=x_{i}}^{+\infty} \!\! \int^{x_{i}=+\infty}_{x_{j}=-\infty}\!\!
\int^{y_{j}=+\infty}_{y_{j}=x_{j}} \\
\delta_{IncStr}(x_{i},y_{i}, x_{j},y_{j}) f_{1}(x_{i},y_{i}) f_{2}(x_{j},y_{j}) dy_{j} dx_{j} dy_{i} dx_{i} \nonumber
\end{eqnarray}
Where $[x_{i},y_{i}]$, $[x_{j},y_{j}]$ are the considered intervals and $\delta_{IncStr}(x_{i},y_{i}, x_{j},y_{j}) $ is the degree of strict inclusion that will allow us to measure the value related to the inclusion of the first interval in the second one.
\\
We will consider that $\delta_{IncStr}(x_{i},y_{i}, x_{j},y_{j})$ is having binary values where:
\begin{eqnarray}
\label{IncStr}
\delta_{IncStr}(x_{i},y_{i}, x_{j},y_{j})= \left\{
\begin{array}{ll}
1, & \mbox{ if } [x_{i},y_{i}] \subseteq[x_{j},y_{j}] \\
0, & \mbox{ otherwise. }
\end{array}
\right.
\end{eqnarray}
If we are in presence of two distributions that do touch each other, there is an intersection between them. The $\delta_{IncStr}(x_{i},y_{i}, x_{j},y_{j})$ will have the value $1$, and the strict inclusion will be weighted by the masses of our continuous belief functions.
Otherwise $\delta_{IncStr}(x_{i},y_{i}, x_{j},y_{j})$ will be null.
\subsection{Partial inclusion}
Considering two $bbds$ represented by two intervals $[x_{i},y_{i}]$ and $[x_{j},y_{j}]$: We say that $[x_{i},y_{i}]$ is partially included in $[x_{j},y_{j}]$ or inversely, if and only if their intersection is different of $\emptyset$.
\\
To represent the partial inclusion we define:
\begin{eqnarray}
\label{IPar}
IncPar(f_{1},f_{2})= \int_{-\infty}^{+\infty}\!\! \int_{y_{i}=x_{i}}^{+\infty} \!\! \int^{x_{}=+\infty}_{x_{i}=-\infty}\!\! \int^{y_{i}=+\infty}_{y_{j}=x_{j}} && \\ \delta_{IncPar}(x_{i},y_{i}, x_{j},y_{j}) f_{1}(x_{i},y_{i}) f_{2}(x_{j},y_{j}) dy_{j} dx_{j} dy_{i} dx_{i} \nonumber
\end{eqnarray}
with $\delta_{IncPar}(x_{i},y_{i}, x_{j},y_{j})$ is the degree of partial inclusion:
\begin{eqnarray}
\label{IncPar}
\!\!\!\!\delta_{IncPar}(x_{i},y_{i}, \!x_{j},y_{j})\!\!=\!\! \frac{max(0, \!min(y_{j},y_{i})\!\!-\!\!max(x_{i},x_{j}))}{y_{i}-x_{i}}
\end{eqnarray}
The degree $\delta_{IncPar}(x_{i},y_{i}, x_{j},y_{j})$ represents the length of the intersection of the two probability density functions $f_{1}$ and $f_{2}$ on the length of $f_{1}$ if we are measuring $IncPar(f_{1},f_{2})$ and the length of $f_{2}$ if we have to calculate the partial inclusion of $f_{2}$ in $f_{1}$: $IncPar(f_{2},f_{1})$
\section{Asymmetry within the inclusion}
Let us consider four probability density functions defined by their means $\mu$, and their standard deviations $\sigma$, like presented in table~\ref{table1} and described in figure~\ref{pdfs}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$pdf$ & 1 & 2 & 3 & 4 \\
\hline
$\mu$ & 0 & 0 & 4 & 4 \\
\hline
$\sigma$ & 1 & 0.5 & 1 & 0.5 \\
\hline
\end{tabular}
\end{center}
\caption{Probability density distributions}
\label{table1}
\end{table}
\begin{center}
\begin{figure}
\center
\includegraphics [scale=0.5]{pdfs4.png}\\
\caption{Modeling probalility distribution functions. \label{pdfs}}
\end{figure}
\end{center}
Once we apply the mathematical formula proposed in equation (15), we obtain the following table~\ref{tableAsyStrict} strict inclusion.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$IncStr$ & $pdf_{1}$ & $pdf_{2}$ & $pdf_{3}$ & $pdf_{4}$ \\
\hline
$pdf_{1}$ & 0.5032 & 0.1437 & 0.009 & 0 \\
\hline
$pdf_{2}$ & 0.8586 & 0.5032 & 0.053 & 0 \\
\hline
$pdf_{3}$ & 0.009 & 0 & 0.5032 & 0.143 \\
\hline
$pdf_{4}$ & 0.537 & 0 & 0.8586 & 0.5032 \\
\hline
\end{tabular}
\end{center}
\caption{Strict inclusion and asymmetry.}
\label{tableAsyStrict}
\end{table}
Otherwise, applying the partial inclusion expressed in equation (17), we obtain table~\ref{tableAsy}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$IncPar$ & $pdf_{1}$ & $pdf_{2}$ & $pdf_{3}$ & $pdf_{4}$ \\
\hline
$pdf_{1}$ & 0.8183 & 0.5498 & 0.0253 & 0.0013 \\
\hline
$pdf_{2}$ & 0.9595 & 0.8183 & 0.0041 & 0.0017 \\
\hline
$pdf_{3}$ & 0.0253 & 0.8247 & 0.8183 & 0.9595 \\
\hline
$pdf_{4}$ & 0.0041 & 0.0017 & 0.5498 & 0.8183 \\
\hline
\end{tabular}
\end{center}
\caption{Partial inclusion and asymmetry.}
\label{tableAsy}
\end{table}
The property of asymmetry between two continuous belief functions can be confirmed when we observe the measures of inclusion in the table~\ref{tableAsyStrict} and table~\ref{tableAsy} .
\\
We witness for strict and partial inclusion that $IncStr(pdf_{2},pdf_{1})=IncStr(pdf_{3},pdf_{4})=0.8586$ and
$IncPar(pdf_{2},pdf_{1})=IncPar(pdf_{3},pdf_{4})=0.9595$ do always have the highest values on both cases. These primary results confirm the distributions presented in Figure~\ref{pdfs},
Meanwhile, for the case of inclusion of $pdf_{1}$ in $pdf_{4}$ and $pdf_{4}$ in $pdf_{1}$, where the difference between the means is very important we are dealing with very small values of inclusion.
Knowing that $\sigma_{1}= \sigma_{3}$ and $\sigma_{2} = \sigma_{4}$, when computing these partial inclusions, the difference between the standard deviations is maximal, generates a considerable value.
\\
For the strict inclusion we have some values where there is no intersection between the distributions and then we naturally obtain a null value, like for $IncStr(pdf_{1},pdf_{4})$.
According to the data in table~\ref{tableAsyStrict} and table~\ref{tableAsy}, the primary results obtained using both of the strict and the partial inclusion, our proposed relation responds to all the properties previously announced (non negativity, asymmetry, upper-bound $<1$).
\subsection{Average of inclusion}
Let us consider $n$ distributions, and $\alpha f $ a set of $bbds$.We can measure the average of inclusion of a $bbd$ ${f_{i}}$ in $\alpha f$.
\\
To do so, we present the following equations the first one related to the strict inclusion and the second one to the partial.
\begin{eqnarray}
IncS(f_{i},\alpha f)=\!\!\!\!\ \displaystyle \frac{1}{n-1} \sum_{j=1, i\neq j} ^{n} IncStr(f_{i},f_{j})
\end{eqnarray}
\begin{eqnarray}
IncP(f_{i},\alpha f)=\displaystyle \frac{1}{n-1} \sum_{j=1, i\neq j} ^{n} IncPar(f_{i},f_{j})
\end{eqnarray}
To model the measurement of the average of inclusion, we will apply the equations (19) and (20) to obtain respectively table~\ref{AvSt} and table~\ref{AvPar}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
IncS & Value \\
\hline
$IncS(f_{1},f_{j})$ & 0.0509 \\
\hline
$IncS(f_{2},f_{j})$ & 0.3038 \\
\hline
$IncS(f_{3},f_{j})$ & 0.0506 \\
\hline
$IncS(f_{4},f_{j})$ & 0.4653 \\
\hline
\end{tabular}
\end{center}
\caption{Average of Strict Inclusion.}
\label{AvSt}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
$IncP$ & Value \\
\hline
$IncP(f_{1},f_{j})$ & 0.1921 \\
\hline
$IncP(f_{2},f_{j})$ & 0.3217 \\
\hline
$IncP(f_{3},f_{j})$ & 0.60317 \\
\hline
$IncP(f_{4},f_{j})$ & 0.1852 \\
\hline
\end{tabular}
\end{center}
\caption{ Average of the Partial Inclusion.}
\label{AvPar}
\end{table}
\section{Illustration of the strict and partial inclusions}
In this section we will present our experimental phase to illustrate both of the strict and partial inclusion between two continuous belief functions.
To illustrate both of the strict and partial inclusion between two continuous belief functions induced using a normal distribution. We decided to fix the value of the first distribution $pdf_{1}$, where, it is characterized by its mean $\mu_{1}=0$ and its standard deviation $\sigma_{1}=1$. For the second distribution $pdf_{2}$, we will vary the values of $\mu_{2}$ in $[0,10]$ and $\sigma_{2}$ in $[0,5]$.
\\
Here, our purpose is to see the behavior of the inclusion when we modify one of the $pdfs$, and the parameters that infer in the obtained results.
\subsection{Strict inclusion between belief densities induced by normal distributions}
This strict inclusion is a natural approach, that allows us to perceive if there exists any intersection between two distributions, how they behave and the parameters that interfere during this process.
In this case, when a degree of inclusion has binary values of $0$ and $1$, we will study this property between two continuous belief functions.
\\
\begin{figure}
\center
\includegraphics [scale=0.6] {Sinc.png}
\caption{ Strict inclusion of $pdf1$ in $pdf2$. \label{IncGlob}}
\end{figure}
The more the value of the second standard deviation grows, the more difference between $\sigma_{1}$ and $\sigma_{2}$ develops.
Here, the strict inclusion between the two distributions, which generates a growth in the behavior of the obtained curve. We are then, in the presence of a growth of the strict inclusion of $pdf_{1}$ in $pdf_{2}$ due to the variation of the second distribution.
\\
Otherwise, we notice in Figure~\ref{IncGlob}, relative to the strict inclusion, that it is not the gap between the means $\mu_{1}$ and $\mu_{2}$ that generates this growth. It is the difference between the two standard deviations, that is in the origin of this growing phenomenon of inclusion.
As shown when $\sigma_{1}>\sigma_{2}$ and $\mu_{2}$ having its maximal value with $\mu_{2}=5$, the strict inclusion is null.
This value is due to the fact that the degree of strict inclusion $\delta_{IncStr}(x_{i},y_{i}, x_{j},y_{j})=0$, which can be explained by the non-existence of any intersection between $pdf_{1}$ and $pdf_{2}$.
\\
The part of the curve where the difference between the two standard deviations is low and at the same time the difference between the means is high. We have a small growing strict inclusion for example when $\mu_{2}=3$ and $\sigma_{2}=1.5$, the strict inclusion $IncStr(f_{1},f_{2})=0.1$.
At the meantime, even when $\mu_{1}=\mu_{2}=0$ $\sigma_{1}=\sigma_{2}=1$, there is an intersection between the two distributions and $IncStr(f_{1},f_{2})>0$.
Moreover, when the gap between $\sigma_{1}$ and $\sigma_{2}$ gets higher, the curve increases, generating a bigger strict inclusion between $pdf_{1}$ and $pdf_{2}$, until rising its maximal value where $IncStr(f_{1},f_{2})=1$, with $\delta_{IncStr}(x_{i},y_{i}, x_{j},y_{j})=1$.
In this specific case, we are in presence of a total inclusion of the first distribution $pdf_{1}$ in the second one $pdf_{2}$.
We also witness a phenomenonin which $\mu_{2}$ gets a high value, the strict inclusion drops considerably. Here we can state that the mean also has an impact on the generated inclusion.
\subsection{Partial inclusion between belief densities induced by normal distributions}
The partial inclusion is defined in order to give us the proportion of the intersection between two $pdfs$.
\subsubsection{Partial inclusion of $pdf1$ in $pdf2$}
\begin{figure}
\center
\includegraphics [scale=0.6] {Pinc.png}
\caption{Partial inclusion of $pdf1$ in $pdf2$ \label{IncPart}}
\end{figure}
During this experimentation, we keep the same values used for the strict inclusion and we obtain Figure~\ref{IncPart}
We notice that, when we are dealing with similar distribution where $\mu_{1}=\mu_{2}=0$ and $\sigma_{1}=\sigma_{2}=1$, the value of the partial inclusion is greater than zero. It is possible to state that when we are in presence of two distributions having the same values, there is not necessarily any total inclusion between them.
We also take note, that as the difference between $\sigma_{1}$ and $\sigma_{2}$ rises, due to the of $pdf_{2}$, the figure obtained grows faster, and reaches its maximal value $IncPar(f_{1},f_{2})=1$, generating a curve more arched than the strict inclusion.
When the difference between $\mu_{1}$ and $\mu_{2}$ increases because of the variation of $pdf_{2}$, the partial inclusion reaches a value of $IncPar(f_{1},f_{2})=0.85$, which becomes lower when the gap between two standard deviations is the highest ($\sigma_{2}=3$), we obtain the maximal value for the partial inclusion: $IncPar(f_{1},f_{2})=1$ like presented in Figure~\ref{IncPart}.\\
In this specific case, we witness a full and total inclusion of $pdf_{1}$ in $pdf_{2}$. This is similar to what we have presented regarding the strict inclusion.
Here we have non-negative inclusions, that respect the lower and upper bounds where the values are $[0,1]$.
Besides, the property of asymmetry is also respected because, the inclusion (what ever is strict or partial) of a distribution in the other does not necessarily involve the inverse case with the same value.
When $IncPar=1$, we have a total inclusion of the first distribution $pdf_{1}$ in the second one $pdf_{2}$, this situation is considered as a strict inclusion where, $pdf_{1}$ is fully included in $pdf_{2}$.
\subsubsection{Partial inclusion of $pdf2$ in $pdf1$}
\begin{figure}
\center
\includegraphics [scale=0.6] {pinc4.png}
\caption{Partial inclusion of $pdf2$ in $pdf1$ \label{Incp2}}
\end{figure}
For this case, we have chosen to measure the partial inclusion of $pdf_{2}$ in $pdf_{1}$, saving the same values for both distributions. Otherwise, the equation 18 becomes:
\begin{eqnarray}
\!\!\!\!\!\!\!\! \delta_{IncPar}(x_{i},y_{i}, \! x_{j},y_{j})\!\!\!=\!\! \frac{max(0, \! min(y_{i},y_{j}) max(x_{j},x_{i}))\!\!\!}{y_{j}-x_{j}}
\end{eqnarray}
We obtain the Figure~\ref{Incp2}. In this case, we notice a different phenomenon comparing to Figure~\ref{IncPart}.
Here, we witness than when the two distributions are totally similar, the value of the partial inclusion equals $0.8183$. This value is the maximal one that is expressed in this, and as stated before, two similar distributions can not be fully included in each other.
We also observe in this figure, that as long as tha values of the second distribution $\mu_{2}$, and $\sigma_{2}$ rise, the value of the partial inclusion of $pdf_{2}$ in $pdf_{1}$ drops.
More the two distributions are getting different from each other, more the partial inclusion decreases. Both parameters; the mean and the standard deviation have considerable impacts in this measure. This is proved by the behavior of the partial inclusion.
\\
The difference between the Figure~\ref{IncPart} and Figure~\ref{Incp2} is very obvious considering the behaviors of the two curves.
This is due to the nature of the focal elements of continuous belief functions which are expressed by intervals.
Specially with the case of the inclusion where each time we measure the inclusion of the intervals of a normal distribution with those of a second normal distribution.
\section{Strict inclusion VS Partial inclusion }
Comparing the results obtained in Figure~\ref{IncGlob} and Figure~\ref{IncPart}, where we have the same values for the two distributions, we notice that the partial inclusion reaches the maximal value faster that the strict one. Besides, its area is bigger and larger. Thus, we can state that the partial inclusion is dominating.
\\
In Figure~\ref{IncGlob} and Figure~\ref{IncPart}, it seems that we have the same phenomena between the two types of inclusion. This can be explained by the fact that we are working with consonant belief functions (where focal intervals are nested).
We can imagine that in presence in other nature of belief functions (categorical, or Bayesian belief functions) we can obtain different behaviors between the strict and the partial inclusion.
We notice the same phenomena that we have seen for the strict, when the values of parameters characterizing the second distribution grow, generating a big difference between the means, and the standard deviations, the partial inclusion decreases significantly and comparing when we are dealing with smaller values of $\mu_{2}$.
Thus, we can say that, both of the mean and the standard deviation do have a real impact on the measurement of the partial inclusion, having the same situation as the strict inclusion.
In the case of two distributions getting more and more different and especially when considering only the standard deviation, the phenomena of inclusion gets bigger. However, if the mean of one distribution is having non similar value than the other we can state than the inclusion has smaller values.
\section{Conclusion}
In this paper, we have emphasized the evaluation of the relation of inclusion between continuous belief functions induced by a normal distribution. We have defined two forms of inclusions: the strict and the partial one.
Before that, we have detailed all preliminary background that will allow us to experiment this kind of relation. We have also provided the approach on which the evaluation of the inclusions will be based.
We have presented a relation of inclusion having normalized values that takes into account the nature of continuous belief functions described using intervals as focal elements.
These two forms of inclusion respond to all the properties that must be satisfied.
We also have shown that both of the mean and the standard deviation have different impacts in this phenomena each one with its specific output.
\bibliographystyle{IEEEtran}
| {
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Lenihan: Bring it on
A busy period is on the horizon and Darragh Lenihan is relishing the feast of festive football
Darragh Lenihan says he's relishing the chance of continuing Rovers' unbeaten run across the festive period.
Wigan Athletic are first up on Monday night at Ewood Park before quickfire encounters with Birmingham City and Huddersfield Town bring 2019 to an end.
The 25-year-old's been crucial to Rovers' fine form in recent weeks, with the team sitting just one point off the play-offs in the Championship.
"We have four games in 10 days or so and then have a cup game after that," the powerful Irishman said in the build up to the clash with Paul Cook's Latics.
"It'll be a busy period but we have a strong squad and will go into the games full of confidence.
"We could have beat Swansea, but overall things have gone well.
"We have a local derby coming up against Wigan and we know what those games are all about, whether you're doing well or not.
"They're always difficult games and you need to be right at it. Hopefully we can win.
"It's disappointing to be injured and to not contribute to the team. You want to get back as soon as you can and chip in by helping the team get points.
"The boys are in good spirits but it's a period that we're all looking forward to."
Blackburn Rovers vs Wigan Athletic on 23 Dec 19
Never too high and never too low
Darragh Lenihan is set for his 150th Rovers appearance this afternoon against Sheffield Wednesday at Hillsborough.
Lenihan: Top six remains our goal
The aim's the same for Darragh
Darragh Lenihan says a top six finish remains the goal for Rovers despite the team's frustrating form over the festive period and the start of 2020.
Leading by example
Armband or no armband, you always know what you're going to get from the defensive powerhouse that is Darragh Lenihan. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,927 |
{"url":"https:\/\/slotsbet.se\/6cqlqn\/samba-ldap---centos-7-b2683f","text":"Categories\n\n# samba ldap centos 7\n\nIn Fallout 3, you could only repair a weapon to a certain condition depending on the players\u00e2\u0080\u0099 level of repair skill. If the player wants to use the strongest unarmed weapon in Fallout New Vegas, then they can look no further after obtaining the Ballistic Fist. See the rules below for more information. So I've decided to make my final playthrough a melee\/unarmed build. I drew me and my buddy in the Mojave Wasteland. If you want a challenge then, the unarmed build is the way to go. Fallout: New Vegas character build guide. That means each pellet will do on average 11.14 + 12*1.5*.3 = 16.54, or 6.54 damage (45.78 total) after armor. First of all, Heavy Handed or Built to Destroy? Reload Speed. $\\begin{matrix}\\text{Initial}\\\\\\text{level}\\%\\end{matrix}=65+(\\text{Agility}+\\text{Strength})\/2$ Example: A starting Agility of 5 and Strengthof 5. in today's video i show of my fallout new Vegas melee\/unarmed build. melee. Increase your weapon damage up to 5% and reveal enemies HP\/DT. Unarmed is a skill in Fallout, Fallout 2, Fallout 3, Fallout: New Vegas, Fallout Tactics, Van Buren and J.E. and out, each at the cost of 20 Action Points, depending on the player's Unarmed skill. All posts must be directly related to Fallout: New Vegas. Unarmed Warrior. DMG: This information comes from the G.E.C.K. No crit build . [\u00e2\u0080\u0093]foodnguns 0 points1 point2 points 5 years ago\u00a0(0 children), If your willing to stay with a limited set of weapons but need less perks melee, The ripper-> chainsaw-> thermal lance Fallout New Vegas is filled to the brim with unique weapons, and Obsidian did a wonderful job designing the special weapons that can be obtained throughout the Mojave Wasteland. [\u00e2\u0080\u0093]raella69PC Master Race 0 points1 point2 points 5 years ago\u00a0(1 child). Yeah it is very Viable to play a melee character in Fallout New Vegas. \u00a9 2020 reddit inc. All rights reserved. Save my name, email, and website in this browser for the next time I comment. Slayer \u00e2\u0080\u0093 24 \u00e2\u0080\u0093 AGI 7, Unarmed 90 \u00e2\u0080\u0093 1 \u00e2\u0080\u0093 The speed of melee and unarmed attacks is increased by 30%. They said 'welcome aboard'\" -- Fantastic. Harder than a ranged weapon build, but much more viable than it was in Fallout 3. The mouthpiece of the gaming generation, The Escapist aims to capture and celebrate the contemporary video gaming lifestyle and the diverse global video game culture by way of in-depth features, thought provoking articles and relevant columns authored by leading video game authorities, as well as cutting-edge video shorts, engaging forums and robust social media elements that incorporate \u2026 The Ballistic Fist doesn\u2019t have any unique effects that are as worthwhile as most of the other unarmed weapons in Fallout New Vegas. Even so, the Corrosive Glove is a great weapon for a critical hit build. Unarmed weapons in Fallout: New Vegas can utilize up to three special attacks both in V.A.T.S. Effect: Extra Critical Chance, Extra Critical Damage, Extra Limb Damage, How To Obtain: Crafted In The Lonesome Road DLC, Requires A 75 Unarmed Skill And Rawr\u2019s Talon Which Can Be Obtained By Defeating Rawr In The Divide. The Corrosive Glove is a scientific weapon that is encountered in the Old World Blues DLC. Specifically, the Paladin Toaster should be used when the player is fighting Brotherhood Of Steel enemies and while completing the Old World Blues DLC. I have literally no idea about which is the most effective but afaik chained\/constant attack melee weapons like the chainsaw\/ripper\/thermo lance have really bad crit rates\/damage, they've probably got stats more suited for a no-crit build (assuming going for more base damage?) And really nothing beats that early game of smashing powder gangers with a bat\/9 iron for me, [\u00e2\u0080\u0093]cereal_number 2 points3 points4 points 5 years ago\u00a0(1 child). I had the idea of also making this into a \"super crit\" build, it relies on having the dlcs. The Ballistic Fist deals 80 damage, and although it doesn\u2019t have a very unique appearance when compared to some of the other hand-to-hand combat weapons, its incredibly high base damage makes up for this lack of appearance. The Paladin Toaster weapon deals 50 extra damage towards robots and 20 extra damage towards most variants of power armor. Unlike the special moves listed above that must be learned before use, these special attacks require only an investment in skill points to unlock use. The 10 Most Powerful Builds In Fallout 4, Ranked. The thing is that you should act sneakier, perform more critical attacks, and find better weapons such as Knuckles ASAP. Also, there is a completely new skill called Survival. All those are automatic weapons,high dur burn no crit damage,BUT ignore any and all armor, If you really want to go unarmed the lonesome road dlc has a power fist with a saw blade on it,basically the unarmed verison of the chain saw -same rules,no crit,high dur burn,but armor ignoring, One point I would make is,all these automatic weapons will not knock down unless you use the powerfist from lonesome road + ranger kick from andy. Fallout New Vegas - Hardcore Build. Effect: Explosion On Critical Hit, Extra Critical Damage, How To Obtain: Sold By Vendortron, Only Obtainable If The Gun Runner\u2019s Arsenal DLC Is Installed. Me and my best friend Lil' Gecko ready for the crush some White-Legs. The motivation to post this was reddit user u\/420kushirino who reached Level 100 with an unarmed build: He wrote: Essential part of the build: Special \u2013 15 Strength, 3 Charisma, 5 Intelligence, 5 Agility , 3 Luck. The article Best Legendary Armors in Fallout 4 goes over some of the armors that are not power armors that might suit the unarmed play style. I understand that there's some good stuff you can unlock via dialog, so is the unarmed build a \u2026 \"No one knew his real name, but my men had a lot of nicknames for that guy. Any reason why you are against critical hits? 1) SPECIAL S 5->9 (IT+1, surgery+1, Reforced Spine+2 after completing OWB,9 Str is required for wielding the main melee wepon Blade of the West) Harder than a ranged weapon build, but much more viable than it was in Fallout 3. RELATED: Fallout New Vegas: 10 Best Mods You Need To Try This trait would be higher on the list if the negative stat didn't affect perk requirements. (I start out with energy weapons exclusively before moving on to unarmed once I gain access to a GRA Power Fist). Your email address will not be published. Fallout: New Vegas has released on Oct 19, 2010 on Steam. REDDIT and the ALIEN Logo are registered trademarks of reddit inc. \u03c0\u00a0Rendered by PID 25281 on r2-app-037e620aba348e218 at 2020-12-09 20:52:29.139217+00:00 running 4423137 country code: US. There are two ways you can go about doing a sniper build in Fallout New Vegas. With all DLCs installed, there is a very powerfull melee\/unarmed build for FNV. Bourbon: +1 Endurance & +1 Strength at the cost of -1 Intelligence. Legend. Unarmed weapons in Fallout New Vegas have unique appearances and are all very different from each other since fist weapons are made out of so many different types of materials. The Fallout: New Vegas Not-So-Ultimate Stealth Character Guide by Stanley E. Dunigan. Should I worry about crit at all with this type of style? Starting another NV playthrough before Cyberpunk drops like. [\u00e2\u0080\u0093]20BabilDPS Manifesto 0 points1 point2 points 5 years ago\u00a0(0 children). Much of Fallout 3's quests can be resolved without killing others. This weapon also deals extra critical hit damage, which will come in handy when attempting to one-shot enemies. Built to Destroy is the best option for one slot, the other isn't too important. I'd like to get some advice on how to best develop my build in New Vegas. Carry Wgt. BBCode (message boards & forums) ... Reddit. Some melee weapons are focused on crits, while other aren't. Even so, the Corrosive Glove is a great weapon for a critical hit build. Repair has been altered fairly drastically in this game. \"I said I had a theoretical degree in physics. Destroyer140. 28 % * crit % multiplier + 9% for non-laser energy weapons and 28 % * crit % multiplier + 19% for lasers. Unarmed weapons have sort of fascinated me ever since Fallout 3 (I haven't played Fallout or Fallout 2). Laser weapons have the highest crit chance of all the weapons in this build. It is, for example, very hard to do an unarmed build that includes explosives and still retain most of it's killing power. [\u00e2\u0080\u0093]SalsaRicePC 0 points1 point2 points 5 years ago\u00a0(0 children). As a courier once left for dead by a mysterious man in a striped suit, the player must now set out to find his assailant and uncover the secrets of the enigmatic ruler of New Vegas. Fallout: New Vegas Unarmed build. The Build Simply put, this build allows you to tank almost anything the game can throw at you, while still pounding everything to scrap around you. Most side missions in the game can be completed with\u00e2\u0080\u00a6 The Corrosive Glove is good for a player that is starting a new playthrough in Fallout New Vegas; however, there are a few better choices for players that want the most powerful unarmed weapon possible. ===== Fallout: New Vegas (PC): Unarmed\/Melee Character Guide, by Erik Fasterius [email ... since they don't need all those VATS and crit-perks that Unarmed characters need. \/r\/falloutlore - In-depth discussion of Fallout lore. $65+(5+5)\/2=70\\%$ Changed the way i approach the game. The post-apocalyptic Fallout universe expands into Nevada in this new title in the franchise. Critical hits do not ignore armor. chainsaws and thermic lances are kings of heavy handed builds, period. The standard crit chance for laser weapons, in the end game, will be 28% affected by the weapon multiplier, not counting true police stories, and 19% not affected by it (9% for the plasma weapons). The fact that this type of damage is uncommon is one of the reasons that the Corrosive Glove is such a unique weapon. Note also, that you can't target limbs in VATS with unarmed\/melee. [\u00e2\u0080\u0093][deleted] 2 points3 points4 points 5 years ago\u00a0(0 children). and join one of thousands of communities. 10 luck, Built to Destroy, Finesse, the First Recon Beret, Ulysses' Duster, and True Police Stories (with Comprehension) will give you a combined base crit rate of 38%. A lot of them are totally viable, but I\u00e2\u0080\u0099d really like to know which guns and energy weapons are the best of the best in the following categories in conjunction with a high crit build: -Annabelle The Rise of Live Roulette During the Internet Age. Melee Dmg. You receive 1% of critical chance for each point placed into Luck. Effect: +50 Damage Towards Robots, +20 Damage Towards Power Armor With The Exception Of NCR Power Armor. I think ninja is more useful for a melee\/unarmed character. Fallout: New Vegas \u00e2\u0080\u0093 Ultimate Sneak Sniper \/ Energy Weapons Crit Build Guide August 16, 2019 Lenusik Guides 1 This is a complete build for a sniper character that duals in energy weapons as a crit build. Melee or Unarmed ? lvl 24, Agility 7, Unarmed or Melee skill 90 + Super Slam: Req. \/r\/falloutcosplayers - Fallout-related cosplay, \/r\/galaxynewsradio - Fallout-sounding music, This subreddit should be night mode compatible. Oh, and this is for a level 50 Ultimate Edition character. - posted in New Vegas Spoilers: I was peeking through the Prima guide earlier and one of the characters (archetypes) seems like a neat and unusual build, at least for me. When the player lands a critical hit on an enemy, then they will be inflicted with acid damage, a damage type which is uncommon in Fallout New Vegas. \/r\/fallout - Reddit's main Fallout community, \/r\/classicfallout - Fallout 1 & 2 community, \/r\/falloutmods - Everything Fallout Modding. Use spoiler tags when applicable. The article Best Legendary Armors in Fallout 4 goes over some of the armors that are not power armors that might suit the unarmed play style. I had tested it at Hard Core and very hard mode. Effect: Inflicts Acid Damage On Critical Hits, How To Obtain: Underneath A Desk To The Right When The Player Enters The Z-43 Innovative Toxins Plant, Obtainable In The Old World Blues DLC. 1) SPECIAL S 5->9 (IT+1, surgery+1, Reforced Spine+2 after completing OWB,9 Str is required for wielding the main melee wepon Blade of the West) As stated in the forum post I didn\u2019t get a sniper rifle until later in the game but when I got it I was killing Deathclaws in one hit if I got a critical hit. Replaces skill book skill bonuses with incremental, small combat bonuses related to the skill in question. Carry Wgt. (self.fnv), submitted 5 years ago by Stealth_CommandoPC(Modded). Fallout New Vegas Vats Gun Build; S-4 P-4 E-6 C-5 I-5 A-7 (8 w\/ small frame) L-8 Small Frame & Built to Destroy ... +15% critical chance with melee & unarmed, +25% damage on sneak. If only someone wrote a guide about playing a melee-based character in Fallout. [\u00e2\u0080\u0093]poastertoasterPC 1 point2 points3 points 5 years ago\u00a0(0 children), Probably the trait lowering crit damage but increasing Melee\/Unarmed damage, [\u00e2\u0080\u0093]Talpanian_Emperor 0 points1 point2 points 5 years ago\u00a0(0 children). -- +5% crit chance to base crit chance of small knifes except the cleavers-- Separates skill requirements (MMUE) + Piercing Strike: Req. Melee is typically higher DPS, but Unarmed has the advantage of all those fighting moves you learn as well as paralyzing palm. However, the Paladin Toaster has one of the most useful effects that is found on any weapon in the Mojave Wasteland. The standard is going to be five-percent chance to score a critical (5%), but by the time you\u2019re at ten LCK (which must happen for this build by the time you\u2019re at the endgame), you will have ten-percent chance to score a critical simply from LCK (10%). [\u00e2\u0080\u0093][deleted] 1 point2 points3 points 5 years ago\u00a0(0 children). How To Obtain: Sold By Blake At The Crimson Caravan Company. ... Armor, it just isn't practical to build a melee- or unarmed-oriented stealth character in Fallout NV. Instead, the Ballistic Fist has the highest base damage of any unarmed weapon in the game. More than 1 in 10 of your posts or comments being self-promotional is spamming. Fallout: New Vegas character build guide. ... (Complete Dead Money, get Veronica to unlock his holotape but keep it for yourself for more crit damage). During character creation, you can add an additional 3% for all weapons by choosing the Built to Destroy trait. High Speech can help you to avoid unnecessary combats. Posts promoting piracy in any way will result in ban. use the following search parameters to narrow your results: For Everything Fallout: New Vegas. Get an ad-free experience with special benefits, and directly support Reddit. This build is pretty close to the previous one but it\u00e2\u0080\u0099s even more complicated. Do not spam. I'll try some Unarmed weapons eventually. I had the idea of also making this into a \"super crit\" build, Chance. Fallout New Vegas Best Build Both melee and unarmed are usually practical for all but a several situations.You will want a weapon for really long variety getting rid of. The Paladin Toaster deals 41 base damage, which is considerably lower than the Two-Step Goodbye. Fallout New Vegas Energy Weapons Build Guide. This was a request for a sniper build in Fallout New Vegas. Let's assume you have ~30% crit chance (roughly where a maxed out crit build with a 1x multiplier would put you) and better criticals. This weapon is another great example of the possibilities for a critical hit build that is focusing on using unarmed weapons. [\u00e2\u0080\u0093]Nickoladze 0 points1 point2 points 5 years ago\u00a0(0 children). So I'd go with 5 and get it up to 6 with an implant before you hit 16. ... Level 24- Slayer (Unarmed 90, AGL 7) The Legion are warriors like Mars and should be feared by all. Depends. Anyways here we go. I prefer unarmed at the beginning of the game once I can get LOVE & HATE and then melee later in the game when I can get my hands on some GRA tier weapons like Gehenna. Also be sure to have something for in case the enemy gets close like a lever action rifle or pistol. Weapons The game offers a plethora of great weapons. and shows the damage caused by a single click of the mouse\/trigger. Bear-Hunter, Nemesis, Steel Guardian. If you really want to go unarmed the lonesome road dlc has a power fist with a saw blade on it,basically the unarmed verison of the chain saw -same rules,no crit\u00e2\u0080\u00a6 The best information I found was in the Fallout: New Vegas (PC): Unarmed\/Melee Character Guide, by Erik Fasterius. The thing is that you should act sneakier, perform more critical attacks, and find better weapons such as Knuckles ASAP. The Corrosive Glove doesn\u2019t deal a lot of base damage compared to some of the other unarmed weapons in the Mojave Wasteland; however, it is still one of the best choices for the player when they are trying to decide on which unarmed weapon to use. Thermal lance and chainsaw don't have crit hits but rule. (I start out with energy weapons exclusively before moving on to unarmed once I gain access to a GRA Power Fist). DMG: This information comes from the G.E.C.K. Image macros\/memes are not permitted as posts. I kept pumping points into melee and explosives, Creating a monster. Do the most effective Melee\/Unarmed weapons have low or high crit multipliers? A melee build can do this relatively easy. Your email address will not be published. Reinforced spine Level 4 \u2013 Educated This is a more selfish version of Thermic lance that you can use pretty decently, as it allows fighting pretty well, and Knock-knock is a good compliment for your need for damage. Rank That Game was created to give gamers more information about their favorite games. Unarmed build in Fallout 4 is extremely difficult. Melee and Unarmed attacks do more damage, but less critical hit damage. I find that toughness isn't as useful in late game in particular because everyone's got weapon penetration. Endurance - \u00e2\u0080\u00a6 Stop pandering up to the myths that bare fist should crit and just dont skill it. Feel free to discuss any aspect of the game you want. ... Armor, it just isn't practical to build a melee- or unarmed-oriented stealth character in Fallout NV. Unarmed build? Grenade rifle, Gauss Rifle, Sniper rifle of selection. Unarmed Dmg. It is only visible to you. Repeat offenders shall be fed to the Deathclaws. But it takes a good long while into the game to get weapons that deal enough damage to be helpful to an unarmed build. The Rundown. Posting memes will lead to temp and further permanent bans. Do not take the heavy handed perk for an Unarmed build. There are 4 unarmed moves to learn which let you do some 'special move' stuff. Using the AER14 Prototype, which has a 2x modifier will give you 76%. NAME CHANGE. Reload Speed. 4. Vital for a \u00e2\u0080\u009ccrit based\u00e2\u0080\u009d unarmed build such as this due to you needing 6 to get the Better Criticals perk at level 16. Oh, and this is for a level 50 Ultimate Edition character. So I've decided to make my final playthrough a melee\/unarmed build. Unarmed is my favourite build for fallout 2. Hateful\/aggressive posts and comments over lore, individual games, or companies will be removed and may result in ban. It won't translate 100% to New Vegas, but hopefully will give you some pointers. -Annabelle The Rise of Live Roulette During the Internet Age. The Fallout: New Vegas Not-So-Ultimate Stealth Character Guide by Stanley E. Dunigan. lvl 12, Unarmed or Melee skill 70 + Slayer: Req. The Unarmed Build is hands down one of the most satisfying builds to run. Unarmed weapons have sort of fascinated me ever since Fallout 3 (I haven't played Fallout or Fallout 2). Chance. Unarmed weapons are a fun weapon type to use because they allow the player to get close to their enemy and defeat them with nothing but their hands and a glove or gauntlet. [\u00e2\u0080\u0093]SpecialRX 0 points1 point2 points 5 years ago\u00a0(0 children). NAME CHANGE. I think the main attraction to unarmed is the different moves you can perform. Big Brained (Moar DT). This build is similar to the Ninja build but with one critical difference, instead of using bladed weapons you are fighting with either a semi-automatic rifle or a full auto outfitted with a suppressor to get stealth kills. The build requires strategies and skills in order to make the best of it. Always follow Reddit guidelines for self-promotion when sharing your own content. on Fallout New Vegas: The 5 Best Unarmed Weapons (& How To Get Them), The Outer Worlds: The 5 Best Long Guns (& How To Get Them), Fallout New Vegas: The 5 Best Implants (& How Much They Cost). Fallout: New Vegas Guide \u2013 Legion Build. All posts and comments, in end, come down to moderator discretion. Two-Step Goodbye is a great weapon because it causes an explosion when the player lands a critical hit. Ever want to kill NCR for being Profligates? On New Vegas you can repair a weapon to 100% regardless, however, the lower your skill the quicker the weapon will degrade in condition with prolonged use. There are two ways you can go about doing a sniper build in Fallout New Vegas. Do the most effective Melee\/Unarmed weapons have low or high crit multipliers? The best information I found was in the Fallout: New Vegas (PC): Unarmed\/Melee Character Guide, by Erik Fasterius. Your unarmed attacks do significantly more limb damage and your critical chance is increased by +3%. I had tested it at Hard Core and very hard mode. Copy one of the following URLs to direct users to this page with your exported build pre-loaded. Reinforced Spine (More STR and DT, and we want to be crippled later). Crit. BBCode (message boards & forums) ... Reddit. First of all, Heavy Handed or Built to Destroy? So that\u2019s pretty much it to being a good sniper in Fallout New Vegas. The build requires strategies and skills in order to make the best of it. (Profligates means slightly more dissolute) Well then let\u2019s begin the Legion Build. Should I worry about crit at all with this type of style? It's partly because I think it makes for a cool tone in the game. If you want to be a beast (Perk wise) you need to get both to 90. Required fields are marked *. Groups are easy, you simply toss a grenade in the middle and pull out your beatstick to finish them off. He would often ambush squads of NCR soldiers by himself, decimating them and leaving nothing but their charred remains. Rendered by PID 25281 on r2-app-037e620aba348e218 at 2020-12-09 20:52:29.139217+00:00 running 4423137 country code: US. I just beat down 15 Deathclaws with \"Oh Baby\" and some reinforced leather armor. Fallout New Vegas melee\/unarmed build help. Melee is, I did an unarmed only build a little bit back and found that most melee weapons would outclass unarmed weapons in terms of pure damage but unarmed weapons often had bonus critical effects and damage. as they don't rely on them for their damage output? Fallout: New Vegas The formulas for critical chance seem unchanged from Fallout 3. \/r\/Wasteland - A subreddit for the Wasteland games. Quarry Junction should\u00e2\u0080\u0099ve been a named quest. in the base game unarmed is still better unless you do a heavy handed build. Copy one of the following URLs to direct users to this page with your exported build pre-loaded. Unarmed Dmg. Combat knife, for example, has good crit % and crit damage, while rebar club has abysmal crit %. Absolutely no harassment, witchhunting, sexism, racism or hate speech will be tolerated. Sawyer's Fallout Role-Playing Game. Basically I want to start leveling up one of those skills and I can not decide which one . Melee or Unarmed ?\n\n-Fatman There doesn't seem to be an equivalent character build thread for New Vegas as there is for FO3 (Forum:Fallout 3 character builds), so I'll try to get things started on this! Requires: Level 20, Adept of the Steel Hand, 80 Unarmed. I tend to forgo endurance for charisma because I love having a full party. With the right perks (and maybe a handful of chems) you can punch death claws to pieces wearing spiked knuckles and light armour. Unarmed build in Fallout 4 is extremely difficult. lvl 8, Strenght 6, Unarmed or Melee skill 45 For an Unarmed Build, focus on whatever grants strength for even MORE damage. Use of this site constitutes acceptance of our User Agreement and Privacy Policy. But New Vegas has some changes in skills, for example, Small Guns and Big Guns are now united in the single Guns category. This was a request for a sniper build in Fallout New Vegas. For most guns, this is a single bullet, but for shotguns it is a single shell (containing several small pellets), for the Gatling laser it is a single beam, and for melee weapons it is a single hit. One of the best things about the Fist Of Rawr is that it is a Deathclaw gauntlet, which makes it one of the most unique unarmed weapons in terms of appearance. If you want a challenge then, the unarmed build is the way to go. I would recommend doing both so you can get the Slayer perk, which requires an Agility of 7 and an Unarmed of 90. But I need a few opinions from you guys before I do so. Energy Weapons are the best for this. Your unarmed attacks are 10% faster and gain 5% crit chance and 50% crit damage while you are not wearing armor. Out of melee and unarmed, I've always found Unarmed to be more fun, since there are interesting power fists that I always enjoy using. Besides an avalanche of detailed advice, I found it educational to read about the distinction between a melee-based build and an unarmed-based build. These are the best unarmed weapons in Fallout New Vegas ranked by how much damage they deal, their special effects, and their appearances. and shows the damage caused by a single click of the mouse\/trigger. Not only does the Fist Of Rawr deal extra critical damage and have a better chance to land a critical hit, but it also deals extra limb damage to enemies. Follow proper Reddiquette when submitting and commenting. Melee Dmg. Which is better ? [\u00e2\u0080\u0093][deleted] 5 points6 points7 points 5 years ago\u00a0(1 child). Anyways here we go. As stated in the forum post I didn\u00e2\u0080\u0099t get a sniper rifle until later in the game but when I got it I was killing Deathclaws in one hit if I got a critical hit. This build, though, has a charisma and intelligence of 2 or so. Apr 11th, 2017. No crit build . See above if you don't know how to hide spoilers. The Two-Step Goodbye weapon deals 70 base damage, which is much more than most of the other unarmed weapons in Fallout New Vegas. The Corrosive Glove is good for a player that is starting a new playthrough in Fallout New Vegas; however, there are a few better choices for players that want the most powerful unarmed weapon possible. The ripper-> chainsaw-> thermal lance All those are automatic weapons,high dur burn no crit damage,BUT ignore any and all armor. The thing is that you should act sneakier, perform more critical attacks, and find better weapons such as Knuckles ASAP. With all DLCs installed, there is a very powerfull melee\/unarmed build for FNV. The latest game in the post-nuclear RPG series is being developed by many members of the Fallout 1 and 2 team at Obsidian Entertainment using the Fallout 3 engine. Whether Meltdown triggers or not, every disintegrated target will temporarily increase your critical chance, making it more likely to score further meltdowns. Any tips ? I'm doing a melee run right now and you need Unarmed to be level 90 to get the Slayer perk and I'm only using melee. Keep it civil and do not make personal attacks to other users. For Fallout: New Vegas on the Xbox 360, a GameFAQs message board topic titled \"Heavy Weapons Build\". Crit. Melee also gives you access to throwing weapons. In terms of pure fun, I would have to say Unarmed. Fallout: New Vegas \u2013 Ultimate Sneak Sniper \/ Energy Weapons Crit Build Guide August 16, 2019 Lenusik Guides 1 This is a complete build for a sniper character that duals in energy weapons as a crit build. All posts and comments, in end, come down to moderator discretion a... Of fascinated me ever since Fallout 3 needing 6 to get weapons that deal enough damage to be crippled )... Some reinforced leather Armor will give you 76 % chance and 50 % chance... I drew me and my best friend Lil ' Gecko ready for the crush some.... Permanent bans ) the Legion build I want to start leveling up of... Giant Mantis style: your strikes seek out enemy vital spots, every disintegrated target will temporarily increase critical. For an unarmed build is hands down one of the game triggers or not, every disintegrated target temporarily. Toughness is n't practical to build a melee- or unarmed-oriented stealth character in Fallout NV is a scientific weapon is... Into Nevada in this New title in the Mojave Wasteland knife, for example, a! Racism or hate Speech will be tolerated weapon to a certain condition depending on the player lands a hit. Best develop my build in New Vegas or hate Speech will be tolerated it more likely to further. Hit damage build '' satisfying builds to run then let \u2019 s begin the Legion are warriors Mars... 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Weapons such as Knuckles ASAP temporarily increase your critical chance, making more... From Fallout 3 cost of -1 Intelligence replaces skill book skill bonuses with incremental small! Style: your strikes seek out enemy vital spots in order to make the of... Is much more viable than it was in Fallout get to be crippled later ) build! Blues DLC: for Everything Fallout Modding do significantly more limb damage and your critical chance is by! Which will come in handy when attempting to one-shot enemies a good sniper in NV... Hands down one of the following URLs to direct users to this page your! Moves you can add an additional 3 % for all weapons by choosing the Built to trait. N'T have crit hits but rule shows the damage caused by a single click of the most useful that. Level 20, Adept of the possibilities for a level 50 Ultimate Edition character, other... As they do n't have crit hits but rule while into the game can be with\u00e2\u0080\u00a6. A request for a cool tone in the game can be completed with\u00e2\u0080\u00a6 Elder Scrolls Fallout! Build, though, has a charisma and Intelligence of 2 or.... Shows the damage caused by a single click of the mouse\/trigger effective melee\/unarmed weapons low... Have low or high crit multipliers to this page with your exported build pre-loaded I do so absolutely harassment! % for all weapons by choosing the Built to Destroy is the different you! Think it makes for a critical hit build be removed and may result in ban weapons deal. Play a melee character in Fallout New Vegas the formulas for critical is! Their critical hits is uncommon is one of the following URLs to direct users to this with! To the previous one but it\u00e2\u0080\u0099s even fallout: new vegas unarmed crit build complicated, making it more likely score! Racism or hate Speech will be removed and may result in ban using the AER14 Prototype, which has charisma... Personal attacks to other users causes an explosion when the player lands a critical hit damage have low or crit... Xbox 360, a GameFAQs message board topic titled heavy weapons ''... At Hard Core and very Hard mode case the enemy gets close like lever! Wise ) you need to get the better Criticals perk at level.... Or comments being self-promotional is spamming perk, which has a charisma and Intelligence 2. If you want to start leveling up one of those skills and I can decide... If only someone wrote a Guide about playing a melee-based build and an unarmed of 90 reinforced Spine ( STR. The base game unarmed is still better unless you do a heavy builds... Minute, * I * wrote a Guide about playing a melee-based character in Fallout.! But my men had a lot of nicknames for that guy personal to. Two-Step Goodbye weapon deals 70 base damage of any unarmed weapon in the Mojave Wasteland into. The different moves you can get the better Criticals perk at level 16 oh Baby and. Two-Step Goodbye is a fallout: new vegas unarmed crit build weapon because it causes an explosion when the lands... Viable to play a melee character in Fallout New Vegas, which is considerably lower than Two-Step! Has the highest base damage, which has a charisma and Intelligence of 2 or so helpful! Tone in the game can be completed with\u00e2\u0080\u00a6 Elder Scrolls and Fallout community character. Get both to 90 play a melee character in Fallout limb damage your! Fact that this type of style using the AER14 Prototype, which has a modifier... Weapons build '' enemy gets close like a lever Action rifle or pistol weapons by the. In VATS with Unarmed\/Melee titled heavy weapons build '' caused by a single click of the reasons the! ) you need to get both to 90 20 Action points, depending on the level... To three special attacks both in V.A.T.S video I show of my Fallout New Vegas, but much more than... 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Satisfying builds to run nicknames for that guy Fallout community, \/r\/falloutmods - Everything Fallout: New Vegas but... Oh, and this is for a \u00e2\u0080\u009ccrit based\u00e2\u0080\u009d unarmed build such as Knuckles ASAP guys before I do.... Benefits, and find better weapons such as Knuckles ASAP Live Roulette During the Internet.! Of critical chance seem unchanged from Fallout 3 ranged weapon build, but hopefully will give you %. Narrow your results: for Everything Fallout Modding chainsaws and thermic lances are kings heavy. ] raella69PC Master Race 0 points1 point2 points 5 years ago ( 0 children ) a in... Jesus and his disciples you guys before I do so: Req Fallout 3 ( I have played! Points7 points 5 years ago ( 1 child ) the AER14 Prototype, which will come in handy when to... Of heavy handed or Built to Destroy is the way to go melee weapons are focused crits... Their charred remains 1 % of critical chance seem unchanged from Fallout 3 this New title the., a GameFAQs message board topic titled heavy weapons build '' damage! Beat down 15 Deathclaws with oh Baby '' and some reinforced leather Armor sniper in Fallout NV added! Is hands down one of those skills and I can not decide which one thermic lances are of. To an unarmed of 90 the AER14 Prototype, which requires an Agility of 7 and an unarmed-based.. Fallout community, \/r\/falloutmods - Everything Fallout: New Vegas on the Xbox 360, a message. Skill it to you needing 6 to get weapons that deal enough to. Fallout Modding formulas for critical chance seem unchanged from Fallout 3 ( I start out energy... Gra Power Fist ) 0 points1 point2 points 5 years ago ( 0 children ) melee-based build and an build! Slayer: Req highest crit chance of all, heavy handed or Built to Destroy is the way go! Vegas the formulas for critical chance is increased by +3 % and this is a. Builds to run moderator discretion the middle and pull out your beatstick to finish them off Baby. Your strikes seek out enemy vital spots end, come down to moderator discretion knock them down then. Fallout 1 & 2 community, \/r\/classicfallout - Fallout 1 & 2 community, \/r\/falloutmods - Everything Fallout New... To play a melee character in Fallout 3, you can go about doing sniper!","date":"2021-04-12 00:10:13","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\": 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\": 0.2998237609863281, \"perplexity\": 7760.077573453607}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2021-17\/segments\/1618038065903.7\/warc\/CC-MAIN-20210411233715-20210412023715-00052.warc.gz\"}"} | null | null |
Radical [] (* 9. September 1981 in Glarus; bürgerlich Peter Baumgartner) ist ein Schweizer Rapper. Er ist Teil der Hip-Hop-Formation Trick77.
Biografie
Radical wuchs im Kanton Glarus auf. 2000 wurde die Glarner Rap-Formation Trick77 gegründet, zu der auch Radical gehörte. In den nächsten Jahren konnte sich die Hip-Hop-Gruppe schweizweit einen Namen machen. Vor und vor allem nach der Veröffentlichung des Demotapes Füühsturm trat Trick77 in der ganzen Schweiz an diversen Konzerten und Festivals auf und waren auf vielen Alben als Gastbeitrag vertreten. 2007 wurde der erste Solosong von Radical veröffentlicht, als der Titel Nägel mit Chöpf auf DJ Don Corleones Album Slangbang Vol. Drüü erschien.
2011 veröffentlichte er sein zweites Soloalbum Blaui Chuglä.
Diskografie
2008: Zwüsched Himmel und Höll
2011: Blaui Chuglä
2015: I zwei Weltä dihei
mit Trick77
2003: Wer interessierts? (Demo)
2004: Füühsturm (Demo)
2005: Huusverbot (Single)
Weblinks
Website
Soundcloud
Mx3
Einzelnachweise
Rapper
Pseudonym
Schweizer
Geboren 1981
Mann | {
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'Eyes of St John' recognised with another award!
January 26, 2016 Assistant Dissent Projects
Dissent Projects' latest documentary, 'Eyes of St John' has won another prestigious Humanitarian Award from The Global Film Awards.
The goal of the Humanitarian Award is to honour filmmakers who are bringing awareness to issues of Ecological, Political, Social Justice, Health and Wellness, Animals, Wildlife, Conservation and Spiritual importance. The nominees are hand picked by the judges and staff from hundreds of entries throughout the year across three international competitions – The Accolade Global Film Competition, The IndieFEST Film Awards and the Best Shorts Competition – the films are brought together under the Global Film Awards umbrella and the winners are chosen from this wide pool of entries.
Information about the Global Film Awards and a list of recent winners can be found at www.GlobalFilmAwards.comIn winning a Humanitarian award from Global Film Awards, Dissent Projects joins the ranks of other high-profile winners of this important award such as Oscar nominee Liam Neesom for narration of Love Thy Nature, multiple Emmy winner Alfre Woodard for Soft Vengence, Oscar winner (student) Emily Kassie for the powerful I Married My Family's Killer and many more.
Rick Prickett, who chairs the competitions under the Global Film Awards umbrella, had this to say about the Humanitarian winners, "It takes great talent to tackle the world's most pressing issues with film and do a great job. It takes an even greater heart. Global Film Awards helps set the standard for Humanitarian filmmaking worldwide. The goal of Global Film Awards is to help winners achieve the recognition they deserve for the incredible job that they do."
Please see see the trailer for 'Eyes of St John' under the video section of our Facebook page.
Previous Post'Eyes of St John' screening held in central LondonNext Post"Eyes of St John" in official selection at London International Short Film Festival | {
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} | 4,087 |
{"url":"https:\/\/imathworks.com\/math\/math-an-urn-contains-n1-white-and-n2-black-balls-from-which-k-are-drawn-without-replacement-find-the-probability-that-the-next-ball-drawn-is-white\/","text":"# [Math] An urn contains N1 white and N2 black balls from which k are drawn without replacement. Find the probability that the next ball drawn is white.\n\ncombinationscombinatoricsprobability\n\nI tried solving the problem but the final expression I fail to simplify. I basically considered each possible cases i.e., drawing 0 white and k black, 1 white and (k-1) black etc and then multiplied by the probability of drawing a white on the next trial. I don't see how we can solve it otherwise. The book provided a solution that doesn't make sense to me. How can it be that simple?\n\nThe counting of \"event points\" in the solution uses the fact that we can use something we know about what happened at the time a particular ball was drawn to deduce what could have happened at the time another particular ball was drawn.\n\nFor the \"total number\" we know that there are $N_1+N_2$ balls in the urn when the first one is drawn. Once we know which of those balls was drawn first, there remain $N_1+N_2-1$ possibilities for which ball was drawn second. (Each \"first ball\" gives us a different set of $N_1+N_2-1$ possibilities for the second ball, but we are just counting the total number of possibilities, so what matters is that there are always $N_1+N_2-1$ of them.) For each of the $(N_1+N_2)(N_1+N_2-1)$ total possibilities for the first two balls, there are $N_1+N_2-2$ possibilities for the third, and so forth. And that's how we end up with $$T = (N_1+N_2)(N_1+N_2-1)(N_1+N_2-2)\\cdots(N_1+N_2-k)$$ event points in the entire probability space.\n\nBut this kind of counting argument actually depends only on the order in which we ask which ball was drawn at each time, not in the order in which the balls actually were drawn. We get the same total $T$ if we first consider the $k+1$st ball to be drawn, and then consider the first ball drawn, then the second, and so forth.\n\nTo count the events in which the $k+1$st ball is white, the book's solution considers the balls in just such a sequence. There are $N_1$ ways in which the $k+1$st ball could be white. For each of these, there are now just $N_1+N_2-1$ possibilities for the first ball drawn (since we already know that a particular ball was not drawn before the $k+1$st drawing), $N_1+N_2-2$ possibilities for the second ball, and so forth. The count of possible event points is still a product of the number of choices available for each ball, and the author has chosen to write these terms in the order in which the balls were drawn rather than the order in which the counting method identified each ball. So the $k+1$st ball, although we \"observed\" its identity first, appears as the last term $N_1$ of the product: $$S = (N_1+N_2-1)(N_1+N_2-2)\\cdots(N_1+N_2-k)N_1.$$\n\nBy symmetry, we assign identical probability to each of the original $T$ event points, and therefore the probability we are looking for is just $S\/T,$ that is, $$\\frac ST = \\frac{(N_1+N_2-1)(N_1+N_2-2)\\cdots(N_1+N_2-k)N_1} {(N_1+N_2)(N_1+N_2-1)(N_1+N_2-2)\\cdots(N_1+N_2-k)}.$$\n\nLook at the ratio on the right. Each term from $(N_1+N_2-1)$ to $(N_1+N_2-k)$ occurs exactly once on top and once on the bottom, so all these terms can be canceled out. The only terms that cannot be canceled are $N_1$ on top and $N_1+N_2$ on the bottom. So we end up with $$\\frac ST = \\frac{N_1}{N_1+N_2}.$$\n\nAs noted in other comments and answer(s), this is not the easy way to find the probability. The easy way is just to imagine what happens if you keep your eyes shut while you draw the first $k$ balls and only open them when you are holding the $k+1$st ball, so that you see it before all the others. You then have a somewhat complicated way of drawing a random ball from an urn, consisting of ejecting $k$ balls without looking before you make your final selection. Every ball in the original urn is equally likely to be the ball selected in this way.","date":"2022-10-01 08:06:46","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\": 0.7004666328430176, \"perplexity\": 135.66395244223537}, \"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-2022-40\/segments\/1664030335573.50\/warc\/CC-MAIN-20221001070422-20221001100422-00574.warc.gz\"}"} | null | null |
Seuratiaceae är en familj av svampar. Enligt Catalogue of Life ingår Seuratiaceae i klassen Dothideomycetes, divisionen sporsäcksvampar och riket svampar, men enligt Dyntaxa är tillhörigheten istället divisionen sporsäcksvampar och riket svampar.
Källor
Sporsäcksvampar
Seuratiaceae | {
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-the-equation-n-times-500-5000","text":"# How do you solve the equation n times 500=5000?\n\nOct 22, 2016\n\n$n = 10$\n\n#### Explanation:\n\nWe have the equation.\n\n$n \\times 500 = 5000 \\Rightarrow 500 n = 5000$\n\nTo solve for n, divide both sides by 500.\n\n$\\frac{\\cancel{500} n}{\\cancel{500}} = \\frac{5000}{500}$\n\n$\\Rightarrow n = 10 \\text{ is the solution}$\n\nOct 22, 2016\n\nn=10\n\n#### Explanation:\n\n$\\textcolor{b l u e}{\\text{Using shortcut method}}$\n\nMove the 500 to the other side of the = and change it to divide\n\n$n = \\frac{5000}{500}$\n\n$n = \\frac{{\\cancel{5000}}^{10}}{{\\cancel{500}}^{1}} \\leftarrow \\text{ canceling out}$\n\n$n = 10$\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n$\\textcolor{b l u e}{\\text{Using first principles method}}$\n\nDivide both sides by 500\n\n$\\frac{n \\times 500}{500} = \\frac{5000}{500}$\n\nBut $10 \\times 500 = 5000$ so we have\n\n$\\frac{n \\times 500}{500} = \\frac{10 \\times 500}{500}$\n\nThis is the same as\n\n$n \\times \\frac{500}{500} = 10 \\times \\frac{500}{500}$\n\nBut $\\frac{500}{500} = 1$\n\n$n \\times 1 = 10 \\times 1$\n\n$n = 10$","date":"2020-08-09 18:04:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 16, \"mathjax_inline_tex\": 1, \"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\": 0.8881508111953735, \"perplexity\": 1486.7427935976318}, \"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-2020-34\/segments\/1596439738562.5\/warc\/CC-MAIN-20200809162458-20200809192458-00080.warc.gz\"}"} | null | null |
Q: How to sort the column in ascending order in R I want to make my column(date) in an ascending order.
I tried below but it didnt worked.
data3 <- data2[order(data2$date_1,decreasing = FALSE)]
It gave me this error:--
Error in [.data.frame(data2, order(data2$date_1, decreasing = FALSE)) :
undefined columns selected
my data is
Sr. date_1 No_of_births
1 1 40255
2 10 41874
3 11 38940
4 12 40320
5 2 36428
6 3 39940
7 4 37641
8 5 39288
9 6 38789
10 7 42148
11 8 42980
12 9 42112
The output i want is as below with other columns also.
(((just showing date column)))
Date
1
2
3
4
5
6
7
8
9
10
11
12
A: Easy dplyr solution:
install.packages('dplyr')
library('dplyr')
data3 <- arrange(data2, date)
A: try data3 <- data2[order(as.integer(data2$date_1),decreasing = FALSE), ]
Plus:
It is a good idea to check class(date2$date_1) before using order. It this data are of class integer or numeric, then you have 1, 2, 3, ..., 10, 11, ...; However, it is character, then without casting/coercing it into integer, you get "1", "10", "11", "12", ... "2".
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,470 |
\section{Introduction} \label{sec:introduction}
\noindent The compatibility between a reactor-grade plasma and the material walls surrounding the plasma is one of the main challenges facing a magnetic fusion device.
The presence of very low levels of high $Z$ impurities in the core plasma may lead to unacceptable levels of radiation losses and fuel dilution.
Also low $Z$ impurities, in the form of Beryllium or Helium-ash, may result in fuel dilution that severely limits the attainable fusion power~\cite{Harte2010}.
Consequently, the transport properties of impurities is a high priority issue in present experimental and theoretical fusion plasma research.
This is emphasised by the the new ITER-like wall experiment in JET~\cite{Matthews2009}, where a Beryllium-clad first wall in the main chamber, combined with carbon and tungsten tiles in the divertor, will be tested for the first time.
The transport of main fuel as well as impurities in the core region of tokamaks is expected to be dominated by turbulence driven by Ion Temperature Gradient (ITG) modes and Trapped Electron (TE) modes.
The main drives for the ITG/TE~mode instabilities are gradients of temperature and density combined with unfavourable magnetic curvature.
Most of the theoretical studies of turbulent particle transport have been devoted to temperature gradient driven ITG~and TE~modes, using both fluid, quasilinear (QL) and nonlinear (NL) gyrokinetic models~\cite{Frojdh1992, Basu2003, Estrada-Mila2005, Naulin2005, Priego2005, Fulop2006, Bourdelle2007, Dubuit2007, Camenen2009, Fulop2010, Futatani2010, Hein2010, Moradi2010, Fulop2011, Nordman2008, Angioni2006, Nordman2007a, Angioni2007, Angioni2009a, Fulop2009, Nordman2011, Skyman2011a}.
Much less effort has been devoted to particle transport in regions with steep density gradients.
The density gradient provides a drive for TE~modes which may dominate the temperature gradient drive for plasma profiles with $R/L_{n_e}>R/L_{T_e}$.
This may occur in connection with the formation of transport barriers, like the high confinement mode edge pedestal, in fusion plasmas.
In the present letter, the turbulent transport of main ion and trace impurities in tokamaks is investigated through nonlinear (NL) gyrokinetic simulations using the GENE code.
The main part considers collisionless TE~modes driven by density gradients but with a transition to temperature gradient driven TE~modes as the density profiles flattens.
The impurity density gradient for zero impurity flux is calculated for varying background electron density gradient drive and for a range of impurity species.
This study complements recent studies~\cite{Nordman2011, Skyman2011a}
on temperature gradient driven TE~and ITG~mode impurity transport.
The results are compared with QL kinetic simulations and a computationally efficient multi fluid model, suitable for use in predictive transport simulations.
Of particular interest is the sign of the impurity convective flux and the degree of impurity peaking in the presence of strong background electron density gradients.
The models used have been described in detail elsewhere, see~\cite{Nordman2011} and references therein, only a brief summary is given here.
The NL and QL GENE simulations were performed in a flux tube geometry, in a low $\beta$ ($\beta=10^{-4}$) s--$\alpha$ equilibrium~\cite{Jenko2000, Dannert2005a, Dannert2005b, Merz2008a}.
In order to ensure that the resolution was adequate, the resolution was varied separately for the perpendicular, parallel and velocity space coordinates, and the effects of this on the mode structure, $k_\perp$ spectra and flux levels were investigated.
The resolution was then set sufficiently high for the effects on these indicators to have converged.
For a typical NL simulation for main ions, fully kinetic electrons, and one trace species, a resolution of $n_{x} \times n_{y} \times n_{z} = 96 \times 96 \times 24$ grid points in real space and of $n_{v} \times n_{\mu} = 48 \times 12$ in velocity space was chosen.
For QL~GENE simulations the box size was set to $n_{x} \times n_{y} \times n_{z} = 8 \times 1 \times 24$ and $n_{v} \times n_{\mu} = 64 \times 12$ respectively.
The impurities were included self-consistently as a third species in the simulations, with the trace impurity particle density $n_Z/n_e = 10^{-6}$ in order to ensure that they have a negligible effect on the turbulence.
For the fluid simulations, the Weiland multi-fluid model~\cite{Weiland2000} is used to derive the main ion, impurity, and trapped electron density response from the corresponding fluid equations in the collisionless and electrostatic limit.
The free electrons are assumed to be Boltzmann distributed.
The equations are closed by the assumption of quasineutrality.
An eigenvalue equation for TE~and ITG~modes is obtained in the presence of impurities.
The eigenvalue equation is solved for general mode width~\cite{Weiland2000}.
Alternatively, a strongly ballooning eigenfunction with $k_\parallel^2=\left(3 q^2 R^2\right)^{-1}$ can be used for magnetic shear $s \sim 1$~\cite{Hirose1994}.
The eigenvalue equation is then reduced to a system of algebraic equations that is solved numerically.
\label{sec:PF}
The main ion and impurity particle fluxes can then be written as:
\begin{equation} \label{eq:Gamma_derivation}
\Gamma_{j} =
\left<\delta n_j v_{\vec{E}\times\vec{B}}\right> =
-n_j \rho_s c_s\left<\widetilde{n}_j\frac{1}{r}\D{\theta}{\widetilde{\phi}}
\right>.
\end{equation}
\noindent The angled brackets imply a time and space average over all unstable modes. Performing this averaging for a fixed length scale $k_\theta\rho_s$ of the turbulence, the particle flux can be written:
\begin{equation}
\label{eq:transport}
\frac{R\Gamma_j}{n_j} = D_j\frac{R}{L_{n_j}} + D_{T_j}\frac{R}{L_{T_j}} + R V_{p,j}.
\end{equation}
The first term in equation~\eqref{eq:transport} corresponds to diffusion, the second to the thermodiffusion and the third to the convective velocity (pinch), where $1/L_{n_j}=-\nabla n_j/n_j$, $n_j$ is the density of species $j$ and $R$ is the major radius of the tokamak
The pinch contains contributions from curvature and parallel compression effects.
These have been described in detail in previous work~\cite{Nordman2007a, Nordman2008, Nordman2011, Angioni2006}.
For trace impurities, equation~\eqref{eq:transport} can be uniquely written $R\Gamma_Z/n_Z =D_Z R/L_{n_Z} + RV_Z$, where $D_Z$ is the impurity diffusion coefficient and $V_Z$ is the impurity convective velocity.
The zero-flux impurity density gradient (peaking factor) is defined as $PF_Z=-RV_Z/D_Z$ for the value of the impurity density gradient that gives zero impurity flux.
Solving the linearised equation~\eqref{eq:transport} for $R/L_{n_Z}$ with $\Gamma_Z = 0$ yields the interpretation of $PF_Z$ as the gradient of zero impurity flux, and it quantifies the balance between convective and diffusive impurity transport.
The main parameters used in the simulations are summarised in table~\ref{tab:parameters}.
The parameters where chosen to represent an arbitrary tokamak geometry at about mid radius, and do not represent any one particular experiment.
A a moderately steep electron temperature gradient ($R/L_{T_e}=5.0$) together with a flatter ion temperature gradient ($R/L_{T_i, Z}=2.0$) were used to promote TE~mode dominated dynamics.
Following~\cite{Ernst2009}, the background density gradient for the base scenario was set higher than the temperature gradient, to ensure density gradient driven dynamics.
In order to preserve quasineutrality $\nabla n_e=\nabla n_i$ was used.
First, the main ion particle flux ($\Gamma_p$) is studied.
Time averaged fluxes are calculated from time series of NL~GENE data after convergence, as illustrated in figure~\ref{fig:time_series}.
The scalings of $\Gamma_p$ with the electron density gradient obtained from NL~GENE and fluid simulations are shown in figure~\ref{fig:Gamma_p}.
The large transport found in NL~GENE simulations is an indication of the stiffness of the gyrokinetic model, and is often seen in fixed-gradient simulations of turbulence.
The fluid model shows a similar scaling of the main ion flux, but the transport is smaller for $R/L_{n_e} > 3.0$.
The main ion density gradient corresponding to zero ion flux ($PF_p$) can be found by similar means to that of $PF_Z$, however, since the trace approximation is not valid for the main ions, the zero-flux gradient has to be found explicitly by varying $\nabla n_p$ until the condition $\Gamma_p=0$ is satisfied.
The NL~GENE results presented in figure~\ref{fig:Gamma_p} indicate that for the present parameters, lower density gradients only result in $\Gamma_p \rightarrow 0$.
The the fluid model gives a small outward flux in the limit $R/L_{n_e} \rightarrow 0$.
Neither model results in flux reversal for TE~mode driven turbulence.
Next, the scaling of the impurity transport with the background density gradient ($R/L_{n_e}$) is investigated.
The results for the impurity peaking factor are shown in figure~\ref{fig:omn_TEM}.
We note that the impurity peaking saturates with $PF_Z\approx 2.0$ for large values of the electron density gradient.
The QL results tend to consistently overestimate the peaking factors, while the fluid model gives results that are somewhat below the NL GENE results for the steeper gradients.
The fluid results show a considerably less dramatic dependency of the peaking factor than the gyrokinetic results, both of which show a strong decrease in $PF_Z$ as the electron density profiles flatten.
This is observed for all values of the impurity charge number.
As the background density profile becomes more peaked, a corresponding increase in impurity transport is expected.
This is illustrated in figure~\ref{fig:D_RV}, where scalings, obtained from NL~GENE simulations, of the diffusivity ($D_Z$) and convective velocity ($R V_Z$) with $R/L_{n_e}$ are shown.
Although $D_Z$ and $R V_Z$ strongly increase with $R/L_{n_e}$, the impurity peaking ($PF_Z=-RV_Z/D_Z$) is only weakly sensitive to the electron density gradient.
For $R/L_{n_e}\lesssim 2.0$ the impurity peaking factor is not well defined, since both $D_Z$ and $R V_Z$ go to zero.
The corresponding linear eigenvalues are displayed in figure~\ref{fig:omn_TEM_eigens}.
The fluid and gyrokinetic results are in qualitative agreement, showing an growthrate that increases uniformly with $R/L_{n_e}$.
The results indicate a smooth transition from density gradient driven to temperature gradient driven TE~mode turbulence, which dominates for $R/L_{n_e} \lesssim R/L_{T_e}$~\cite{Ernst2009}.
The scaling of the impurity peaking factor with impurity charge ($Z$), with $R/L_{n_e}$ as a parameter, is illustrated in figure~\ref{fig:Z}.
The models show only a very weak scaling, with $PF_Z$ falling toward saturation for higher $Z$.
The results are similar to those for the temperature gradient driven TE~mode reported in~\cite{Skyman2011a}.
Notably, the QL~GENE simulations overestimate the peaking factors, whereas the fluid results are lower than the peaking factors obtained from NL~GENE simulations.
The trend observed for low $Z$ impurities is reversed compared to trends reported in e.g.~\cite{Nordman2011} for ITG~mode driven impurity transport.
The qualitative difference can be understood from the $Z$-dependent thermodiffusion in equation~\eqref{eq:transport}, which is outward for ITG~modes and inward for TE~modes.
In summary, the turbulent transport of main ion and trace impurities in regions of steep density gradients has been investigated through nonlinear (NL) gyrokinetic simulations using the GENE code.
The main part has considered collisionless TE~modes driven by density gradients but with a transition to temperature gradient driven TE~modes as the density profiles flattens.
The results for the impurity density gradient of zero particle flux (peaking factor) have been compared with QL kinetic simulations and a reduced and computationally efficient multi-fluid model, suitable for use in predictive transport simulations.
For the parameters studied, qualitative agreement between gyrokinetic and fluid results has been obtained for the scaling of the impurity peaking factor with both the background density gradient and the impurity charge.
In the region of steep electron density gradients, it was shown that the impurity peaking factor saturates at values significantly smaller than the driving electron density gradient.
It was noted that for the chosen length scales, the QL~GENE results generally overestimate the peaking factor, whereas the fluid results are close to or lower than the NL~GENE results.
The scaling of the peaking factor with impurity charge was observed to be weak, with a slight increase in the impurity peaking factor observed in the gyrokinetic results for low impurity charge numbers.
\section*{Acknowledgements}
The simulations were performed on resources provided on the Lindgren~\cite{Lindgren} and HPC-FF~\cite{HPC-FF} high performance computers, by the
Swedish National Infrastructure for Computing (SNIC) at Paralleldatorcentrum
(PDC) and the European Fusion Development Agreement (EFDA), respectively.
The authors would like to thank Frank Jenko, Tobias G\"orler, M.~J. P\"uschel, and the rest of the GENE~team~\cite{GENE} at IPP--Garching for their help with the gyrokinetic simulations.
\bibliographystyle{unsrtnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,038 |
\section{Introduction}
We encounter phase transitions under various incarnations in every-day life, such as in the case of boiling water or the melting of ice.
From a thermodynamic perspective, these processes manifest as a nonanalytic behavior upon tuning an externally-controlled parameter~\cite{1987imsm.book.....C,1998AmJPh..66..164C,2011qpt..book.....S}, such as the temperature.
In the case of boiling water for instance the free energy $F(T)$ exhibits a nonanalytic structure upon increasing the temperature $T$, as illustrated in Fig.~\ref{fig.1}B.
However, all these processes take their time.
Suppose that we monitor the temperature $T$ during the actual heating process in real time, then we would observe the behavior displayed in Fig.~\ref{fig.1}C.
After an initial growth, $T$ levels off to the boiling temperature $T_B$.
A further increase can only occur, when the heating mechanism has provided the latent heat necessary to cross this first-order phase transition.
And since a realistic heating machine can only provide a finite power, this takes time.
Importantly, monitoring the temporal heating process of crossing the transition is now smooth in contrast to the characteristic nonanalytic behavior as a function of the control parameter.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{Fig1.pdf}
\caption{{\bf A} Schematic illustration of the first-order boiling transition of water. {\bf B} The free energy $F(T)$ exhibits nonanalytic behavior as a function of the external control parameter $T$ at the boiling temperature $T_B$. {\bf C} When monitored as a function of real time the temperature $T$ changes smoothly upon heating up the liquid.}
\label{fig.1}
\end{figure}
In the quantum world, remarkably, this can be different: a quantum many-body system can undergo a dynamical quantum phase transition (DQPT) with physical quantities becoming nonanalytic as a function of real time~\cite{2013PhRvL.110m5704H,2018RPPh...81e4001H}, see Fig.~\ref{fig.2} for some recently obtained experimental measurements.
At such a DQPT a system can therefore show nonsmooth properties caused solely by drastic internal changes and not imposed by the exterior.
In this article we give an overview over this phenomenon including both a summary of the theoretical and experimental developments as well as a discussion on open challenges and future prospects.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig2.pdf}
\caption{At a dynamical quantum phase transition physical quantities can become nonanalytic as a function of time. Here, the data from a trapped ion experiment shows that the Loschmidt echo rate function $\lambda(t)$ for a long-range transverse-field Ising model can exhibit kinks as a function of a rescaled dimensionless time $\tau$~\cite{2017PhRvL.119h0501J}.}
\label{fig.2}
\end{figure}
These DQPTs occur in closed quantum many-body systems during unitary real-time evolution, so that the influence of an environment can be neglected.
In addition to fundamental theoretical aspects~\cite{2011RvMP...83..863P, 2015NatPh..11..124E, 2015ARCMP...6..383A, 2015ARCMP...6...15N, 2017NatPh..13..424M } the motivation to study such nonequilibrium unitary dynamics originates to a large extent in experimental advances in so-called quantum simulators over the last years~\cite{2008RvMP...80..885B, 2012NatPh...8..267B, 2012NatPh...8..277B, 2014RvMP...86..153G}.
In platforms such as ultra-cold atoms or trapped ions among many others it has become possible to experimentally realize and probe such scenarios with a high degree of control and precision, which has lead to the observation of inherent dynamical phenomena without equilibrium counterparts.
This includes the observation of many-body localization~\cite{2015Sci...349..842S, 2016Sci...352.1547C, 2016NatPh..12..907S}, prethermalization~\cite{2012Sci...337.1318G,2017SciA....3E0672N,2018arXiv180905554S}, particle production in lattice gauge theories~\cite{2016Natur.534..516M}, discrete time crystals~\cite{2017Natur.543..221C, 2017Natur.543..217Z}, or dynamical quantum phase transitions~\cite{2017PhRvL.119h0501J, 2018NatPh..14..265F, 2017Natur.551..601Z, 2018arXiv180609269G, 2018arXiv180704483T, 2018arXiv180610871W, 2018arXiv180803930X}.
The theoretical description and understanding of such nonequilibrium quantum states is, however, facing major challenges.
On a fundamental level, these states don't exhibit a description in terms of a free energy and are therefore beyond thermodynamics.
This, however, might not only be seen as an obstacle but rather as an opportunity to generate new quantum states beyond equilibrium constraints such as the principle of equal a priori probabilities in the microcanonical ensemble.
As a consequence the field of nonequilibrium quantum physics admits a framework to generate many-body states with novel properties impossible to realize within conventional thermodynamics.
A prominent example in this direction constitutes the celebrated time crystal~\cite{2016PhRvL.116y0401K, 2016PhRvL.117i0402E, 2017NatPh..13..424M}, which cannot exist in equilibrium states~\cite{2013PhRvL.110k8901B,2015PhRvL.114y1603W}.
From an alternative point of view the main challenge in the understanding of nonequilibrium quantum states is that now it is not sufficient to understand the properties of Hamiltonians.
Instead, we have to characterize time-evolution operators:
\begin{equation}
U(t) = \mathcal{T}e^{-i \int_0^t dt' \, H(t')}\, .
\end{equation}
Here, $\mathcal{T}$ denotes the time-ordering prescription and $H(t)$ the, in general, time-dependent Hamiltonian.
Crucially, the propagator $U(t)$ contains one additional scale, which is time itself.
As we will discuss in the remainder of these Perspectives, this additional scale can lead to new physics.
However, it remains a central question how to extract general principles in time-evolution operators, i.e., quantum dynamics, and to which extent we can describe nonequilibrium quantum many-body states in a unified manner.
It is one of the main purpose of this article to summarize how the theory of DQPTs can contribute to this field.
\section{Dynamical quantum phase transitions}
A DQPT is a phase transition in time driven by sharp internal changes in the properties of a quantum many-body state and not driven by an external control parameter such as temperature or pressure.
The central object for the theory of DQPTs is the so-called Loschmidt amplitude
\begin{equation}
\mathcal{G}(t) = \langle \psi_0 | \psi_0(t)\rangle = \langle \psi_0| U(t) | \psi_0 \rangle\, ,
\label{eq:def_G}
\end{equation}
denoting the overlap between the initial state $|\psi_0\rangle$ and the time-evolved one $|\psi_0(t)\rangle$.
Alternatively, one can interpret $\mathcal{G}(t)$ as a matrix element of the time-evolution operator $U(t)$.
In this way, the study of $\mathcal{G}(t)$ serves the purpose of characterizing $U(t)$ instead of Hamiltonians and their respective thermal states, as is done in equilibrium.
In the following we will consider for convenience a specific nonequilibrium protocol, a so-called quantum quench.
For a quantum quench the initial condition $|\psi_0\rangle$ is chosen as the ground state of an initial Hamiltonian $H_0$ and the Hamiltonian driving the time evolution as $H(t)=H$.
Such a scenario results from a sudden switching of a system parameter at time $t=0$.
The Loschmidt amplitude $\mathcal{G}(t)$ then acquires the following form
\begin{equation}
\mathcal{G}(t) = \langle \psi_0 | e^{-iHt} |\psi_0 \rangle \, .
\end{equation}
While we will restrict the discussion to such quantum quenches, the theory of DQPTs is formulated in a much more general context~\cite{2018RPPh...81e4001H}, as one can already see from Eq.~(\ref{eq:def_G}), which does not rely on a specific nonequilibrium protocol.
Although we will focus here on the case of initial pure states, extensions to mixed states have been discussed in the literature recently~\cite{2016PhRvB..93j4302A,2017PhRvB..96r0303B,2017PhRvB..96r0304H,2018PhRvB..97q4401L,2018NatSR...811921B}.
While real-time nonanalyticities in the Loschmidt echo have been first found in Ref.~\cite{2010PhRvE..81b0101P}, the interpretation as a DQPT and the connection to conventional phase transitions has been formulated in Ref.~\cite{2013PhRvL.110m5704H}.
It is one of the most important insights that Loschmidt amplitudes on a formal level resemble partition functions in conventional statistical physics~\cite{2013PhRvL.110m5704H}.
While this might appear already evident from, e.g. the defining equation of the partition function $Z$ of the canonical ensemble $Z=\mathrm{Tr} e^{-\beta H}$, there is an even stronger formal analogy to so-called boundary partition functions $Z_B$ which have the structure $Z_B=\langle \psi_A | e^{-RH} | \psi_B \rangle$~\cite{1995NuPhB.453..581L}.
This class of partition functions describes systems subject to boundary conditions on two ends, which are encoded in the boundary states $|\psi_{A/B}\rangle$.
The spatial distance between the two boundaries is proportional to $R$ and $H$ denotes the bulk Hamiltonian.
In this context one can interpret the Loschmidt amplitude $\mathcal{G}(t)$ as a boundary partition function in time with $|\psi_0\rangle$ implementing the respective temporal boundary conditions.
As the free energy corresponding to an equilibrium partition function becomes nonanalytic at a phase transition, so can therefore
\begin{equation}
g(t) = -\frac{1}{N} \log \big[ \mathcal{G}(t) \big] \, ,
\end{equation}
which can be viewed as a dynamical counterpart to a free energy density up to a differently chosen normalization with $N$ denoting the number of degrees of freedom.
A point in time $t^\ast$, where such a nonanalyticity occurs, we define in the following as a DQPT.
It will be useful to also consider the quantity
\begin{equation}
\lambda(t) = -\frac{1}{N} \log \big[ \mathcal{L}(t) \big] \, , \quad \mathcal{L}(t) = \big| \mathcal{G}(t) \big|^2 \, ,
\end{equation}
which is the analog of $g(t)$ for the probability $\mathcal{L}(t)$ associated with $\mathcal{G}(t)$.
Evidently, $\lambda(t) = 2 \mathrm{Re}[g(t)]$ such that a nonanalytic behavior in $g(t)$ directly translates into a nonanalytic behavior in $\lambda(t)$.
In Fig.~\ref{fig.2} one can see a DQPT in $\lambda(t)$ for the paradigmatic Ising model upon quenching a transverse field.
In the meantime DQPTs have been observed in a variety of different platforms addressing diverse aspects and physical phenomena. This includes experiments in ultra-cold atoms in optical lattices~\cite{2018NatPh..14..265F}, trapped ions~\cite{2017PhRvL.119h0501J}, quantum walks~\cite{2018arXiv180610871W, 2018arXiv180803930X}, nanomechanical oscillators~\cite{2018arXiv180704483T}, and superconducting qubits~\cite{2018arXiv180609269G}.
It will be the goal of the remainder of this article to discuss the physical meaning of these real-time nonanalyticities and to explore their implications for the understanding of the dynamics in quantum many-body systems.
\section{General implications}
While the real-time nonanalyticities themselves might be regarded already as an intriguing phenomenon, it remains a central question to understand the physical implications of DQPTs.
On a more general note, let us emphasize that the dynamical analog $g(t)$ to the free energy density is not a thermodynamic potential.
In particular, derivatives of $g(t)$ are not related to measurable quantities as compared to the equilibrium case where, for example, the second derivative of the free energy with respect to temperature yields the specific heat.
As a consequence, the observation of a temporal nonanalyticity in $g(t)$ does not immediately imply a measurable signature in a physical observable.
A DQPT rather indicates a point in time where the time-evolution operator $U(t)$ exhibits a drastic change in its properties without providing insights into the character of this changes, which requires a further analysis.
Let us take the chance at this point to draw an analogy to an equilibrium scenario sharing some similarities.
Quantum phase transitions for 1D systems can be detected through those points in parameter space where the area law of the entanglement entropy is violated~\cite{2003PhRvL..90v7902V}.
This method for detection has found various applications in recent years since it represents a system-independent general-purpose tool that doesn't require detailed knowledge about the physical system.
The study of the entanglement entropy alone, however, does not provide the full physical picture, for which we still need to identify the respective order parameters for instance.
In a similar way, a DQPT implies a point in time with a radical change in the time-evolution operator and therefore represents an analogous system-independent indicator.
The nonanalytic temporal behavior in $g(t)$, however, does not specify the physical origin of the DQPT, which requires further analysis.
These considerations naturally lead to the question: what do we learn from DQPTs then?
Clearly, this field is still developing and new facets are likely to explored in the future.
However, currently, two major aspects are worthwhile mentioning here:
(i) it is, remarkably, possible for some classes of models to obtain information about ground state phase diagrams from the study of DQPTs although being driven far away from equilibrium;
(ii) the theory of DQPT provides general principles of quantum dynamics, which allow us to understand classes of nonequilibrium scenarios instead of analyzing individual problems.
\section{Relation to underlying equilibrium phase transitions}
In many cases it has been recognized that DQPTs are directly connected to the underlying equilibrium phase transitions of the considered models.
However, it has turned out that this connection is, in general, not to be understood as a one-to-one correspondence~\cite{2014PhRvB..89p1105V, 2014PhRvB..89l5120A, 2015PhRvB..92g5114S, 2017PhRvB..96m4427H}.
Therefore, DQPTs constitute a genuine nonequilibrium phenomenon.
The relation between the appearance of DQPTs and the underlying ground state properties of the Hamiltonian are most extensively understood for topological two-band models~\cite{2015PhRvB..91o5127V,2016PhRvB..93h5416B,2016PhRvL.117h6802H}.
A quantum quench across a topological ground state phase transition in one dimension (1D) always leads to a DQPT in real-time evolution, which is why these DQPTs are also termed topologically protected~\cite{2015PhRvB..91o5127V, 2016PhRvL.117h6802H}.
The reverse, however, is not always true.
It can happen that a system undergoes a DQPT without crossing an underlying equilibrium transition.
These DQPTs are therefore 'accidental' and upon smoothly changing the Hamiltonian parameters, they can be made disappear without closing a gap along the way.
This strong connection between equilibrium critical points and DQPTs can be used in 1D, remarkably, to map out ground state phase diagrams based just on the study of DQPTs~\cite{2016PhRvB..93h5416B}.
In this way, information about ground states can be inferred via nonequilibrium dynamics, which necessarily takes place at elevated energy densities.
Specifically, the dynamics of a dynamical topological order parameter~\cite{2016PhRvB..93h5416B}, which can be defined on general grounds for these models, is capable to distinguish uniquely, whether an underlying quantum phase transitions has been crossed or not.
This has, for example, be used in split-step quantum walks to experimentally obtain information about phase boundaries~\cite{2018arXiv180803930X}.
For systems other than the topological ones discussed in the paragraph before, it is not possible to rigorously connect equilibrium and dynamical phase transitions on general grounds.
For many models, however, the appearance of DQPTs is nevertheless directly related, so that also here DQPTs can be utilized to map out equilibrium ground state phase diagrams, by keeping in mind, however, that this might be subject to change upon continuously deforming the Hamiltonian even without gap closing.
Importantly, a weak symmetry-preserving perturbation is, however, not sufficient to achieve this, see also the discussion on the robustness of DQPTs below.
Finally, there are also models for which quantum quenches across equilibrium transitions do not lead to DQPTs because of kinetic constraints~\cite{2014PhRvB..89p1105V, 2014PhRvB..89l5120A}.
The discussion up to now has only considered the case of ground state phase transitions.
How transitions at nonzero temperature reflect dynamically in DQPTs is currently much less known, in particular because such transitions require at least 2D, which is much more challenging to theoretically address with some notable exceptions~\cite{2014PhRvL.113z5702C, 2015PhRvL.115n0602H, 2018PhRvL.120m0601Z, 2017PhRvB..96m4427H, 2017PhRvB..96m4313W, 2017PhRvL.119h0501J,2018arXiv180807874H}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig3.pdf}
\caption{Dynamical counterpart to quantum critical regions in the vicinity of DQPTs. {\bf A} Schematic illustration of the energy density-time plane, where energy is measured with respect to the initial Hamiltonian and $\varepsilon=0$ corresponds to the ground state energy density. A DQPT occurs along $\varepsilon=0$ at a critical time $t^\ast$. The nonanalytic properties of the DQPT at $t^\ast$ can extend to $\varepsilon>0$, schematically depicted here as the green area. The average energy $\varepsilon_\mathrm{av}(t)$ is included as the dotted red line.
%
{\bf B} Such dynamical analogs of quantum critical regions can, for some specific cases, be measured experimentally~\cite{2017PhRvL.119h0501J}, see {\bf (c)} where the energy- and time-resolved magnetization $M_x(\varepsilon,t)$ is shown on a color scale.
%
The dashed gray lines indicate the locations of two subsequent DQPTs, occurring in the unitary dynamics of realized long-ranged transverse-field Ising chain, see {\bf (a)}.
%
In the vicinity of both of the two DQPTs an area, which is controlled by the DQPT (white), extends to $\varepsilon>0$ and intersects $\varepsilon_\mathrm{av}(t)$, which allows to conclude that these DQPTs control the nonequilibrium dynamics of the considered magnetization, as one can indeed observe as vanishing values for the mean magnetization $M_x$ in {\bf (b)}.
}
\label{fig.4}
\end{figure}
\section{Dynamical analogs to quantum critical regions}
The identification of DQPTs as nonequilibrium phase transitions has been argued initially on the basis of the formal similarity of the Loschmidt amplitude with equilibrium partition functions.
However, it is important to note again that $g(t)$ does not represent a thermodynamic potential despite of the formal similarities to free energy densities.
Derivatives of $g(t)$ cannot be connected directly to observables or correlation functions, leading immediately to the following open questions:
(i) Is there an indirect connection to observables or correlation functions then?
(ii) What do these DQPTs mean when the quantity $g(t)$, which entails the defining real-time nonanalyticities, does not function as a thermodynamic potential?
While many models and nonequilibrium protocols have been identified, where temporal nonanalyticities occur, for the understanding of DQPTs these two questions still remain as major challenges.
In the following, a physical interpretation of DQPTs, useful in many cases, will be summarized.
This interpretation is based on viewing the Loschmidt amplitude and echo as measures to probe the time-evolved quantum many-body state $|\psi_0(t)\rangle$ in the ground state manifold of the initial Hamiltonian $H_0$, since both $\mathcal{G}(t)$ and $\mathcal{L}(t)$ project $|\psi_0(t)\rangle$ back onto the ground state $|\psi_0\rangle$ of $H_0$.
Along these lines, let us decompose $|\psi_0(t)\rangle$ in the full eigenbasis $|\psi_\nu\rangle$ of the initial Hamiltonian:
\begin{equation}
|\psi_0(t)\rangle = \sum_{\nu} a_\nu(t) | \psi_\nu \rangle \, , \quad a_\nu(t) = \langle \psi_\nu | \psi_0(t) \rangle \, ,
\end{equation}
where $\nu=0$ corresponds to the initial ground state, i.e., $a_{\nu=0}(t) = \mathcal{G}(t)$.
From the perspective of this expansion $\mathcal{G}(t)$ quantifies one of the exponentially many amplitudes $a_\nu(t)$.
Thus, how can this single overlap be important, when most of the weights $a_\nu(t)$ are in $\nu\not=0$?
This can only be the case when $\nu=0$ does not represent a singular point, but rather when the properties of $\mathcal{G}(t)=a_0(t)$ extend to other amplitudes $\nu\not=0$.
An important example in equilibrium, where the properties of a single quantum many-body state extend to a significant portion of Hilbert space, is that of a quantum phase transition~\cite{2011qpt..book.....S}.
Although nonanalyticities as a function of a control parameter can be found only in the ground state, the zero-temperature critical point controls the whole quantum critical region in the temperature-control parameter plane.
It is the goal of the following discussion to argue that an analog to a quantum critical region can also exist for DQPTs, see also Fig.~\ref{fig.4}.
From an operational point of view, $\mathcal{L}(t) = | a_0 (t)|^2$ is the result of a projective measurement of the energy $E$ with the initial Hamiltonian $H_0$, where as a measurement outcome we have obtained $E=0$ upon choosing the zero of energy accordingly.
We might, however, also consider other possible measurement outcomes and probe the state's properties also at excited energy densities $\varepsilon=E/N>0$.
In case $a_0(t)$ does not represent a singular point, the DQPT occurring at $t=t^\ast$ along $\varepsilon=0$ extends also to $\varepsilon>0$, as illustrated in Fig.~\ref{fig.4}A.
Accordingly, the temperature-control parameter plane is replaced by an energy density-time plane with a potential dynamical analog to a quantum critical region.
Importantly, this can be measured even experimentally~\cite{2017PhRvL.119h0501J}, see also Fig.~\ref{fig.4}.
Despite of the various suggestive analogies to equilibrium quantum critical regions, there remains one central difference.
In equilibrium the temperature can, at least in principle, be chosen at will.
In the dynamical case discussed here this is not the case for the energy density.
Expectation values of local observables rather get their dominant contributions only from a limited set of states with energy densities close to the mean value $\varepsilon_\mathrm{av}(t) = \int d\varepsilon \, \varepsilon \, P(\varepsilon;t)$ due to a central limit theorem~\cite{2014PhRvL.113t5701H}, where $P(\varepsilon;t)$ denotes the energy density distribution function at time $t$.
In this way the energy density, at which we probe our system with local observables, is fixed by the nonequilibrium protocol itself via $\varepsilon_\mathrm{av}(t)$.
Whenever $\varepsilon_\mathrm{av}(t)$ crosses the dynamical analog of the critical region, the green area in Fig.~\ref{fig.4}A, one can expect that the underlying DQPT controls also the dynamics of local observables.
It might, however, also occur that $\varepsilon_\mathrm{av}(t)$ enters a non-universal regime at elevated energy densities, where an underlying DQPT might then not have a significant influence on observables.
Importantly, such dynamical analogs of quantum critical regions can be computed~\cite{2014PhRvL.113t5701H} and even measured~\cite{2017PhRvL.119h0501J}, see also Fig.~\ref{fig.4}B.
However, these quantitative calculations can only be done for fine-tuned models~\cite{2014PhRvL.113t5701H}.
While the picture from Fig.~\ref{fig.4}A is not tied to these particular problems, it remains a significant challenge for the future to develop a more general framework to quantitatively compute such critical regions.
Such a framework might be also particularly important for exploring whether other major properties of quantum critical regions take over to the dynamical case, such as scaling, for example.
For the models, in which critical regions as in Fig.~\ref{fig.4} can be computed, the question of scaling cannot be unambiguously addressed, since the exponents associated with the DQPT are all integer-valued and therefore can only hardly be distinguished from a trivial scaling.
\section{General properties}
The real-time nonanalyticities of DQPTs and the formal similarity of Loschmidt amplitudes to complex partition functions have initially motivated the notion of a dynamical quantum phase transition.
However, it is important to emphasize that equilibrium phase transitions are much more than just nonanalytic behavior.
For example, a continuous equilibrium transition separates two phases characterized by an order parameter, and inherits the powerful concepts of scaling and universality, which allow for a macroscopic description independent of microscopic details.
In the following we summarize the progress of the theory of DQPTs in connecting to such important equilibrium concepts.
\subsection{Dynamical order parameters}
\begin{figure}
\includegraphics[width=\columnwidth]{Fig4.pdf}
\caption{Observation of a dynamical order parameter in an ultra-cold atom experiment~\cite{2018NatPh..14..265F}. {\bf(b)} Subsequent snapshots of a momentum-dependent phase profile across the 2D Brioullin zone for increasing times from (i) to (ix). At some points in time, here between (ii) and (iii), suddenly pairs of vortices appear enclosed by the red circles. At a later time between (vii) and (viii), they recombine and annihilate. {\bf (e)} The number of dynamically generated vortices constitutes a dynamical order parameter for DQPTs happening at those points in time where the vortices a created and annihilated.}
\label{fig.3}
\end{figure}
Order parameters for DQPTs have been reported for noninteracting topological two-band models both in 1D and 2D~\cite{2016PhRvB..93h5416B, 2017PhRvB..96r0303B, 2017PhRvB..96a4302B, 2017PhRvB..96r0304H, 2018NatPh..14..265F}.
These dynamical order parameters are topological quantum numbers detecting topological defects in phase profiles across the Brioullin zone.
Importantly, these have been measured in various different platforms~\cite{2018NatPh..14..265F, 2018arXiv180610871W, 2018arXiv180704483T, 2018arXiv180803930X}, one example is shown in Fig.~\ref{fig.3}.
Recently, a dynamical order parameter in a momentum-time plane of Green's functions has been reported in the context of gauge theories, which can be used for interacting and noninteracting systems on equal footing~\cite{2018arXiv180807885Z}.
All of these mentioned examples constitute order parameters of nonlocal nature associating a global topological quantum number to the time-evolved nonequilibrium wave function.
To which extent also local order parameters can exist is currently unknown.
However, it is clear from general principles that certain scenarios are impossible due to constraints induced by locality and causality.
Since DQPTs occur at a finite time, long-ranged quantum correlations cannot develop at a DQPT due to Lieb-Robinson bounds~\cite{1972CMaPh..28..251L, 2006JSP...124....1N, 2006CMaPh.265..781H}.
And therefore a conventional local order parameter associated to such long-range correlations cannot be formulated.
However, this leaves open the possibility still for order parameters which are non-local or global, as discussed for the topological systems and gauge theories before.
In this context let us note that non-local order parameters can also be identified for some quantum quenches in systems with symmetry-broken phases in equilibrium~\cite{2014PhRvL.113t5701H, 2018PhRvL.120m0601Z, 2017PhRvB..96m4313W, 2018arXiv180807874H} after a non-local projection onto a suitable low-energy manifold~\cite{2014PhRvL.113t5701H}.
Let us also take the chance at this point to repeat the saying that "choosing an order parameter is an art"~\cite{1992cond.mat..4009S}.
Having found a suitable one, implies a high degree of understanding of the studied transition, which, on general level, has not yet been achieved for DQPTs as compared to the equilibrium case.
This leaves open a promising avenue for future developments in this field.
\subsection{Scaling and universality}
Continuous phase transitions in equilibrium are associated with a divergent correlation length, as a consequence of which macroscopic properties become independent of microscopic details~\cite{1987imsm.book.....C,1998AmJPh..66..164C,2011qpt..book.....S}.
Whether such universal behavior can be found also for DQPTs is not known on a general level.
However, for a transverse-field Ising chain it has been established rigorously that the DQPTs appearing in this model are associated with an unstable fixed point of an exact renormalization group transformation (RG)~\cite{2015PhRvL.115n0602H}.
Thus, in the very equilibrium sense, this DQPT results from a divergent correlation length appearing in the Loschmidt amplitude.
Note, that this should not be confused with a divergent length scale in correlation functions, although these also show indicative features of the DQPT in their dynamics for these models~\cite{2015PhRvL.115n0602H,2018ScPP....4...13S}.
Further, for DQPTs in a 2D transverse-field Ising model strong indications of a divergent correlation length have been found.
Specifically, the nonanalytic real-time behavior follows the same scaling as for the equilibrium nonzero-temperature critical point of the 2D classical Ising model~\cite{2015PhRvL.115n0602H}.
While this suggests that scaling and universality also hold for this case, it has up to now not been possible to settle this rigorously, for example, by identifying the corresponding RG fixed point.
\subsection{Landau and effective field theories}
The universal properties at continuous equilibrium phase transitions are determined solely by macroscopic properties such as symmetries or dimensionality.
These are sufficient to construct Landau or effective field theories, which describe the universal properties in the vicinity of the transition.
Although for one particular model system a Landau theory for DQPTs has been derived from microscopics~\cite{2018PhRvB..97q4303T}, for a general understanding of DQPTs it will be of central importance to explore ways that can approach DQPTs from a macroscopic perspective.
As emphasized also before, the theory of DQPTs is facing a major challenge in this context, since it is the central property of the considered nonequilibrium quantum states, that they defy a thermodynamic description and therefore a description in terms a free energy in an equilibrium sense.
On the other hand, this might not only be seen as an obstacle but rather as the defining feature of these quantum states which grant them potentially new properties, but might require a redefined notion of free energies.
\subsection{Robustness}
While DQPTs have been initially mostly studied for exactly solvable models, it has been soon recognized as an important question to which extent they are robust against perturbations~\cite{2013PhRvB..87s5104K, 2014PhRvB..90l5106K, 2015PhRvB..92j4306S}, in particular, those perturbations which break the exact solvability and make the models ergodic.
From the study of individual model systems, the phenomenology of equilibrium transitions has been recovered in the sense that DQPTs appear to be stable against weak symmetry-preserving perturbations~\cite{2013PhRvB..87s5104K, 2014PhRvB..90l5106K, 2015PhRvB..92j4306S} on the accessible transient time scales.
It is, however, not known how such perturbations can influence DQPTs occurring on long time scales, where weak perturbations can change significantly the dynamics by making for instance a nonergodic system thermalizing~\cite{2016AdPhy..65..239D}.
For the previously discussed case of a transverse-field Ising chain, where the DQPT is associated with scaling and universality, such a robustness is the consequence of an unstable RG fixed point~\cite{2015PhRvL.115n0602H}.
Upon including a symmetry-breaking perturbation, however, the character of the fixed point can change and the DQPT can then be transformed to a first-order transition without divergent correlation length~\cite{2018PhRvB..97q4303T}.
For some models it has been reported that symmetry-breaking perturbations can even lead to a smoothing of DQPTs~\cite{2010PhRvE..81b0101P}, although this might not be a general rule~\cite{2013PhRvB..87s5104K,2018PhRvB..97q4303T}.
\section{Prospect}
Summarizing, the field of DQPTs has advanced significantly in recent years.
Nevertheless, it still awaits challenges.
While some have been already pointed throughout the prior presentation, it will be the purpose in the following to discuss further open questions and prospects as well as potential future research directions.
As highlighted before, it is currently unclear to which extent DQPTs can be captured from a macroscopic perspective.
In different words: Is it possible to describe the main properties in the vicinity of a DQPT by concepts analogous to a Landau or effective field theory, which requires as an input only a few macroscopic properties such as symmetries or dimensionality?
This might also be important for classifying DQPTs on a general level, thereby extending previous approaches~\cite{2015PhRvB..91o5127V,2014PhRvL.113z5702C}.
A further intriguing prospect is to study DQPTs in 2D and 3D beyond the 1D cases addressed mostly up to now, where novel critical phenomena might appear.
This expectation is driven mainly from the equilibrium perspective, where compared to 1D it is now possible to have, for example, phase transitions at nonzero temperature or fractional and irrational critical exponents.
How these more complex critical properties express themselves in the context of DQPTs is largely unknown, which is mainly due to a methodological challenge.
Accessing quantum dynamics in such higher dimensional systems by itself is already difficult.
Further, on a technical level Loschmidt amplitudes or echos share the complexity of full partition functions at complex parameters, whose calculation is in addition much more demanding in most cases than determining local observables or correlation functions.
However, for example, projected-entangled pair states (PEPS) promise to provide access to 2D systems both in and out of equilibrium~\cite{2008AdPhy..57..143V}.
In this context it is important to emphasize a central advantage of DQPTs in that they occur at transient and intermediate time scales.
For PEPS for instance this implies that the entanglement production can be still limited to a tractable extent.
Another promising approach for studying quantum dynamics in 2D and 3D is to use quantum many-body state encodings on the basis of classical~\cite{2018ScPP....4...13S,2018arXiv181004178D} or artificial neural networks~\cite{2017Sci...355..602C}.
Here, however, it is not known, in general, how to compute Loschmidt amplitudes except for specific cases~\cite{2018ScPP....4...13S}.
The theory of DQPTs exhibits also important open questions in their physical interpretation.
As discussed already before in these Perspectives, DQPTs reflect a drastic change in the properties of the time-evolution operator without, however, being specific about what that change implies on physical grounds.
In equilibrium, phase transitions are characterized by order parameters marking the two phases separated by the transition.
While dynamical order parameters of topological nature have been formulated for some specific problems recently~\cite{2016PhRvB..93h5416B,2018NatPh..14..265F,2017PhRvB..96a4302B,2017PhRvB..96r0303B,2017PhRvB..96r0304H,2018arXiv180807885Z}, this might be rather seen as a first step towards a more general understanding, since order parameters are still missing for a multitude of observed DQPTs.
Thus, in the future it will be particularly important to develop new approaches for characterizing the two phases separated by a DQPT.
One promising route might be to utilize recent progress in applying machine learning approaches to quantum many-body problems, i.e., quantum phase recognition~\cite{2017PhRvX...7c1038C,2017NatSR...7.8823B}, which are not only limited to equilibrium states.
\acknowledgments
The author is thankful for the stimulating discussions and collaborations on topics related to this review with various colleagues including Debasish Banerjee, Rainer Blatt, Jan Budich, Sebastian Diehl, Nick Fläschner, Philipp Hauke, Yi-Ping Huang, Petar Jurcevic, Ben Lanyon, Achilleas Lazarides, Stefan Kehrein, Michael Knap, Marcus Kollar, Roderich Moessner, Anatoli Polkovnikov, Christian Roos, Markus Schmitt, Alessandro Silva, Daniele Trapin, Dominik Vogel, Matthias
Vojta, Simon Weidinger, Christof Weitenberg, Peter Zoller, and Bojan Zunkovic. Financial support by the Deutsche Forschungsgemeinschaft via the Gottfried Wilhelm Leibniz Prize program is gratefully acknowledged.
\bibliographystyle{eplbib}
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The modification of the second l\u2026Read more\n\u2022 \u00a022\n##### Model Selection, Simplicity, and Scientific Inference with William L. Harper Philosophy of Science\u00a069\u00a0(S3). 2002.\nThe Akaike Information Criterion can be a valuable tool of scientific inference. This statistic, or any other statistical method for that matter, cannot, however, be the whole of scientific methodology. In this paper some of the limitations of Akaikean statistical methods are discussed. It is argued that the full import of empirical evidence is realized only by adopting a richer ideal of empirical success than predictive accuracy, and that the ability of a theory to turn phenomena into accurate,\u2026Read more\n\u2022 \u00a0247\n##### Chasing Chimeras British Journal for the Philosophy of Science\u00a060\u00a0(3):\u00a0635-646. 2009.\nEarman and Ruetsche ([2005]) have cast their gaze upon existing no-go theorems for relativistic modal interpretations, and have found them inconclusive. They suggest that it would be more fruitful to investigate modal interpretations proposed for \"really relativistic theories,\" that is, algebraic relativistic quantum field theories. They investigate the proposal of Clifton ([2000]), and extend Clifton's result that, for a host of states, his proposal yields no definite observables other than mul\u2026Read more\n\u2022 \u00a0321\n\u2022 \u00a0492\n##### Computability in Quantum Mechanics In Werner De Pauli-Schimanovich, Eckehart K\u00f6hler & Friedrich Stadler (eds.), Vienna Circle Institute Yearbook, Kluwer Academic Publishers. pp. 33-46. 1995.\nIn this paper, the issues of computability and constructivity in the mathematics of physics are discussed. The sorts of questions to be addressed are those which might be expressed, roughly, as: Are the mathematical foundations of our current theories unavoidably non-constructive: or, Are the laws of physics computable?\n\u2022 \u00a0105\n##### On the Evidential Import of Unification Philosophy of Science\u00a084\u00a0(1):\u00a092-114. 2017.\nThis paper discusses two senses in which a hypothesis may be said to unify evidence. One is the ability of the hypothesis to increase the mutual information of a set of evidence statements; the other is the ability of the hypothesis to explain commonalities in observed phenomena by positing a common origin for them. On Bayesian updating, it is only mutual information unification that contributes to the incremental support of a hypothesis by the evidence unified. This poses a challenge for the vi\u2026Read more\n\u2022 \u00a0116\n##### Lessons of Bell's Theorem: Nonlocality, yes; Action at a distance, not necessarily In Shan Gao Mary Bell (ed.), Quantum Nonlocality and Reality: 50 Years of Bell's Theorem, Cambridge University Press. pp. 238-260. 2016.\nFifty years after the publication of Bell's theorem, there remains some controversy regarding what the theorem is telling us about quantum mechanics, and what the experimental violations of Bell inequalities are telling us about the world. This chapter represents my best attempt to be clear about what I think the lessons are. In brief: there is some sort of nonlocality inherent in any quantum theory, and, moreover, in any theory that reproduces, even approximately, the quantum probabilities for \u2026Read more\n\u2022 \u00a098\n##### A Bayesian Account of the Virtue of Unification Philosophy of Science\u00a070\u00a0(2):\u00a0399-423. 2003.\nA Bayesian account of the virtue of unification is given. On this account, the ability of a theory to unify disparate phenomena consists in the ability of the theory to render such phenomena informationally relevant to each other. It is shown that such ability contributes to the evidential support of the theory, and hence that preference for theories that unify the phenomena need not, on a Bayesian account, be built into the prior probabilities of theories.","date":"2021-11-30 15:41:05","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\": 0.5517287254333496, \"perplexity\": 1157.1044236099588}, \"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-2021-49\/segments\/1637964359037.96\/warc\/CC-MAIN-20211130141247-20211130171247-00083.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.rdocumentation.org\/packages\/spatstat\/versions\/1.57-1\/topics\/kernel.factor","text":"# kernel.factor\n\n0th\n\nPercentile\n\n##### Scale factor for density kernel\n\nReturns a scale factor for the kernels used in density estimation for numerical data.\n\nKeywords\nmethods, smooth, nonparametric\n##### Usage\nkernel.factor(kernel = \"gaussian\")\n##### Arguments\nkernel\n\nString name of the kernel. Options are \"gaussian\", \"rectangular\", \"triangular\", \"epanechnikov\", \"biweight\", \"cosine\" and \"optcosine\". (Partial matching is used).\n\n##### Details\n\nKernel estimation of a probability density in one dimension is performed by density.default using a kernel function selected from the list above.\n\nThis function computes a scale constant for the kernel. For the Gaussian kernel, this constant is equal to 1. Otherwise, the constant $c$ is such that the kernel with standard deviation $1$ is supported on the interval $[-c,c]$.\n\nFor more information about these kernels, see density.default.\n\nA single number.\n\n##### See Also\n\ndensity.default, dkernel, kernel.moment, kernel.squint\n\n##### Aliases\n\u2022 kernel.factor\n##### Examples\n# NOT RUN {\nkernel.factor(\"rect\")\n# bandwidth for Epanechnikov kernel with half-width h=1\nh <- 1\nbw <- h\/kernel.factor(\"epa\")\n# }\n\nDocumentation reproduced from package spatstat, version 1.57-1, License: GPL (>= 2)\n\n### Community examples\n\nLooks like there are no examples yet.","date":"2020-02-19 07:02:57","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\": 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\": 0.3529038727283478, \"perplexity\": 8387.571652390678}, \"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-2020-10\/segments\/1581875144058.43\/warc\/CC-MAIN-20200219061325-20200219091325-00240.warc.gz\"}"} | null | null |
Nazione costitutiva è un termine utilizzato dalle istituzioni ufficiali per designare quelle nazioni che insieme ad altre formano un'entità o un gruppo più ampio, come uno stato sovrano. L'Organizzazione per la Cooperazione e lo Sviluppo Economico (OCSE) ha impiegato questa formula per indicare le repubbliche socialiste della Jugoslavia, mentre l'Unione Sovietica l'ha usata per indicare le repubbliche sovietiche.
Le istituzioni europee come il Consiglio d'Europa usavano spesso questa dicitura per riferirsi ai Paesi dell'Unione europea. In italiano l'espressione è utilizzata usualmente per riferirsi a quegli stati aventi una propria Costituzione autonoma che tuttavia riconoscono l'autorità di una Casa reale regnante su più nazioni. La parola costitutivo è semplicemente un aggettivo privo di valenza legale, pertanto non ha alcun significato al di fuori del contesto in cui si adopera, cioè per indicare un componente di un gruppo alternativo di stati.
Regno Unito di Gran Bretagna e Irlanda del Nord
Le nazioni costitutive (inglese: constituent country) del Regno Unito di Gran Bretagna e Irlanda del Nord sono:
Inghilterra
Irlanda del Nord
Status distintivi
Spesso, anche ufficialmente, si parla delle nazioni costitutive del Regno Unito come di "nazioni dentro una nazione". Nella dicitura ufficiale del Regno Unito, il termine "Gran Bretagna" si riferisce all'insieme di Inghilterra, Scozia e Galles, dato che quest'isola comprende i territori di queste tre nazioni.
La complessa storia dell'Irlanda del Nord ha portato ad una differente visione del suo status; la moderna Assemblea dell'Irlanda del Nord fu stabilita nel 1998 ed è attualmente attiva dopo un numero di periodi di sospensione. Scozia e Galles adottarono governi decentrati negli anni '90, e sono descritte come "nazioni" nel loro stesso diritto. Anche se l'Inghilterra manca di un governo decentrato per se stessa, ha un proprio sistema legale (diritto inglese) ed è quasi universalmente pensata come paese o nazione.
Tutte e quattro le nazioni hanno sempre avuto e continuano ad avere variazioni caratterizzanti nello status legislativo e amministrativo e Inghilterra e Scozia erano originariamente stati indipendenti.
Cittadinanza
Tutte e quattro sono generalmente considerate possedenti nazionalità distinte (un attributo di società civile), anche se non hanno cittadinanza distinta (un attributo di stato): il passaporto standard del Regno Unito include la dicitura "nationality: British". I cittadini britannici possono considerare sé stessi di nazionalità inglese, irlandese, nordirlandese, scozzese, gallese o semplicemente britannica.
Valuta
Tutte e quattro le nazioni costitutive del Regno Unito condividono la stessa valuta: la sterlina britannica.
Regno dei Paesi Bassi
Le nazioni costitutive (landen) del Regno dei Paesi Bassi sono:
(nel continente americano)
(nel continente americano)
(nel continente europeo)
(nel continente americano)
Status distintivi
Ognuna delle quattro nazioni costitutive ha la sua costituzione: la Costituzione del Regno dei Paesi Bassi (Grondwet van het Koninkrijk der Nederlanden) che si applica in toto nei Paesi Bassi e solo in alcuni aspetti nelle altre nazioni costitutive; la Costituzione di Aruba (Staatsregeling van Aruba); la Costituzione di Curaçao (Staatsregeling van Curaçao); la Costituzione di Sint Maarten (Staatsregeling van Sint Maarten). Ognuna delle quattro parti costituenti ha anche una propria amministrazione e un parlamento. Insieme formano una federazione sotto il potere del monarca come di un singolo capo di stato.
Cittadinanza
Il Regno dei Paesi Bassi è membro dell'Unione europea; tuttavia Aruba, Curaçao e Sint Maarten non ne sono considerate parte, ma hanno lo status di Paesi e territori d'oltremare (PTOM, in lingua olandese LGO's, Landen en Gebiedsdelen Overzee). Dato che la cittadinanza fa capo al regno, e non è distinta per le quattro nazioni costituenti, i cittadini di tutte e quattro sono anche cittadini dell'Unione Europea, ma i residenti di Aruba, Curaçao e Sint Maarten non hanno diritto di voto alle elezioni per il Parlamento Europeo.
Valuta
Le quattro nazioni costitutive non hanno la medesima valuta. I Paesi Bassi hanno l'euro e, nelle proprie municipalità speciali, il dollaro statunitense; Aruba usa il fiorino di Aruba; Curaçao e Sint Maarten il fiorino delle Antille olandesi.
Regno di Danimarca
A giugno 2009, il Regno di Danimarca è ufficialmente composto da tre nazioni costitutive:
(nel continente europeo)
(nel continente europeo)
(nel continente americano)
Cittadinanza
Nel Regno di Danimarca è presente solo la Costituzione della Danimarca (in danese Danmarks Riges Grundlo) che ha validità nelle tre nazioni del regno.
Status distintivi
La Danimarca è membro dell'Unione europea. Tuttavia le Isole Fær Øer non sono considerate parte dell'UE, ma hanno lo status di regioni ultraperiferiche, mentre la Groenlandia è stata membro dell'allora Comunità Economica Europea (oggi Unione europea) fino al 1982. Gli abitanti delle isole Fær Øer e della Groenlandia sono cittadini UE in virtù della loro cittadinanza danese, ma non hanno diritto di voto al Parlamento Europeo.
Valuta
Le tre nazioni costitutive non hanno la medesima valuta. La Danimarca e la Groenlandia utilizzano la Corona danese, le isole Fær Øer usano la Corona delle Isole Fær Øer, che ha lo stesso valore della Corona danese e ha circolazione ufficiale nelle isole.
Note
Voci correlate
Nazione
Stato
Devoluzione (politica)
Diritto internazionale
Politica dei Paesi Bassi
Politica del Regno Unito
Politica della Danimarca | {
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Q: Windows Phone 8.1 Share Contract I wrote a Windows Phone 8.1 (WINRT) App. I am trying to share an image from my app which is in LocalStorage of the app. I am using Windows Phone 8.1 Share Contract.
private async void OnShareDataRequested(DataTransferManager sender, DataRequestedEventArgs _dataRequestedEventArgs)
{
_dataRequestedEventArgs.Request.GetDeferral();
List<StorageFile> ListObject = new List<StorageFile>();
Uri UriObject = new Uri(FileLocation,UriKind.RelativeOrAbsolute);
_dataRequestedEventArgs.Request.Data.Properties.Title = "Dr. App";
_dataRequestedEventArgs.Request.Data.Properties.Description = "Photo from my Dr. App Album.";
StorageFolder StorageFolderObject;
StorageFile StorageFileObject;
try
{
StorageFolderObject = await Windows.Storage.ApplicationData.Current.LocalFolder.GetFolderAsync(LocalCache);
StorageFileObject = await StorageFolderObject.GetFileAsync(FileNameSaved);
_dataRequestedEventArgs.Request.Data.Properties.Thumbnail = RandomAccessStreamReference.CreateFromFile(StorageFileObject);
_dataRequestedEventArgs.Request.Data.SetBitmap(RandomAccessStreamReference.CreateFromFile(StorageFileObject));
ListObject.Add(StorageFileObject);
_dataRequestedEventArgs.Request.Data.SetStorageItems(ListObject);
}
catch(Exception ex_)
{
}
finally
{
_dataRequestedEventArgs.Request.GetDeferral().Complete();
}
}
protected override void OnNavigatedFrom(NavigationEventArgs e)
{
DataTransferManager.GetForCurrentView().DataRequested -= OnShareDataRequested;
base.OnNavigatedFrom(e);
}
protected override void OnNavigatedTo(NavigationEventArgs e)
{
DataTransferManager.GetForCurrentView().DataRequested += OnShareDataRequested;
base.OnNavigatedTo(e);
}
private void Button_Click(object sender, RoutedEventArgs e)
{
DataTransferManager.ShowShareUI();
}
I am getting PREPARING CONTENT TO SHARE and then it vanishes in a second. ShareUI doesnt open.
A: The documentation states that the asynchronous work has an upper limit of 200 ms. Are you violating this premise?
DataRequest.GetDeferral(): Use this method when you want to use an asynchronous function call to generate the DataPackage during a share operation. This function must return a DataPackage object within 200ms to prevent the operation from timing out. If your app shares content that takes more time to package, such as a collection of files or photos, don't use this method. Instead, use the SetDataProvider method to assign a delegate to a DataPackage and return that DataPackage to the target app.
Another thing that stands out to me when I look at your code is that you invoke the GetDeferral method twice instead of saving the result from the first invocation.
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package com.recallq.parseweblog.fieldparser;
import com.recallq.parseweblog.SingleResultFieldParser;
import org.joda.time.format.DateTimeFormatter;
import org.joda.time.format.ISODateTimeFormat;
/**
* Converts a nginx log date to number of seconds since 1970.
*
* @author Jeroen De Swaef
*/
public class ISO8601DateFieldParser extends SingleResultFieldParser {
@Override
public Object parse(String input) {
DateTimeFormatter parser2 = ISODateTimeFormat.dateTimeNoMillis();
return String.valueOf(parser2.parseDateTime(input).toDate().getTime());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,881 |
Meme and You
Consider These
On December 10th the club held its Christmas dinner at Duza's Kitchen. We had a good dinner and an equally good time. My apologies to anyone that came in late who I may have missed. I lose track of the camera after a couple of glasses of wine. Check out the pictures.
Additional photos below taken by and offered by Jeannie Herford. Thanks, Jeannie.
We want to thank Cody O'Donnel, our newest member, who owns and operates Drop Tine Construction for his incredible support of Donald Trump in the Light Parade on Main Street on Saturday, December 7th. His truck and trailer were festooned with Trump support signs. We witnessed the feeble attempt by Democrats to muster support for their cause with their float. The Dems got tepid support at best. When Cody rolled by you could hear the rousing support ripple like a wave down the parade route. Below are some pictures of his effort on behalf of our club and our president.
From our club meeting held 9/30/19:
Tim Horn, pictured below, spoke to club members in attendance about the pitfalls of the push from the left to adopt replacing the Electoral College with the popular vote to elect our presidents. Tim is a passionate defender of conservative values. He is also a published author. You may want to purchase his book, available on Amazon, entitled "Ruling The Elite." You may also recognize Tim from his YouTube presence. Millions of people know him as "Joe American." Check out https://www.youtube.com/watch?v=1XYyT4Rx7Dc
RIM COUNTRY REPUBLICAN CLUB | {
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Q: Basic (Fake) Raycasting on a 2D Heightmap Basically what I'm trying to do is shade a 2D heightmap using a very very basic raycasting system that basically just checks if the ray is intercepted before it should be to shade it. However it's not working correctly and I've been banging my head for several hours now on this so I figured it couldn't hurt to turn it over to you guys, because I think it's probably something either so blindingly obvious that I won't see it or so complex that I'll never wrap my head around it.
I have a map like this:
And the raycasting is giving me this (keep in mind it's just debug colors; red is ray interception, but before intended position (so shading), blue would be ray interception in the correct place (so highlights or just as-is), and yellow means that point had no ray interaction at all before the while loop cut-out).
The result should be with red on backfacing slopes and areas behind large mountains (shadows) and blue on sun-facing slopes (highlights). There should not be any yellow. So this image indicates that either all of the rays are hitting the wrong place, or the rays are being intersected ALWAYS somewhere else before they reach their target, which is impossible.
At this point I highly suspect the problem is with my trig.
Here's the Ray class:
class Ray
{
public Vector2 Position;
public Vector2 Direction; // Think in XZ coordinates for these (they are on a perpendicular plane to the heightmap)
// Angle is angle from horizon (I think), and height is height above zero (arbitrary)
public float Angle, Height;
private TerrainUnit[,] Terrainmap;
private float U, V;
public Ray(ref TerrainUnit[,] Terrainmap, float height, float angle)
{
this.Terrainmap = Terrainmap;
this.Angle = angle;
this.Height = this.V = height;
// Create new straight vector
this.Direction = new Vector2(0, 1);
// Rotate it to the values determined by the angle
this.Direction = Vector2.Transform(Direction, Matrix.CreateRotationX(Angle));
//this.Direction = new Vector2((float)Math.Sin(angle), -(float)Math.Cos(angle));
// Find the horizontal distance of the origin-destination triangle
this.U = V / (float)Math.Tan(Angle);
// Bleh just initialize the vector to something
this.Position = new Vector2(U, V);
}
public void CastTo(int x, int y)
{
// Get the height of the target terrain unit
float H = (float)Terrainmap[x, y].Height;
// Find where the ray would have to be to intersect that terrain unit based on its angle and height
Position = new Vector2(x - U, H + V);
float Z = 1000 * (float)Terrainmap[0, y].Height;
// As long as the ray is not below the terrain and not past the destination point
while (Position.Y > Z && Position.X <= x)
{
// If the ray has passed into terrain bounds update Z every step
if (Position.X > 0) Z = 1000 * (float)Terrainmap[(int)Position.X, y].Height;
Position.X += Direction.X;
Position.Y += Direction.Y;
}
Terrainmap[x, y].TypeColor = Color.Yellow;
if ((int)Position.X == x) Terrainmap[x, y].TypeColor = Color.Blue;
else Terrainmap[x, y].TypeColor = Color.Red;
}
}
Also just as a formality, the function that is casting each ray and how I am calling that:
if (lighting) CastSunRays(1f, MathHelper.PiOver4);
private void CastSunRays(float height, float angle)
{
Ray ray = new Ray(ref Terrainmap, height, angle);
for (int x = 0; x < Width; x++)
for (int y = 0; y < Height; y++)
ray.CastTo(x, y);
}
A: I ended up using a much simpler approach with Bresenham's Line Algorithm to find the intercept point; I imagine it's much faster and more efficient than the way I was trying to do it would have been.
A: My guess is that when your Direction vector is applied to Position, it oversteps the lower limit (Position.Y > -1) before it has a chance to hit the surface (Position.Y <= Terrainmap[(int)Position.X, y].Height).
You could try to decrease the lower limit, or re-order your if/while tests.
Another problem might be that the Direction Vector is too large in comparison to your height-range. The distance between two neighboring pixels is 1, while the whole range of height differences is contained in the range (-1,1). This gives a very flat surface from the ray-casters point of view. When the Direction vector is applied to the Position vector is takes a relatively small step over the length, and a relatively large step over the height.
A: @Maltor: I actually wanted to comment your own answer, but due to my reputation am not currently able to.
I also used the bresenham's line approach and decreased calculation time to 1/10!
A running example of that can be viewed at my github project TextureGenerator-Online.
The terrain tool uses this approach.
See function setTerrainShadow() at tex_terrain.js
| {
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\section*{Introduction}
\addcontentsline{toc}{section}{Introduction}
Let $X_0$ be a connected variety defined over a finite field $k=\bF_q$, equipped with a point $x \in X_0(k)$, and let $l$ be a prime not dividing $q$. If we set $X:=X_0\ten_k\bar{k}$, the embedding $i: \pi_1(X,\bar{x}) \into W(X_0,x)$ of the fundamental group into the Weil group gives a conjugation action of the Weil group on the fundamental group. The Weil conjecture for $\H^1(X,\Q_l)$ can be re-expressed by considering $\H^1(X,\Q_l)^{\vee}$ as the $l$-adic Weil representation $V$, universal among continuous Weil-equivariant group homomorphisms
$$
\pi_1(X,x) \to V,
$$
and then stating that this Weil representation is mixed.
We consider a non-abelian version of this, by defining the pro-$\Q_l$-algebraic group ${}^W\!\algpia$ to be the
universal object classifying continuous $W(X_0,x)$-equivariant homomorphisms
$$
\pi_1(X,\bar{x}) \to G(\Q_l),
$$
where $G$ ranges over all algebraic groups $G$ over $\Q_l$ equipped with continuous $W(X_0,x)$-actions. Representations of $\w\algpia$ are precisely $\pi_1(X,\bar{x})$-subrepresentations of $W(X_0,x)$-representations.
We say that an algebraic Weil action on a pro-algebraic group $G$ is mixed if the structure sheaf $O(G)$ is a sum of mixed Weil representations.
The Levi decomposition for pro-algebraic groups allows us to write
$$
{}^W\!\algpia \cong \wmypia\rtimes \w\redpia,
$$
where $\wmypia$ is the pro-unipotent radical of ${}^W\!\algpia$ and ${}^W\!\redpia$ is the pro-reductive completion of $\w\algpia$. This decomposition is unique up to conjugation by $\wmypia$.
In Section \ref{del}, we use Lafforgue's Theorem to show that for any variety, the Weil action on $\w\redpia$ is pure of weight zero. Deligne's Weil II theorems then show that, if $X$ is smooth or proper, the Weil action on ${}^W\!\algpia$ is mixed. This can be thought of as a direct analogue of the non-abelian Hodge theorems of \cite{Simpson}. One consequence is that for any morphism $f:X \to Y$ of varieties over $\bF_q$, with $X$ smooth and $\bV$ any semisimple constructible $\Q_l$-local system underlying a Weil sheaf on $Y$, the pullback $f^{-1}\bV$ is semisimple.
The rest of the paper is dedicated to studying the Weil action on $\wmypia$ when $X$ is smooth or proper, and thus establishing restrictions on the structure of the fundamental group.
In order to study the pro-unipotent extension $\w\algpia \to \w\redpia$, we use deformation-theoretic machinery. The group $\wmypia$ is the universal deformation
$$
\rho: \pi_1(X,\bar{x}) \to U \rtimes {}^W\!\redpia
$$
of the canonical representation
$$
\rho_0:\pi_1(X,\bar{x}) \to {}^W\!\redpia,
$$
for $U$ pro-unipotent. In \cite{higgs}, a theory of deformations over nilpotent Lie algebras with $G$-actions was developed, and this enables us to analyse our scenario.
In Section \ref{final}, we use Deligne's Weil II theorems to study $\wmypia$.
If $X$ is smooth and proper, then the weight decomposition on $\wmypia$ splits the lower central series filtration, and it is quadratically presented, in the sense that its Lie algebra can be defined by equations of bracket length two. If $X$ is merely smooth, then $\wmypia$ is defined by equations of bracket length at most four. Since rigid representations of the fundamental group extend to Weil representations, these properties are used to give new examples of groups which cannot occur as fundamental groups of smooth varieties in finite characteristic.
This generalises the results of \cite{paper1} on deforming reductive representations of the fundamental group. In this paper, we are taking a reductive representation
$$
\rho_0:\pi_1(X,\bar{x}) \to G,
$$
and considering deformations
$$
\rho: \pi_1(X,\bar{x}) \to U \rtimes G
$$
of $\rho_0$, for $U$ unipotent. Effectively, \cite{paper1} considers only $U=\exp(\mathrm{Lie}(G) \ten \m_A)$, for $\m_A$ a maximal ideal of an Artinian local $\Q_l$-algebra. Since taking $U=\mypia$ pro-represents this functor when $G=\redpia$, all examples can be understood in terms of the structure of $\mypia$.
The structure result in the smooth and proper case is much the same as those established in \cite{malcev} and \cite{higgs} for fundamental groups of compact K\"ahler manifolds. Likewise, \cite{paper1} was the analogue in finite characteristic of Goldman and Millson's results on K\"ahler representations (\cite{gm}).
\section{The pro-algebraic fundamental group as a Weil representation}\label{del}
\subsection{Algebraic actions}
All pro-algebraic groups in this paper will be defined over fields of characteristic zero (usually $\Q_l$). All representations of pro-algebraic groups will be finite-dimensional.
\begin{definition}
Given a pro-algebraic group $G$, let $O(G)$ denote global sections of the structure sheaf of $G$. This is a sum of $G\by G$-representations, the actions corresponding to right and left translation. Let $E(G)$ be the dual of $O(G)$ --- this is a pro-$G\by G$-representation. In fact, since any coalgebra is the sum of its finite-dimensional subcoalgebras, $E(G)$ is an inverse limit of finite-dimensional (non-commutative) algebras.
$E(G)$-modules then correspond to pro-$G$-representations, and for a morphism $G \to H$ and a pro-$G$-representation $V$, we define
$$
\Ind_{G}^{H}V:=V\hat{\ten}_{E(G)}E(H).
$$
\end{definition}
\begin{definition}
Given a discrete group $\Gamma$ acting on a pro-algebraic group $G$, we define ${}^{\Gamma}\!G$ to be the maximal quotient of $G$ on which $\Gamma$ acts algebraically. This is the inverse limit $\Lim_{\alpha} G_{\alpha}$ over those surjective maps
$$
G \to G_{\alpha},
$$
with $G_{\alpha}$ algebraic, for which the $\Gamma$-action descends to $G_{\alpha}$.
\end{definition}
\begin{lemma}\label{repchar}
The representations of ${}^{\Gamma}\!G$ are precisely those $G$-representations which arise as $G$-subrepresentations of (finite-dimensional) $G\ltimes \Gamma$-representations.
\end{lemma}
\begin{proof}
Given ${}^{\Gamma}\!G \xra{\theta} \GL(V)$, there must exist an algebraic quotient group $G_{\alpha}$ of $G$ to which $\Gamma$ descends, with $\theta$ factoring as ${}^{\Gamma}\!G \to G_{\alpha} \to \GL(V)$. Now, since $G_{\alpha}$ is an algebraic group, $\Aut(G_{\alpha})$ is also, and there is a homomorphism $G\ltimes \Gamma \to H_{\alpha} :=G_{\alpha} \ltimes \Aut(G_{\alpha})$. Since $G_{\alpha}\into H_{\alpha}$, the $G_{\alpha}$-representation $V$ is a subrepresentation of the pro-$H_{\alpha}$-representation $\Ind_{G_{\alpha}}^{H_{\alpha}}V $, so for some quotient representation $\Ind_{G_{\alpha}}^{H_{\alpha}}V \to W$, the composition $V \to W$ must be injective. Thus $V$ is a subrepresentation of the $G\ltimes \Gamma$-representation $W$.
Conversely, Let $V\le W$ be $G$-representations, with $W$ a $G\ltimes \Gamma$-representation. If we let $G_{\alpha}$ be the image of $G \to \GL(W)$, then the adjoint action of $\Gamma$ on $\GL(W)$ restricts to an action on $G_{\alpha}$. Since the action of $G$ on $W$ preserves $V$, there is an algebraic map $G_{\alpha} \to \GL(V)$, as required.
\end{proof}
\begin{definition}
Given a pro-algebraic group $G$, we will denote its reductive quotient by $G^{\red}$; this is the universal object among quotients $G \to H$, with $H$ reductive algebraic. Representations of $G^{\red}$ correspond to semisimple representations of $G$. We write $\Ru(G)$ for the kernel of $G \to G^{\red}$ --- this is called the pro-unipotent radical of $G$.
\end{definition}
\begin{lemma}
${}^{\Gamma}(G^{\red})=({}^{\Gamma}G)^{\red}$. We will hence denote this group by ${}^{\Gamma}G^{\red}$.
\end{lemma}
\begin{proof}
Note that in both cases, representations correspond to those semisimple $G$-representations which arise as $G$-subrepresentations of (finite-dimensional) $G\ltimes \Gamma$-representations.
\end{proof}
The Levi decomposition, proved in \cite{Levi}, states that for every pro-algebraic group $G$, the surjection $G \to G^{\red}$ has a section, unique up to conjugation by $\Ru(G)$, inducing an isomorphism $G\cong \Ru(G)\rtimes G^{\red}$.
\begin{lemma}\label{innerworks}
Given a pro-algebraic group $G$, an automorphism $F$ of $G$, and an element $g \in G$, the action of $F$ on $G$ is algebraic if and only if the action of $\ad_g \circ F$ is algebraic.
\end{lemma}
\begin{proof}
First note that we have an isomorphism from $G\rtimes\langle \ad_g \circ F \rangle$ to $G\rtimes\langle F \rangle$ fixing $G$, given by sending $\ad_g \circ F$ to $g\cdot F$. Hence, by Lemma \ref{repchar}, ${}^{F}\!G={}^{\ad_g \circ F}\!G$.
\end{proof}
\begin{corollary}\label{algboth}
The action of $F$ on $G$ is algebraic if and only if the corresponding actions on $G^{\red}$ and $\Ru(G)$ are.
\end{corollary}
\begin{proof}
Without loss of generality, by the previous lemma, we may assume that $F$ must preserve the Levi decomposition (following conjugation by a suitable element of $\Ru(G)$). Write $F=F^{\red}F^u$, for $F^u:\Ru(G) \to \Ru(G)$, and $F^{\red}:G^{\red} \to G^{\red}$. By Lemma \ref{repchar} and Tannakian duality, ${}^{F}\!G$ is the image of $G \to (G\rtimes \langle F \rangle)^{\alg}$, the latter group being the pro-algebraic completion of $G\rtimes \langle F \rangle$.
Then note that we have an embedding
$$
(G\rtimes\langle F \rangle)^{\alg} \into (\Ru(G)\rtimes\langle F^u \rangle)^{\alg} \rtimes (G^{\red}\rtimes\langle F^{\red} \rangle)^{\alg},
$$
so the map from $G$ to the group on the left is an embedding if and only if the maps from $G^{\red},\Ru(G)$ to the groups on the right are embeddings.
\end{proof}
\begin{lemma}\label{algchar} Let $F$ act on $G\ltimes U$, for $G$ reductive and $U$ pro-unipotent, with $F$ preserving and acting algebraically on $G$. If we also assume that $\Hom_G(V,U/[U,U])$ is finite-dimensional for all $G$-representations $V$, then $F$ acts algebraically on $G\ltimes U$.
\end{lemma}
\begin{proof}
By the previous lemma, it suffices to show that $F$ acts algebraically on $U$. Let $S$ be the set of isomorphism classes of irreducible representations of $G$. Since $F$ acts algebraically on $G$, the $F$-orbits in $S$ are all finite. Let $\fu:=\Lie(U)$, and take the canonical decomposition $\fu=\prod_{s\in S}\fu_s$ of $\fu$ as a $G$-representation. Let $T=S/F$ be the set of $F$-orbits in $S$, giving a weaker decomposition $\fu=\prod_{t\in T}\fu_t$, where $\fu_t =\prod_{s \in T}\fu_s $.
$F$ is then an automorphism of $\fu$ respecting this decomposition; let $H$ be the group of all such automorphisms. We then have an embedding
$$
U\rtimes\langle F \rangle \into U \rtimes H,
$$
so it suffices to show that the group $H$ is pro-algebraic, since this embedding must then factor through $(U\rtimes\langle F \rangle)^{\alg}$.
Choose a $G$-equivariant section to the map $\fu \to \fu/[\fu,\fu]$, and let its image be $V$. The group $H$ is a closed subspace of the space of all linear maps $\Hom_T(V,\fu)$ preserving the $T$-decomposition. The hypothesis implies that $V_s$ is finite-dimensional for all $s \in S$, so $V_t$ must be finite-dimensional for all $t \in T$, the $F$-orbits being finite. Thus $H$ is an affine group scheme, i.e. a pro-algebraic group, as required.
\end{proof}
\begin{lemma}\label{dual} If $G$ is a pro-algebraic group, and we regard $O(G)$ as a sum of $G$-representations via the left action, then for any $G$-representation $V$, $V^{\vee}\cong \Hom_G(V, O(G))$, with the $G$-action on $V^{\vee}$ coming from the right action on $O(G)$.
\end{lemma}
\begin{proof}
This follows immediately from \cite{tannaka} II Proposition 2.2, which states that $G$-representations correspond to $O(G)$-comodules. Under this correspondence, $\alpha \in V^{\vee}$ is associated to the morphism which sends $v \in V$ to the function $g \mapsto \alpha(g\cdot v)$.
\end{proof}
\begin{lemma}\label{fdual}
If an endomorphism $F$ acts on a pro-algebraic group $G$ and compatibly on a $G$-representation $V$ (i.e. $F(g\cdot v)=(Fg)\cdot (Fv)$), then the dual action of $F$ on $V^{\vee}$ corresponds to the action on $\Hom_G(V, O(G))$ which sends $\theta$ to the composition
$$
V \xra{F} V \xra{\theta} O(G) \xra{F^*} O(G)
$$
\end{lemma}
\subsection{Weil actions}\label{weilact}
Let $k=\bF_q$, take a connected variety $X_0/k$, and let $X=X_0\ten_k\bar{k}$. Fix a closed point $x$ of $X$, and denote the associated geometric point $x\ten_{k(x)}\bar{k} \to X$ by $\bar{x}$. Without loss of generality (increasing $q$ if necessary), we assume that $k(x) \subset \bF_q$. Let $l$ be a prime not dividing $q$, and consider the pro-$\Q_l$-algebraic completion $\algpia$ of the \'etale fundamental group $\pi_1(X,\bar{x})$ of $X$. This is the universal object classifying continuous homomorphisms
$$
\pi_1(X,\bar{x}) \to G(\Q_l),
$$
where $G$ ranges over all algebraic groups $G$ over $\Q_l$.
Recall that the Frobenius element gives a canonical generator of $\pi_1(\Spec k)\cong \hat{\Z}$, and that the Weil group $W(X_0,x)$ is defined by
$$
W(X_0,x) = \pi_1(X_0,\bar{x})\by_{\hat{\Z}}\Z,
$$
which has $\pi_1(X,\bar{x})$ as a normal subgroup.
Observe that the conjugation action of $W(X_0,x)$ on $\pi_1(X,\bar{x})$ then extends by universality to an action of $W(X_0,x)$ on $\algpia$. Let $F_x \in W(X_0,x)$ be the Frobenius element associated to $x$.
\begin{lemma}\label{wf}
If $W:=W(X_0,x)$ and $F:=F_x$, then $\w\algpia={}^{F}\!\algpia$, with representations of this group being those continuous $\pi_1(X,\bar{x})$-representations which arise as $\pi_1(X,\bar{x})$-subrepresentations of Weil representations.
\end{lemma}
\begin{proof}
By Lemma \ref{repchar}, representations of $\w\algpia$ are continuous $\pi_1(X,\bar{x})$-subrepresentations of $\pi_1(X,\bar{x})\ltimes W(X_0,x)$-representations. These are precisely $\pi_1(X,\bar{x})$-subrepresentations of $W(X_0,x)$-representations. Since $W(X_0,x)=\pi_1(X,\bar{x})\ltimes \langle F_x\rangle$, these are the same as representations of ${}^{F}\!\algpia$. By Tannakian duality (\cite{tannaka}), this determines the quotient groups $\w\algpia,{}^{F}\!\algpia$ of $\algpia$, which must then be equal.
\end{proof}
\begin{lemma}
$\w\algpia$ is the image of the homomorphism $\algpia\xra{i} \mathpzc{W}(X_0,x)$, where $\mathpzc{W}(X_0,x)$ is the pro-algebraic completion of the Weil group $W(X_0,x)$.
\end{lemma}
\begin{proof}
Representations of $\im(i)$ are those $\algpia$ representations $V$ for which $V \to \Ind_{\algpia}^{\mathpzc{W}(X_0,x)}$ is injective. By Lemmas \ref{repchar} and \ref{wf}, these are the same as $\w\algpia$-representations.
\end{proof}
\begin{definition}
Given a pro-$\Q_l$-algebraic group $G$, equipped with an algebraic action of the Weil group $W(X_0,x)$, we will say that this Weil action on $G$ is mixed (resp. pure of weight $w$) if $O(G)$ is a sum of finite-dimensional Weil representations which are mixed (resp. pure of weight $-w$). Note that if $O(G)$ is pure, then it is pure of weight $0$, since the unit map $\Q_l \to O(G)$ must be Weil equivariant, so we always have a subspace of weight $0$.
\end{definition}
\begin{theorem}\label{weilred}
The natural Weil action on $\w\redpia$ is pure (of weight $0$).
\end{theorem}
\begin{proof}
Since $\w\redpia$ is reductive, its category of representations is generated under addition by the irreducible representations. Tannakian duality (\cite{tannaka}) states that $ O(\w\redpia)$ must then be dual to the pro-vector space of endomorphisms of the fibre functor from the category of representations to the category of vector spaces. Similarly, $ O(\w\redpia)\ten_{\Q_l}\bar{\Q}_l$ classifies $\bar{\Q}_l$-representations, and is dual to the fibre functor from representations over $\bar{\Q}_l$. By Schur's Lemma, scalar multiplications are the only endomorphisms of irreducible representations over $\bar{\Q}_l$.
If we write $\End(V)$ for the space of endomorphisms of the vector space underlying $V$,
there is then an isomorphism of $\w\redpia\by \w\redpia$-representations
$$
O(\w\redpia)\ten_{\Q_l}\bar{\Q}_l\cong \bigoplus_{V \in T} \End(V),
$$
where $T$ is the set of all isomorphism classes of irreducible representations $V$ of $\w\redpia$ over $\bar{\Q}_l$. By Lemma\ref{wf}, it follows that $V$ is an irreducible representation of $\pi_1(X,\bar{x})$ which is a subrepresentation of some $W(X_0,x)$-representation. This is the same as underlying a $W(X_{\bF_q^n},x)$-representation for some $n$, since $W(X_{\bF_q^n})= \pi_1(X,\bar{x})\ltimes \langle F_x^n\rangle$.
From Lafforgue's Theorem (\cite{Weil2} Conjecture 1.2.10, proved in \cite{La} Theorem VII.6 and Corollary VII.8), every irreducible Weil representation over $\bar{\Q_l}$ is of the form
$$
V \cong P \ten \bar{\Q_l}^{(b)},
$$
for some pure representation $P$ of weight zero. Now,
$$
\End(V) \cong V^{\vee} \ten V \cong P^{\vee} \ten P,
$$
which is a pure $W(X_{\bF_q^n},x)$-representation of weight $0$. Therefore
$$
\sum_{i}\End((F_x^{\sharp})^iV) =\sum_{i=0}^{n-1}\End((F_x^{\sharp})^iV) \le O(\w\redpia)\ten \bar{\Q_l})
$$
is a pure Weil subrepresentation of weight $0$. Hence $O(\w\redpia)\ten_{\Q_l}\bar{\Q}_l$ and $O(\w\redpia)$ are also pure of weight $0$, as required.
\end{proof}
\begin{lemma}\label{weiluniv}
If $X$ is a smooth or proper variety, then $\w\algpia$ is the universal group $G$ fitting in to the diagram
$$
\algpia \to G \to \w\redpia,
$$
with $\ker(G \to \w\redpia)$ pro-unipotent.
\end{lemma}
\begin{proof}
Since $G$ and $\w\algpia$ are both quotients of $\algpia$, with $G \to \w\algpia$, it suffices to show that the composition $G \to \mathpzc{W}(X_0,x)$ is an embedding, or equivalently that the Frobenius action on $G$ is algebraic. By Lemma \ref{algchar}, it then suffices to show that $\Hom_{\w\redpia}(V,U/[U,U])$ is finite-dimensional for all $\w\redpia$-representations $V$, where $U$ is the pro-unipotent radical of $G$.
By studying derivations, we will see in Lemmas \ref{factorset} and \ref{radcoho}, and Proposition \ref{gpxcoho}, that $\Hom_{\w\redpia}(U/[U,U],V)$ is just
$$
\H^1(\algpia, V)\cong\H^1(\pi_1(X,\bar{x}), V)\cong\H^1(X, \vv),
$$
which is finite-dimensional.
\end{proof}
\begin{proposition}\label{weillevi}
The Weil action on $G$ is mixed if and only if the the induced actions on $G^{\red}$ and on the continuous dual vector space $(\Ru(G)/[\Ru(G),\Ru(G)])^{\vee}$ are mixed.
\end{proposition}
\begin{proof}
We first choose a Levi decomposition $G=G^{\red}\ltimes\Ru(G)$. The Weil action will not usually preserve this decomposition. However, for each $y \in X$, we may choose an element $u_y \in \Ru(G)$ such that $F_y':=\ad_{u_y} \circ F_y$ does preserve this Levi decomposition. The key point is that $u_y$ acts unipotently on $O(G)$.
Now, for any Weil representation $V$, the weight $a$ subrepresentation $\cW_a(V)$ of $V$ is defined as the intersection of the weight $n(y)a$ $F_y$-subrepresentations $\cW_{n(y)a}(V,F_y)$ of $V$, for all $y \in X$ and $|k(y)|=q^{n(y)}$. Since $\ad_{u_y} $ acts unipotently on $O(G)$, we deduce that
$$
\cW_{n(y)a}(V,F_y)=\cW_{n(y)a}(V,F_y'),
$$
for all $y \in X$.
If we write $\fu$ for the (pro-nilpotent) Lie algebra of $\Ru(G)$, and let $\fu^{\vee}$ denote its continuous dual, then the isomorphism $\Ru(G)\cong \exp(\fu)$ and the Levi decomposition give us an isomorphism
$$
O(G)\cong O(G^{\red})[\fu^{\vee}]= \bigoplus_n O(G^{\red})\ten \Symm^n(\fu^{\vee}),
$$
which is $F_y'$ equivariant for all $y \in X$.
To say that a Weil representation is mixed is the same as saying that
$$
V=\bigoplus_{a\in \Z} (\bigcap_{y \in X} \cW_{n(y)a}(V,F_y)),
$$
and we have seen that for $V=O(G)$ it is equivalent to replace $F_y$ by $F_y'$. Since $O(G^{\red})$ is mixed, and this property is respected by sums and tensor operations, it suffices to show that $\fu^{\vee}$ is mixed for the $F_y'$. This is the same as being mixed for the natural action of the $F_y$ on $\fu^{\vee}$, so it suffices to show that the latter is a mixed Weil representation.
Consider the lower central series filtration $\Gamma_n \fu$ of $\fu$ given by
$$
\Gamma_1 \fu:=\fu,\quad \Gamma_{n+1} \fu=[\fu,\Gamma_n \fu],
$$
so that $\fu=\Lim \fu/\Gamma_n \fu$. If $\fu_n^{\vee}:= (\fu/\Gamma_{n+1} \fu)^{\vee}$, then $\fu^{\vee}=\sum \fu_n^{\vee}$, and it only remains to show that the latter are mixed. Now there is a canonical map
$$
\fu_n^{\vee}/\fu_{n-1}^{\vee} \into \mathrm{CoLie}_n(\fu^{\vee}_1),
$$
where $\mathrm{CoLie}_n$ is the degree $n$ homogeneous part of the free co-Lie algebra functor. Since this is a tensor operation, the right-hand side is mixed ($\fu^{\vee}_1$ being mixed by hypothesis).
We next observe that if
$$
0 \to V' \to V \to V''\to 0
$$
is a short exact sequence of ind-Weil representations with any two mixed, then the third is; this completes the proof.
\end{proof}
\begin{theorem}\label{mixed} If $X$ is smooth or proper, then the natural Weil action on $\w\algpia$ is mixed of non-positive weight.
\end{theorem}
\begin{proof}
By Theorem \ref{weilred} and Proposition \ref{weillevi}, it suffices to show that the Weil action on $(\Ru(\w\algpia)/[\Ru(\w\algpia),\Ru(\w\algpia)])^{\vee}$ is mixed of non-negative weight. By Lemma \ref{weiluniv} and Lemma \ref{dual}, we may alternatively describe this as
$$
\H^1(X, \bO(\w\redpia)),
$$
where $\bO(\w\redpia)$ is the sheaf on $X$ corresponding to the vector space $O(\w\redpia)$ equipped with its left $\redpia$-action. The $\pi_1(X,\bar{x})$-action on $(\Ru(\w\algpia)/[\Ru(\w\algpia),\Ru(\w\algpia)])^{\vee}$ then comes from the right action on $O(\w\redpia)$, and by Lemma \ref{fdual} the Frobenius action comes from the natural Frobenius action on $O(\w\redpia)$.
Now, as in Theorem \ref{weilred}, we may write
$$
O(\w\redpia)\ten_{\Q_l}\bar{\Q}_l\cong \bigoplus_{V \in T} \End(V),
$$
where $T$ is the set of all isomorphism classes of irreducible representations of $\w\redpia$. This is a sum of Weil representations, and each $V$ extends to a representation of $W(X_{\bF_q^n},x)$ for some $n$, automatically compatible with the Frobenius action on $O(\w\redpia)$ (which then corresponds to the adjoint action). Since a Weil representation is pure of weight $w$ if and only if the restricted $W(X_{\bF_q^n},x)$-representation is so, it suffices to show that the $W(X_{\bF_q^n},x)$-representation
$$
\H^1(X, \vv^{\vee})\ten V
$$
is mixed for each irreducible $\pi_1(X,\bar{x})$-representation with $(F^n)^*V \cong V$.
The group $W(X_{\bF_q^n},x)$ acts on $\H^1(X, \vv^{\vee})$ by composing the canonical map $W(X_{\bF_q^n},x) \to \Z$ with the Frobenius action arising from the Weil structure of $V$. By Lafforgue's Theorem, we may assume that $V$ is pure of weight zero (by Schur's Lemma, note that different choices of Frobenius action on $V$ all give the same adjoint action on $\End(V)$). From Deligne's Weil II theorems (\cite{Weil2} Corollaries 3.3.4 -- 3.3.6), it then follows that $\H^1(X, \vv^{\vee})$ is mixed of non-negative weight, so $\H^1(X, \vv^{\vee})\ten V$ must also be mixed of non-negative weight, $V$ being pure of weight $0$.
\end{proof}
\begin{corollary}
If $X$ is smooth, then the quotient map $\w\algpia \to \w\redpia$ has a unique Weil-equivariant section.
\end{corollary}
\begin{proof}
In this case, the weights of $\H^1(X, \vv^{\vee})\ten V$ are strictly positive ($1$ or $2$), so
$O(\w\algpia)/O(\w\redpia)$ is of strictly positive weights, giving us a decomposition
$$
O(\w\algpia)= \cW_0O(\w\algpia) \oplus \cW_{+}O(\w\algpia).
$$
Projection onto $\cW_0O(\w\algpia)=O(\w\redpia)$ yields the section.
\end{proof}
\begin{corollary}\label{pullback}
If $f:X \to Y$ is a morphism of connected varieties over $\bar{\bF}_p$, with $X$ smooth, and $\bV$ a semisimple constructible $\Q_l$-local system underlying a Weil sheaf on $Y$, then $f^{-1}\bV$ is semisimple.
\end{corollary}
\begin{proof}
If $\bV$ is of rank $n$, then it corresponds to a homomorphism $\w\varpi(Y,\bar{y})^{\red} \to \GL(n,\Q_l)$, or equivalently
$$
O(\GL_n) \to O(\w\varpi(Y,\bar{y})^{\red})\le \cW_0O(\w\varpi(Y,\bar{y})),
$$
so $f^{-1}\bV$ must correspond to
$$
O(\GL_n) \to \cW_0O(\w\algpia)=O(\w\redpia),
$$
as $f$ commutes with Frobenius. Therefore $f^{-1}\bV$ is semisimple.
\end{proof}
\section{Structure of the fundamental group}\label{final}
\subsection{Comparison of cohomology groups}
Fix a pro-finite group $\Gamma$, a reductive pro-algebraic group $R$ over $\Q_l$, and a Zariski-dense continuous representation $\rho\co \Gamma \to R(\Q_l)$.
We adapt the following definition from \cite{malcev} to pro-finite groups:
\begin{definition}
Define the Malcev completion $(\Gamma)^{\rho,\mal}$ of $\Gamma$ relative to $\rho$ to be the universal diagram
$$
\Gamma \to (\Gamma,\rho)^{\mal} \xra{p} R,
$$
with $p$ a pro-unipotent extension, and the composition equal to $\rho$.
\end{definition}
\begin{remark}
Observe that if $\Gamma= \pi_1(X,\bar{x})$ and $R=\w\redpia$, with $\rho$ the canonical map, then Lemma \ref{weiluniv} shows that
$$
(\Gamma,\rho)^{\mal}=\w\algpia.
$$
\end{remark}
\begin{lemma}\label{factorset}
For any finite-dimensional $R$-representation $V$, the canonical maps
$$
\H^i(\Gamma^{\rho,\mal},V) \to \H^i(\Gamma,V),
$$
are bijective for $i=0,1$ and injective for $i=2$.
\end{lemma}
\begin{proof}
In both cases $\H^0(V)=V^R$ and $\H^1(V)$ is the set of continuous derivations from $\Gamma$ to $V$, which coincides with the definition of the tangent space. We now adapt the argument of \cite{haintorelli} \S 5.
Writing $G:=\Gamma^{\rho,\mal}$, we know that $\H^2(G,V)$ is the set of isomorphism classes of extensions
$$
0 \to V \to E \to G\to 1,
$$
which pulls back to give the extension
$$
0 \to V \to E(\Q_l)\by_G\Gamma \to \Gamma\to 1
$$
of topological groups. It follows from \cite{W} 6.11.15 that $\H^2(\Gamma,V)$ classifies such extensions.
If this extension is trivial, then we have a section $\Gamma \to E(\Q_l)$, and hence a section $G \to E$, establishing injectivity.
\end{proof}
\begin{remarks}
\begin{enumerate}
\item In the terminology of \cite{higgs}, note that the vector space-valued functors on $\Rep(R)$ given by
$$
V \mapsto \H^i(G, V)
$$,
for a pro-unipotent extension $G \to R$,
are the tangent space and universal obstruction space of the functor
$$
U \mapsto \Hom(G, R\ltimes U)_R/U
$$
on $\cN(R)$.
For the latter, observe that if we have a small extension $U'\to U$ with kernel $V$, and a map $f:G \to R\ltimes \U$ over $R$, then $G \by_{ R\ltimes U}R\ltimes U'$ is an extension of $G$ by $V$, which splits if and only if $f$ lifts to $ R\ltimes U'$.
\item
Note that the cohomological comparison maps above can be defined in terms of derived functors, by comparing the categories of $G$-representations, $\Gamma$-representations over $\Q_l$ and $\Gamma$-representations over $\Z_l$. In particular, this means that they respect cup products.
\end{enumerate}
\end{remarks}
\begin{remark} For a pro-unipotent group $U$ equipped with an $R$-action, note that
in the terminology of \cite{higgs}, the vector space-valued functors on $\Rep(R)$ given by
$$
V \mapsto \H^i(U, V)^R
$$
are the tangent space and universal obstruction space of $\fu$ for $i=1,2$ respectively.
For the final observation, note that if we have a small extension $\fh'\to \fh$ with kernel $V$, and a map $f:\Gamma \to R\ltimes \fh$, then $\Gamma \by_{ R\ltimes \fh}R\ltimes \fh'$ is an extension of $\Gamma$ by $V$, which splits if and only if $f$ lifts to $\fh'$.
\end{remark}
\begin{lemma}\label{radcoho}
If $G=R\ltimes U$, for $U$ pro-unipotent, then for any $R$-representation $V$,
$$
\H^i(G,V)\cong(\H^i(U,\Q_l)\ten V)^R.
$$
\end{lemma}
\begin{proof}
The Hochschild-Serre spectral sequence gives
$$
\H^a(R, \H^b(U,V))\abuts \H^{a+b}(G,V),
$$
but $R$ is reductive, so cohomologically trivial, giving
$$
\H^i(G,V)\cong \H^i(U, V)^R\cong(\H^i(U,\Q_l)\ten V)^R,
$$
the last isomorphism following since $V$ is an $R$-representation.
\end{proof}
\begin{lemma}\label{liecoho}
If $U$ is a pro-unipotent algebraic group with associated Lie algebra $\fu$, then
$$
\H^*(U,\Q_l)\cong \H^*(\fu,\Q_l).
$$
\end{lemma}
\begin{proof}
Observe that the categories of $U$-representations and of $\fu$-representations are equivalent.
\end{proof}
\begin{proposition}\label{gpxcoho}
Given a pointed connected algebraic variety $(Z, \bar{z})$ with \'etale fundamental group $\Gamma$, and a constructible $\Q_l$-local system $\vv$ on $Z$, the canonical maps
$$
\H^i(\Gamma, \vv_{\bar{z}}) \to \H^i(Z, \vv)
$$
are bijective for $i=0,1$, and injective for $i=2$.
\end{proposition}
\begin{proof}
It suffices to prove this for finite local systems $\ww$. If we write $\Gamma_{\alpha}$ for the finite quotients of $\Gamma$, then the canonical $\Gamma_{\alpha}$-torsors $Y_{\alpha}$ form an inverse system of varieties over $Z$, giving a Leray spectral sequence
$$
\varinjlim \H^a(\Gamma_{\alpha}, \H^b(Y_{\alpha}, \ww)) \abuts \H^{a+b}(Z, \ww).
$$
The result now follows from the observation that for $\alpha$ sufficiently large, $\H^0(Y_{\alpha}, \ww)\cong \ww_{\bar{z}}$ and $\H^1(Y_{\alpha}, \ww)=0$.
\end{proof}
\subsection{Frobenius actions}\label{frob
We retain the conventions of \S \ref{weilact}, assuming furthermore that $X$ is either proper or smooth.
\begin{definition}
As in Theorem \ref{mixed}, $\bO(\w\redpia)$ is the sheaf of algebras on $X$ corresponding to the vector space $O(\w\redpia)$ equipped with its left $\redpia$-action. From now on, we will simply denote this sheaf by $\bO$. This is a pure Weil sheaf of weight $0$. The Frobenius actions on the cohomology groups $\H^i(X,\bO)$ combine with the right $\redpia$-actions to make them mixed Weil representations.
\end{definition}
\begin{theorem}\label{nfrobhull}
There is an isomorphism
$$
\Lie(\wmypia) \cong L(\H^1(X,\bO)^{\vee})/(f(\H^2(X,\bO)^{\vee})),
$$
where $L(V)$ is the free pro-nilpotent Lie algebra on generators $V$, and
$$
f:\H^2(X,\bO)^{\vee} \to \Gamma_2L(\H^1(X,\bO)^{\vee})
$$
is $R$-equivariant and preserves the (Frobenius) weight decompositions of \cite{Weil2} 3.3.7. The resulting weight decomposition on $\wmypia$ is the same as the natural Weil weight decomposition of Theorem \ref{mixed}.
Moreover, for $\ff:=L(\H^1(X,\bO)^{\vee})$, the quotient map
$$
f:\H^2(X,\bO)^{\vee} \to \Gamma_2\ff/\Gamma_3\ff \cong {\bigwedge}^2(\H^1(X,\bO)^{\vee})
$$
is dual to the cup product
$$
\H^1(X,\bO) \by \H^1(X,\bO) \xra{\cup} \H^2(X,\bO).
$$
\begin{proof}
Write $G:= \w\algpia$, $R:= \w\redpia$, $U:= \wmypia$ and $\fu:= \Lie(U)$. By Lemmas \ref{radcoho}--\ref{liecoho}, Lemma \ref{dual} and Proposition \ref{gpxcoho}, we know that there is a canonical isomorphism
$$
\H^1(\fu,\Q_l) \cong \H^1(X,\bO)
$$
of $R$-representations. Since this isomorphism is functorial, it is Frobenius-equivariant.
We now make use of Theorem \ref{mixed}, which gives a weight decomposition on $\fu$ (which is $R$-semilinear). In fact, the theorem gives a weight decomposition on $G$, so we have an action of $\bG_m \ltimes R$ on $U$.
To see that the Frobenius decomposition corresponds to the Weil decomposition, note that the action of $F_x \in W(X_0,x)$ determines the Weil decomposition.
We may choose a lift of the map
$$
\fu \to \fu/[\fu,\fu]= \H^1(\fu,\Q_l)^{\vee}\cong \H^1(X,\bO)^{\vee}
$$
as a $\bG_m \ltimes R$-representation. Writing
$\ff:=L(\H^1(X,\bO)^{\vee})$, this gives the surjection $\ff \onto \fu$.
If $J$ is the kernel of this surjection, then
$$
J/[\ff,J] \cong \H^2(\fu,\Q_l)^{\vee},
$$
and this isomorphism is also $\bG_m \ltimes R$-linear. Once again, we may use reductivity of $\bG_m \ltimes R$ to choose a lift, giving
$$
\H^2(\fu,\Q_l)^{\vee} \to J,
$$
and we define $f:\H^2(X,\bO)^{\vee} \to J$ to be the composition of this with the maps of Lemma \ref{factorset} and Proposition \ref{gpxcoho}.
Finally, the characterisation of the cup product is a standard result in Lie algebra cohomology, being dual to the map $J/[\ff,J]\to [\ff,\ff]/[\ff,[\ff,\ff]]$.
\end{proof}
\end{theorem}
\begin{corollary}\label{quadratic}
If $X$ is smooth and proper, then
$$
\Lie(\wmypia)
$$
is quadratically presented.
In fact, there is an isomorphism of Weil representations
$$
\Lie(\wmypia) \cong L(\H^1(X,\bO)^{\vee})/(\check{\cup}\,\H^2(X,\bO)^{\vee})),
$$
where $\check{\cup}$ is dual to the cup product.
\begin{proof}
This follows since, under these hypotheses, \cite{Weil2} Corollaries 3.3.4--3.3.6 imply that $\H^1(X,\bO)$ is pure of weight $1$, and $\H^2(X,\bO)$ is pure of weight $2$. This makes the choices of lifts unique, and hence Frobenius-equivariant.
\end{proof}
\end{corollary}
\begin{corollary}
If $X$ is smooth and proper, there is a canonical equivalence of categories between:
\begin{enumerate}
\item the full subcategory $\C$ of the category of constructible local systems over $\Q_l$ on $X$ whose objects are subsystems of Weil sheaves, and
\item the category of pairs $(\ww,\alpha)$, for $\ww \in \C$ semisimple and $\alpha \in \H^1(X, \End(\ww))$ with $\alpha\cup \alpha =0$.
\end{enumerate}
\end{corollary}
\begin{corollary}\label{pi1quadratic}
If $X$ is smooth and proper, then the pro-unipotent Malcev completion
$$
\pi_1(X,\bar{x})\ten\Q_l
$$
is quadratically presented.
In fact,
$$
\cL(\pi_1(X,\bar{x}),\Q_l) \cong L(\H^1(X,\Q_l)^{\vee})/(\check{\cup}\,\H^2(X,\Q_l)^{\vee})),
$$
where $\check{\cup}$ is dual to the cup product.
\end{corollary}
\begin{proof}
The pro-unipotent completion $\pi_1(X,\bar{x})\ten\Q_l $ is just the maximal quotient $\theta_{\sharp}\mypia$ of $\mypia$, for $\theta:\redpia \to 1$, on which $\pi_1(X,\bar{x})$ acts trivially.
\end{proof}
\begin{example}
This implies that, for $X$ smooth and proper, the pro-$l$ quotient $\pi_1^l(X,\bar{x})$ of $\pi_1(X,\bar{x})$ cannot be the Heisenberg group
$$
\cH_3(\Z_l)=\left\{ \begin{pmatrix} 1 & x & y\\0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} \in \GL_3(\Z_l)\right\},
$$
since this is not of quadratic presentation (in particular, this can be inferred from the non-vanishing of the Massey triple product on $\H^1(\pi_1(X,\bar{x}),\Q_l)$ ---
see \cite{Am} Ch.3 \S 3 for criteria for a Lie algebra to be quadratically presented).
\end{example}
\begin{corollary}
If $X$ is smooth and proper, then
$$
\cL(\pi_1(X,\bar{x}))/\Gamma_3(\cL(\pi_1(X,\bar{x}))) \ncong L(V)/\Gamma_3(L(V)),
$$
for any free Lie algebra $L(V)$.
\end{corollary}
\begin{proof}
As for \cite{Am} Proposition 3.25, making use of the Hard Lefschetz Theorem (\cite{Weil2} Theorem 4.1) to see that $\H^1(X,\Q_l)\by \H^1(X,\Q_l)\to \H^2(X,\Q_l)$ must be non-degenerate.
\end{proof}
\begin{corollary}\label{quartic}
If $X$ is smooth, then
$$
\Lie(\wmypia)
$$
is a quotient of the free pro-nilpotent Lie algebra $L(\H^1(X,\bO)^{\vee})$ by an ideal which is finitely generated by elements of bracket length $2,3,4$.
\begin{proof}
This follows from Theorem \ref{nfrobhull} since, under these hypotheses, \cite{Weil2} Corollaries 3.3.4--3.3.6 imply that $\H^1(X,\bO)$ is of weights $1$ and $2$, while $\H^2(X,\bO)$ is of weights $2, 3$ and $4$.
\end{proof}
\end{corollary}
\begin{corollary}\label{pi1quartic}
If $X$ is smooth, then
$$
\pi_1(X,\bar{x})\ten\Q_l
$$
is a quotient of the free Lie algebra $L(\H^1(X,\Q_l)^{\vee})$ by an ideal which is finitely generated by elements of bracket length $2,3,4$.
\end{corollary}
\begin{example}
Thus $\pi_1^l(X,\bar{x})$ cannot be the group
$$
\left\{ \begin{pmatrix} 1 & * & *\\0 & \ddots & * \\ 0 & 0 & 1 \end{pmatrix} \in \GL_5(\Z_l)\right\}.
$$
\end{example}
\begin{remark} If $X$ is singular and proper, weights tell us nothing about the structure of the fundamental group, since zero weights are permitted, so any equations may arise.
\end{remark}
\subsection{Further examples}
We will now show how Theorem \ref{nfrobhull} can be used to establish stronger restrictions on the fundamental group.
\begin{corollary}\label{criterion} Let $G$ be an arbitrary reductive $\Q_l$- algebraic group, acting on a unipotent $\Q_l$-algebraic group $U$ defined by homogeneous equations, i.e. $\fu \cong \gr \fu$ as Lie algebras with $G$-actions.
\begin{enumerate}
\item
If $X$ is smooth and proper, and
$$
\rho_2: W(X_0,x) \to (U/[U,[U,U]]) \rtimes G
$$
is a Zariski-dense representation, then
$$
\rho_1: \pi_1(X,x) \to (U/[U,U]) \rtimes G
$$
lifts to a representation
$$
\rho: \pi_1(X,x)\to U\rtimes G.
$$
\item
If $X$ is merely smooth, and
$$
\rho_4: W(X_0,x) \to (U/\Gamma_5 U) \rtimes G
$$
is a Zariski-dense representation, then
$$
\rho_1: \pi_1(X,x) \to (U/[U,U]) \rtimes G
$$
lifts to a representation
$$
\rho: \pi_1(X,x)\to U\rtimes G.
$$
\end{enumerate}
\begin{proof}
As for \cite{higgs} Corollary \ref{higgs-criterion}.
\end{proof}
\end{corollary}
\begin{remarks}
Note that Corollaries \ref{quadratic} and \ref{quartic} imply the results of \cite{paper1}:
The problem considered in \cite{paper1} is to fix a reductive representation \mbox{$\rho_0:W(X_0,x) \to G(\Q_l)$,} and consider lifts $ \rho:\pi_1(X,\bar{x}) \to G(A)$, for Artinian rings $A$. The hull of this functor is the functor
$$
A \mapsto \Hom_{\pi_1(X,\bar{x})}(\mypia, \exp(\g\ten \m_A)),
$$
where $\g$ is the Lie algebra of $G$, regarded as the adjoint representation. It follows that this hull then has generators $\Hom_{\pi_1(X,\bar{x})}(\g, H_1)$, and relations
$$
\Hom_{\pi_1(X,x)}(\g, H_2) \to \Symm^2 \Hom_{\pi_1(X,x)}(\g, H_1)
$$
given by composing the coproduct and the Lie bracket, where
$$
H_i:=\H^i(X,\bO(\redpia))^{\vee}.
$$
\end{remarks}
\begin{definition}
A representation $\rho:\Gamma\to G$ of a pro-finitely generated group $\Gamma$ is said to be rigid if the orbit $G(\rho) \subset \Hom(\Gamma,G)$ under the conjugation action is open in the $l$-adic topology. Observe that this is equivalent to the condition that $\H^1(\Gamma, \Lie(G))=0$, since this is the dimension of the quotient space at $[\rho]$.
A representation is properly rigid if the representation to the Zariski closure of its image is rigid.
\end{definition}
The following lemma is inspired by the observation in \cite{Simpson} that rigidity ensures that a local system on a complex projective variety is a variation of Hodge structure.
\begin{lemma}\label{rigid}
Every properly rigid representation $\rho:\pi_1(X,\bar{x})\to G$ extends to a representation of $W(X_{k'},x)$, for some finite extension $k\subset k'$.
\begin{proof}
Replace $G$ by the Zariski closure of the image of $\rho$.
If we give the set $\N$ the multiplicative ordering, then it becomes a poset, and ${F_x^n\rho}_{n \in \N}$ is a net in $\Hom(\pi_1(X,\bar{x}),G)$. Since $F_x^n \to 1$, this net tends to $\rho$. Since $G(\rho)$ is an open neighbourhood of $\rho$, there exists an $n$ for which $F_x^n\rho \in G(\rho)$; let $F_x^n =\ad_g(\rho)$. We may now define a representation
$$
\pi_1(X,\bar{x}) \rtimes \langle F_x^n\rangle \xra{(\rho,g)} G,
$$
noting that the former group is $W(X_{k'},x)$, for $k\subset k'$ a degree $n$ extension.
\end{proof}
\end{lemma}
\begin{remark}
Observe that the lemma remains true under the weaker hypothesis that $\im(\H^1(\pi_1(X,\bar{x}), \Lie(\im \rho)) \to \H^1( \pi_1(X,\bar{x}), \Lie(G)))=0$.
\end{remark}
\begin{proposition}
If $X$ is smooth and proper, and $\Gamma:=\pi_1(X,\bar{x}) =\Delta \rtimes \Lambda$, let $H$ be the Zariski closure of the image of $\Lambda$ in $\Aut(\Delta\ten \Q_l)$. If $H$ is reductive, $\H^1(\L, \Lie(H))=0$, and $\Hom_{\L}(\Delta/[\Delta,\Delta], \Lie(H))=0$, then $\Delta\ten \Q_l$ is quadratically presented.
\begin{proof}
First observe that the representation $\rho:\Gamma \to \Lambda \to H$ is rigid. This follows because the condition $\H^1(\L, \Lie(H))=0$ ensures that $\L \to H$ is rigid, so any representation $\Gamma \to H \ltimes \eps\Lie(H)$ (for $\eps^2=0$) must be conjugate to one which restricts to $\rho$ on $\L$. The image of $\Delta$ must also lie in $\Lie(H)$, so the representation is determined by an element of $\Hom_{\L}(\Delta/[\Delta,\Delta], \Lie(H))=0$, so it must be $\rho$. Therefore, by Lemma \ref{rigid}, $\rho$ extends to a Weil representation (possibly after changing the base field).
Hence $\rho$ factors as $\pi_1(X,\bar{x}) \to\w\redpia \xra{\theta} H$.
Now, by Corollary \ref{quadratic}, we know that
$$
\theta_{\sharp}Lie(\wmypia)
$$
must be quadratically presented. The proof now proceeds as in \cite{higgs} Proposition \ref{higgs-crit}.
\end{proof}
\end{proposition}
\begin{example} Let $\mathfrak{d}$ be the free $\hat{\Z}$-module
$$
\mathfrak{d}:=\hat{\Z}x \oplus \hat{\Z}y \oplus \half \hat{\Z}[x,y],
$$
which has the structure of a Lie algebra, with $[x,y]$ in the centre.
The Campbell-Baker-Hausdorff formula enables us to regard $\Delta:=\exp(\mathfrak{d})$ as the profinite group with underlying set $\mathfrak{d}$ and product
$$
a\cdot b=a + b +\half [a,b],
$$
since all higher brackets vanish.
Let $\exp(\fh):=\Delta\ten \Q_l$; this is isomorphic to the three-dimensional $l$-adic Heisenberg group.
Observe that $\Aut(\Delta \ten \Q_l) \cong \GL_2(\Q_l)$, and that $\SL_2(\hat{\Z})$ acts on $\Delta$ by the formula:
$$
A(v,w) := (Av, (\det A) w)=(Av,w),
$$
for $v \in \hat{\Z}x \oplus \hat{\Z}y$ and $w \in \half \hat{\Z}[x,y]$.
The group
$$
\Gamma:=\Delta \rtimes \SL_2(\hat{\Z});
$$
cannot be the geometric fundamental group of any smooth proper variety defined over the algebraic closure of a finite field.
\begin{proof}
We wish to show that $\Gamma \to \Aut(\Delta \ten \Q_l)$ is properly rigid. For this, it will suffice to show that $\L \to \Aut(\Delta \ten \Q_l)$ is properly rigid, and that $\Hom_{\L}(\Delta/[\Delta,\Delta], \mathfrak{sl}_2(\Q_l))=0$.
To prove the first, observe that
$$
\SL_2(\hat{\Z})=\prod_{\nu \text{ prime}}\SL_2(\Z_{\nu}),
$$
and that only pro-$l$ groups contribute to cohomology. We need to show that the only derivations $\SL_2(\Z_l) \to \mathfrak{sl}_2(\Q_l)$ are inner derivations. Now, for $N$ sufficiently large, $\exp:l^N\mathfrak{sl}_2(\Z_l) \to \SL_2(\Z_l)$ converges, and it follows from the simplicity of $\mathfrak{sl}_2(\Z_l)$ that any derivation must agree with an inner derivation when restricted to $\exp(l^N\mathfrak{sl}_2(\Z_l))$. Since this is a subgroup of finite index, and $\mathfrak{sl}_2(\Q_l)$ is torsion-free, the derivation and inner derivation must agree on the whole of $\SL_2(\Z_l)$, as required.
To prove the second, observe that $\Q_l^2$ and $\mathfrak{sl}_2(\Q_l)$ are distinct irreducible $\SL_2(\Z_l)$-representations.
We therefore conclude from the previous proposition that $\Gamma$
cannot be the fundamental group of any smooth proper variety defined over the algebraic closure of a finite field, since the action of $\SL_2(\hat{\Z})$ on $\fh$ is semisimple, hence reductive, and $\fh$ is not quadratically presented.
Alternatively, we could use Corollary \ref{criterion} to prove that $\Gamma$ is not such a group. Let $G=\SL_2(\Q_l)$, $\fu=L(\Q_l^2)$ and $U=\exp(\fu)$. Observe that $\fh \cong \fu/[\fu,[\fu,\fu]]$, and let $\rho_2$ be the standard embedding
$$
\rho_2: \Delta \rtimes \SL_2(\hat{\Z}) \to \exp(\fh) \rtimes \SL_2(\Q_l),
$$
which extends to a Weil representation by Lemma \ref{rigid} and the above calculation.
Since all triple commutators vanish in $H$, this does not lift to a representation
$$
\rho: \Delta \rtimes \SL_2(\hat{\Z}) \to U \rtimes \SL_2(\Q_l).
$$
\end{proof}
Note that Corollary \ref{pi1quartic} cannot be used to exclude this group --- the abelianisation of $\Gamma$ is a torsion group, as $\SL_2$ acts irreducibly on the abelianisation of $\fh$, so $\Gamma\ten \Q_l =1$, which is quadratically presented.
\end{example}
\begin{example}
Let $\mathfrak{d}$ be the free $\hat{\Z}$-module on generators
$$
x,y, \frac{1}{2}[xy], \frac{1}{12}[x[xy]], \frac{1}{12}[y[xy]], \frac{1}{24}[x[x[xy]]],\frac{1}{24}[x[y[xy]]],\frac{1}{24}[y[y[xy]]];
$$
this has a Lie algebra structure, with all quintuple commutators vanishing. We define $\Delta:=\exp(\mathfrak{d})$, the group whose underlying set is $\mathfrak{d}$, given a group structure via the truncated Campbell-Baker-Hausdorff formula:
$$
a\cdot b=a+b+\frac{1}{2}[a,b]+\frac{1}{12}([a,[a,b]]-[b,[a,b]]) -\frac{1}{24}[a,[b,[a,b]]].
$$
Again, let $\L:=\SL_2(\hat{\Z})$, acting in the natural way on $\hat{\Z}x \oplus \hat{\Z}y$, with the action extending to the whole of $\Delta$ via the laws of Lie algebras. Then
$$
\Gamma:=\Delta \rtimes \SL_2(\hat{\Z})
$$
cannot be the geometric fundamental group of any smooth variety defined over the algebraic closure of a finite field.
\begin{proof}
Let $G=\SL_2(\Q_l)$, and let $\fu=L(\Q_l^2)$ and $U=\exp(\fu)$. Observe that
$$
\fh:=\cL(\Delta, \Q_l) \cong \fu/\Gamma_5\fu,
$$
and let $\rho_4$ be the standard embedding
$$
\rho_4: \Delta \rtimes \SL_2(\hat{\Z}) \to \exp(\fh) \rtimes \SL_2(\Q_l),
$$
which extends to a Weil representation by Lemma \ref{rigid} and calculation in the previous example.
Since all quintuple commutators vanish in $H$, this does not lift to a representation
$$
\rho: \Delta \rtimes \SL_2(\hat{\Z}) \to U \rtimes \SL_2(\Q_l),
$$
which gives a contradiction, by Corollary \ref{criterion}.
\end{proof}
\end{example}
\bibliographystyle{alphanum}
\addcontentsline{toc}{section}{Bibliography}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,822 |
Accenture Federal Services has announced the appointment of Bryan Rich as managing director of analytics and artificial intelligence (AI) solutions, with a focus on integrating AI into next-gen applied technology solutions and services for federal agencies.
In his new role he will work with the data science and technology teams inside Accenture Federal Services to build the innovative design, architecture and project management capabilities of the portfolio, including advanced automation and machine learning.
"Our federal clients will benefit from Bryan's ability to leverage our global firm's unique reach into the core technology vendors and platforms which we've deployed countless times to solve similar at-scale problems for Fortune 100 firms," said Tom Greiner, senior managing director of the Technology Business at Accenture Federal Services.
Rich brings outstanding operational experience delivering innovative analytics solutions for global customers, the company said. He joins Accenture from Novetta Solutions, a Carlyle Group-owned business, where he served as senior vice president of analytics strategy. A former TV and radio producer, Bryan founded an analytics-as-a-service technology business focused on automating pattern detection and analytics in real-time open source data.
"The hybridization of data science and management consulting coupled with the pace of automation signals a generational disruption in our market and I look forward to working with the talented professionals at Accenture to drive the convergence of artificial intelligence and data analytics into game-changing real-time solutions for federal customers," Rich said. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,142 |
Q: Flex: Custom context menu for a component I have a Flex application, running with Flash Player, not AIR, that contains a Tree that I would like to put a custom context menu on.
Tried just doing <mx:Tree ... contextMenu="{MyClassWithStatic.menu}">, but that didn't do anything.
Went searching, and found this quote from some Adobe docs somewhere
In Flex or Flash Builder, only top-level components in the application can have context menus. For example, if a DataGrid control is a child of a TabNavigator or VBox container, the DataGrid control cannot have its own context menu.
so went upwards, trying each parent element until I reached my <Application>-element, which is consistent with what they wrote.
Tried making a Flex component, based on Group (the default) which contained my tree, and the context menu on the top-level element there, hoping it would work, but to no avail.
Is there any other way to manage this that I haven't found yet?
The code I use to create the menu:
var menuItems:Array = [];
var rename:ContextMenuItem = new ContextMenuItem("Rename");
rename.addEventListener(ContextMenuEvent.MENU_ITEM_SELECT, renameSelectedHandler);
menuItems.push(rename);
menu.customItems = menuItems;
menu.hideBuiltInItems();
A: You're right, the contextmenu only works on top level components. It's a limitation of Flex which is annoying and shouldn't be there in the first place. There's not much you can do since there is no way to capture the event other than using some Javascript trickery, but even then, it doesn't tell you where you were clicking.
If I were you, I would just forget the concept and go away from using right click altogether if possible.
A: I can't be sure, as all the code isn't' there. But you seem to have ignored your own research. Don't use your new component, or anything which "contains" your tree. Then just stick the Tree in your application.
Also I've a memory of TreeItemRenderer not being the same as in other UIcomponents. Maybe, test your "menu" code with a Datagrid first and make sure it works. Good luck
A: I did not try it myself, but after reading the comments on http://michael.omnicypher.com/2007/02/flex-trees-with-context-menu_14.html it looks like you could add a context menu to the tree's item renderer.
The article and comments at http://blog.arc90.com/2008/04/21/adding-a-contextmenu-to-a-flex-tree/ are worth a look too.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,348 |
У́лица Ширшо́ва — небольшая улица на севере Москвы в районе Лианозово Северо-Восточного административного округа, от улицы Байдукова до улицы Чкалова.
Происхождение названия
Названа в 1960 году в честь Героя Советского Союза, академика Петра Петровича Ширшова (1905—1953), участника первой в мире научно-исследовательской станции на Северном полюсе.
Ссылки
Официальный сайт управы Лианозово
Улицы Москвы, названные в честь людей
Улицы Москвы, появившиеся в 1960 году | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,338 |
Tain Museum Image Library
No: 343 Contributor: Tain & District Museum Year: 0
The Grove, Tain
The Grove is situated off Cunarder Lane (which links the High Street and Kirksheaf Road). This picture was taken before part of the grounds were used for the existing Post Office car park. It is currently owned by the Arthur family. Shona Arthur's father, Alex Macleod ("Dow") was a photographer and used parts of the building as a shop and a studio. He also rented out part of the building to a dentist, David Geekie whose son still lives in Tain. Past owners of The Grove were Mrs Murray Cameron (Mary), Kate Mackenzie, James Cree (Road Surveyor) and Dr. Kennedy.
Picture added on 07 February 2006
My great great grandfarther was the Dr Kennedy mentioned. I wud be grateful for any up to date pictures of the Grove.
Added by Jason Kennedy on 08 May 2007
My grandparents are Shona and Jim Arthur (current owners), so I'll see what I can do.
Added by Andrew Mackenzie on 29 May 2007
I am Dr Kennedy's great grand daughter and I have photograph of a tea party in the Grove garden, taken when they lived there. I also have a painting of Mrs Kennedy sitting in the garden under a Gean tree. Is it still there? The painting is by my great aunt Janet C. Kennedy who taught Art in Tain for many years and was a well known character in the town. If the current owners would like copies I will send them to you.
Added by Louise Dickson on 30 May 2007
I think it would be lovely to get a photo of a tea party in The Grove garden AS IT ONCE WAS . I genuinely regret having to confess that during the last few years the garden has been allowed to run wild. My husband and I have never been gardeners and since he is now eighty and I am --sshh!--77, we don't even have the energy to keep the weeds down as we used to do.
Added by Shona Arthur on 09 June 2007
For Andrew, Good to see your name there! Love, Grandma
For Jason, You might be a bit disappointed by up-to-date pictures of The Grove, but if you would still like one, I'll be happy to send or e-mail one to you.
For Louise: see separate 'comment'
Louise, I'm sure the museum would love to have a copy of the painting. Janet Kennedy was a great friend of Rosemary Mackenzie (nee Munro), who founded the museum, and her mother who was also an artist and came to Tain to teach art at the Academy. The Munros lived a couple of hundred yards along at Alderbrae.
Added by Estelle Quick on 10 June 2007
I will copy as many pictures as possible and bring them to Tain when I am coming North in September. I brought the general pictures I had of Tain to the museum some years ago but kept the family relevant pictures.
Added by Louise Dickson on 12 June 2007
Not only David Geekie but his sister June (Walker) still live in Tain.
Could anyone give me any details about James Cree (Road Surveyor) who lived at the Grove Tain. We know he was appointed Road Surveyor for Ross-shire in 1930.
Added by Rhona Munro on 19 April 2010
We have a fairly comprehensive list of ' dispositions' of The Grove, including those transacted in the 1930s, but Mr. Cree's name does not appear on any of these. Shona Arthur
Added by Shona Arthur on 25 April 2010
As far as I know The Grove was being run in the 1930s as a 'Board Residence' ( I am quoting from an advertisement in a tourist book of the 1930s called ' The Book of the Highlands' ), proprietor Mrs. Mackenzie. Perhaps James Cree was one of the paying guests....?
Thank you for your comments - I will try and look into it a bit more.
Another advert, dated May 1935 states "Mrs Mackenzie is no longer in The Grove which has been sold." Perhaps James Cree bought it from her?
Added by Margaret Urquhart,Tain & District Museum on 25 April 2010
Alas! The owners, who always expected to live out their days in the Grove, have had to move out. The deterioration due to old age, their own and that of the house, has made staying in it no longer a viable option.
Added by Shona Arthur on 27 July 2010
My name is Sasha and I once lived in Tain down by the shore. Sometime around 1996/7 some nasty
men scared me away, and I found a nice car to live under. The car owner's name was Jim a very nice
gentleman who didn't mind me staying there. The name of his house was called The Grove and it was
a lovely place with green grass and lots of voles for me to eat, with great big tree's for me to climb and
nice and quiet, away from all the noisy traffic. A kind lady called Shona who was Jim's wife was there
too and I'll always remember my happy times there before my owner found me and took me away on
a long journey to live with her and my other friends on the Isle of Skye. I'll always remember the
beautiful place called The Grove.
Added by Sasha on 27 June 2011
Thank you for those kind words, Sasha. Trust you are continuing happy on the Misty Isle. Shona
Anonymous comment added on 27 June 2011
19 year old Jessie Ross in the 1891 census, from Parish of Tarbat (Arboll), was cook at The Grove for the Kennedy family. Anyone know what became of her? She was a sister of my Grandfather.
Added by Donald MacDonald-Ross on 28 April 2014
Sad to say that The Grove burnt down today. What a shame that the building became derelict and unloved, a target for vandals & squatters.
Added by John Smith on 02 October 2014
The Grove remained fairly respectable for five years after we moved out, and we considered that a great blessing in this day and age. Jim made any small repairs that cropped up as a result perhaps of ball-throwing. However, that all changed within the last year when entry was gained by vandals. Everything that happened to it after that was really sad . It was sad not because we didn't love the dear old house enough, but because we did.
Added by Shonaarthur@yahoo.co.uk on 05 October 2014
I appreciate how much The Grove meant to you both Shona and everyone who knows you will agree. I loved the stories you shared concerning The Grove in the TDPP which will stay with our readers - a piece of history. So unfortunate what happened and we were all saddened for you. We can't be vigilantes 24/7 on a property that isn't occupied to prevent potential culprits they will always find ways to gain access. Just sad that we had such vandals in our midst.
Added by Maggie Mercer (editor Tain & District Picture Post) on 26 November 2014
Oh, THANK YOU for that, Maggie ! The people of Tain have been very very supportive, and that has been most helpful to Jim and myself.
Added by Shona Arthur on 16 January 2015
Have been doing some family research and it seems my great aunt Alice McLeod who was born in Kildonan was a general domestic servant aged 19 to the Kennedy family in 1901. Does anyone have any more information.
Added by Joanna Harvie on 18 March 2015
Kate Mackenzie was my Grandmother Chrissie's sister Katherine - affectionally know as Auntie Katie by my father John who often told tales of her during her time in Tain. Her husband was John Robertson Mackenzie, a taylor originally from Glasgow, who died, aged 44 in 1912. I have no idea if they lived in the Grove together or if she moved in after his death. I think that she may have run a B&B business there. She died in Crosshills in 1955 aged 85.
Added by John Fraser on 24 June 2015
Next Random Pic | Full List | My Album Admin Login | Sponsor this Site | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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package example;
import javax.faces.event.ActionEvent;
/**
* Created by IntelliJ IDEA.
* User: John Ellis
* Date: Dec 28, 2005
* Time: 4:49:32 PM
* To change this template use File | Settings | File Templates.
*/
public class TestSubBean {
private String index = "" + Math.random();
public void processTestButton(ActionEvent event) {
int i = 1;
i++;
}
public String getIndex() {
return index;
}
public void setIndex(String index) {
this.index = index;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/www.woundsource.com\/blog\/how-determine-if-cellular-andor-tissue-based-product-worth-it","text":"# How to Determine If That Cellular and\/or Tissue-Based Product Is Worth It\n\nKeywords:\nBlog Category:\n\nby Thomas E. Serena MD, FACS, FACHM, FAPWCA\n\n\"Price is what you pay. Value is what you get\"\n-Warren Buffet1\n\nLawrence Mills introduced the concept of Value Analysis to the manufacturing industry a half century ago. The basic idea entails analyzing the function and importance of the various parts of a product as they relate to cost. He derived the following equation2:\n\n$\\large&space;Value&space;=&space;\\frac{Function}{Cost}$\n\nI admire my colleagues in economics and industry, although I admit that in my younger years, I dismissed all but the biological sciences as the impractical folly of academicians. What possible use would I have for economics, of all fields, in the practice of medicine? My misguided arrogance as a surgeon has since afforded me a fine taste for crow. There is a seriously pressing need for value analysis in medicine, most especially in selecting from the dazzling array of Cellular and\/or Tissue-based Products (CTPs).\n\n## Determining Which Cellular and\/or Tissue-base Product to Use\n\nCTPs for the treatment of chronic wounds have undoubtedly improved patient outcomes; however, the Cambrian-like proliferation of these products has led to considerable confusion as to which to choose for a particular patient or wound type. I reckon from my last count that there are more than 70 CTPs approved for use through a variety circuitous FDA pathways. Daily, I face the question from my fellow woundologists: which product should I use? Of greater concern is that many of us are victims of our last success. CTP \"X\" worked recently on a difficult patient; therefore, I am prone to using it on the next 10 patients, often times forgetting my own admonition to practice evidence-based, fiscally responsible medicine. I believe this conundrum can be resolved through a value analysis process uniquely adapted for our discipline.\n\nI have humbly defined \"value\" (in deference to my economic colleagues with far greater skill) in the wound care marketplace as the strength of the clinical evidence multiplied by a constant plus a factor to account for the number of grafts required plus product reimbursement.\n\n$\\large&space;CTP\\:&space;Value&space;=&space;(E\\,*\\,&space;S)\\,&space;+\\,&space;R\\,&space;+\\,&space;(2\\,&space;-\\,&space;A)$\n\nFirst of all, products with randomized, controlled trials (RCT) will be highly favored in this formula. This narrows the field substantially from 70 products to 8 that have RCT evidence. Clinical evidence (CE) is evaluated using a scale ranging from -2 to +3,\n\nCalculating Clinical Evidence Score (E):\n-2: Mechanism-based reasoning\n-1: Case-series, case-control studies, or historically controlled studies\n0: Non-randomized controlled cohort\/follow-up study\n+1: Randomized trial single\n+2: Randomized trial(s) and comparative effectiveness trial\n+3: Systematic review of randomized trials\n\nThe CE score is then multiplied by a constant in order to give greater weight to clinical evidence over reimbursement. I have called this the \"Serena Constant\" (S). It is completely arbitrary; therefore, if anyone is to take the blame for determining its value it is me. Reimbursement is the cost of the product minus its average blended net payment. Medicare's introduction of the high and low bundles has made this process much simpler. A score is given to the reimbursement and it is inserted into the equation. We have proposed the following scale for reimbursement,\n\nCalculating Reimbursement Score (R):\nThe facility profits from the product: +1\nThe product is budget neutral: 0\nThe facility does not profit from the product: -1\n\nFinally the number of graft applications required (A) to achieve closure will also be taken into consideration in this formula.\n\nThe calculation of CTP Value can be performed per wound type. Potentially it could be performed for each individual facility and clinician as well.\n\nFor example, CTP \"Y\" has an RCT demonstrating efficacy in the treatment of diabetic foot ulcers. It has a single RCT giving it a CE score of +1. This is then multiplied by the S constant of 3. CTP \"Y\" is in the high bundle and the cost of a 1.5 x 1.5 sheet is $800. The reimbursement is$1407 netting a profit to the center of \\$607. The reimbursement score would be +1. Finally, CTP \"Y\" requires an average of 2.5 applications to achieve complete closure.\n\n$\\large&space;CTP\\:&space;Value&space;=&space;(1\\,*\\,3)\\,&space;+\\,&space;(1)\\,&space;+\\,&space;(2\\,&space;-\\,2.5)&space;=&space;3.5$\n\nA total score of +3.5 would suggest that CTP \"Y\" should result in a good evidenced-based clinical outcome in a fiscally responsible manner. In addition, CTP \"Y\" could now be compared to all of the other CTPs allowing physicians to choose the best product for his or her patient.\n\n## A Call for Wound Care Community Input\n\nThe final step is to publish the results annually for all of the CTPs. This formula is a first draft. I would invite my fellow clinicians or industry partners interested in this venture to join me. Please e-mail me at serena@serenagroups.com.\n\nReferences:\n1. Warren Buffet. 1930. American Investment Entrepreneur\n2. Adapted by Kenneth Crow DRM Associates. Value analysis and function analysis system technique. Accessed at http:\/\/www.npd-solutions.com\/va.htm 10\/15\/2015.","date":"2019-05-26 15:54:45","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\": 3, \"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\": 0.3541143536567688, \"perplexity\": 3641.007501502248}, \"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-2019-22\/segments\/1558232259316.74\/warc\/CC-MAIN-20190526145334-20190526171334-00101.warc.gz\"}"} | null | null |
PT Aviastar Mandiri, que opera como Aviastar es una aerolínea indonesia de vuelos regulares domésticos del Yakarta Oriental, Yakarta, Indonesia.
Historia
La aerolínea fue fundada en 2003 por el capitán Sugeng Triyono.
Su base de operaciones principal se localiza en Yakarta (CGK) con bases de operaciones secundarias en Palangkaraya (PKY), Balikpapan (BPN), Makassar (UPG), Nabire (NBX) y Denpasar (DPS).
Destinos
Aviastar opera a los siguientes destinos domésticos de Indonesia :
Balikpapan - Puruk Cahu
Balikpapan - Samarinda - Melak
Batam - Tembilahan
Yakarta - Ketapang - Pontianak
Nabire - Biak
Nabire - Enarotali - Timika
Nabire - Sugapa
Palangkaraya - Muara Teweh
Palangkaraya - Tumbang Samba
Flota
La flota de Aviastar incluye las siguientes aeronaves (mayo de 2020):
Incidentes y accidentes
El 9 de abril de 2009, un BAe 146-300 de Aviastar registro PK-BRD, se estrelló contra una montaña cerca de Wamena, Papua, Indonesia, tras una segunda aproximación fallida de aterrizaje en el aeropuerto Wamena.
Referencias
Enlaces externos
Página web oficial
Actualización de aerolínea
Datos de Maskapai Dirjen Perhubungan Udara Dephub
Aerolíneas de Indonesia
Aerolíneas fundadas en 2003 | {
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Loyola Clinical Centers
Earn Your Master's at Loyola
Masters Student Admissions and Outcome Data
Masters Curriculum
Student Admissions, Outcomes, and Other Data
Field Placements
Externship Sites
Psy.D. Advisory Board
Additional Faculty
Graduate Scholarships & Fellowships
David G. Crough, Ph.D
Evolutionary psychology and its contributions to forensic psychology especially in the understanding and prevention of violence.
Leo Mickey Fenzel, Ph.D.
Lifespan human development, counseling and clinical psychology, higher education administration, stress and coping, research on urban school effectiveness and substance abuse, research on spirituality and the benefits of having an active spiritual life, and quantitative research design and analysis.
Faith D. Gilroy, Ph.D.
Business applications of psychology, attribution theory, conformity, attitudinal measurement (general), women's issues, gerontology, career patterns, gender choice of offspring.
Charles LoPresto, Ph.D.
Homophobia/homonegativity (e.g., etiological, personality and cognitive correlates), the etiology and construction of sexual orientation, stigma associated with sexual minority status, adolescent treatment issues, cognitive-behavioral approaches to treatment, cross-cultural psychology and men's issues.
Martin F. Sherman, Ph.D.
Terror management theory, five factor model of personality, psychopathology and personality, disgust sensitivity, survey and attitude research, gender role research-sexual orientation, the emotional state of elevation, and spiritual transcendence.
Steven A. Sobelman, Ph.D.
Non-verbal communication, stress disorders and stress management, therapeutic process and outcome, psychoanalytic psychotherapy, the nature of sleep and dreaming, sexual deviance, emotions, hypnosis, sport psychology.
Application Requirements and Deadlines
Meet a graduate student in the Clinical Professional Counseling Program who aims to spread mental health awareness through her practice as a clinician | {
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Pecan Deluxe Candy staff support cancer charity fund
Vince Bamford · 18 June, 2018
Inclusions supplier Pecan Deluxe Candy has raised more than £3,500 for a cancer charity.
Staff and management at the Leeds-based business are supporting Jacqui's Million, a charity set up by a terminal cancer sufferer to raise £1m for Leeds Cancer Centre.
The charity was founded by Jacqui Drake, who has personal ties to the company. She was first diagnosed with melanoma in 1991 and despite setbacks has beaten the odds for survival.
Activity by Pecan Deluxe Candy has included support for a book launch, an 80s night, donation of auction prizes and staff initiatives including a Christmas Jumper Day and Festive Buffet.
The company is sponsoring the Positive Vibes Cabaret Concert in Leeds on 30 June, and many factory and office staff at the business are running in the Yorkshire Marathon Relay on 14 October. Workers have formed two teams to take part in aid of the charity.
"Having known Jacqui for some time we were very proud to announce Jacqui's Million as our chosen corporate charity six months ago," said Pecan Deluxe Candy MD Graham Kingston. "We're all delighted to see how Jacqui's boundless positivity and contributions from many supporters have already helped to bring in nearly £90,000 towards her target."
"I also have nothing but admiration for our teams of colleagues who will be tackling the Yorkshire Marathon – especially as more than one of them has never run an organised race before in their lives."
Established in the UK in 1999, Pecan Deluxe Candy (Europe) Ltd is a fully owned UK subsidiary of US-based Pecan Deluxe Candy Company, a family owned business founded in 1950. | {
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3 new multi-concept restaurants to eat, play and work at
Consider having your next business retreat at The Summerhouse, FOC Sentosa and VLV.
Meryl Koh
Tucked away on a tree-lined road and amid a cluster of charming black and white bungalows, The Summerhouse could easily be mistaken for an idyllic hideaway in the English countryside. Until the humidity sets in and you realise this sprawling property – all 1.18 million sq ft – is the latest addition to the aerospace park surrounding the former Seletar Airbase.
It has a garden, patisserie/cafe and florist on ground level, and a restaurant-bar on the upper fl oor. Consider the air-conditioned venue, with total capacity of 460, for your next business retreat – if you're looking for a place that doesn't require your staff boarding a plane.
PHOTOS: The Summerhouse
More than just sweet treats and fresh flowers, The Summerhouse also offers essentials like projectors and sound systems to meet your business needs.
It also has its own edible garden.
So, besides getting fresh flowers and sweet treats on the table, companies who book the venue for a corporate do will also have access to meeting essentials like projectors, screens, a solid sound system and plenty of writing materials. The layout of the venue furniture can be customised accordingly as well, which means The Summerhouse can also double as an art gallery or a seminar room, depending on what the customer wants.
To be sure, more restaurants are turning into multi-concept spaces. FOC Sentosa, for instance, is another that marries good food with entertainment. The restaurant opened late last year along Tanjong Beach Walk, with the idea of bringing Barcelonian beachside culture to Singapore.
PHOTOS: FOC Sentosa
Cod fish and spinach caldoso rice
Eggplant with sobrassada and parmesan
King crab cannelloni
Interior of FOC Sentosa
"We are also looking to fill a gap in the market with a purpose-built multifunctional venue on the beach that can cater for corporate off -sites, weddings, and team-building activities," says a representative from FOC Restaurant Group.
At FOC Sentosa, guests have access to a pool deck and lounge, bar, and a private dining room on the second floor that is furnished with projectors and screens. The various break-out areas provide for endless possibilities when planning for events, from meetings to product launches and mega parties that can spill over to the beach.
But, if a 30-minute drive from town to either place seems like too much hassle, VLV at Clarke Quay is another multi-concept venue in a more central location. At the historic River House (formerly Indochine Group's restaurant-bar Forbidden City) is a 140-seater modern Chinese restaurant that serves luxe offerings such as Canadian lobster wonton, a lounge, and an outdoor riverfront dining space. It is owned by chief executive Dolores Au, who used to head global branding for the Ku De Ta club, now known as Ce La Vi, at Marina Bay Sands.
PHOTOS: VLV
"We have had guests use our space as discussion rooms. These were largely casual, round-table discussions," says Jessica Loo, senior manager of marketing communications at VLV. "It's a great opportunity to take meetings to a more informal space which could be conducive to more creative and productive sessions," she says.
And, in case you're wondering, free Wi-Fi is always a given.
These three venues might check all the boxes when it comes to business essentials, but how does the food measure up? We find out.
THE SUMMERHOUSE
With chef Florian Ridder (formerly from one Michelin star Alma at Goodwood Park Hotel) helming the kitchen, expect simple produce elevated by refined techniques. Many ingredients here are sourced locally as much as possible, with garnishes coming from the restaurant's edible garden. Goes well with a wide selection of freshly pressed juices. 3 Park Lane.
FOC SENTOSA
If there is a sure way to bring the intensity of a business meeting down a notch, it's introducing childhood favourites like fish and chips to snack on while brainstorming. And FOC Sentosa does a delicious version. The fish and chips here is made with tender, flaky sea bass coated in a light, crisp batter, served with elegantly thin slices of potatoes. 110 Tanjong Beach Walk.
VLV
Depending on the time of day for your meeting, the menu here could go from dim sum during tea to a set lunch with restaurant signatures like General Tso's chicken with fried rice. For large-scale functions, there are bite-sized items like chicken rice sushi and house-made carrot cake that make mingling easy. #01-02, 3A River Valley Road.
DiningFOC SentosaMulticoncept RestaurantsSingaporeThe SummerhouseVLV
Restaurants and bars to catch the National Day fireworks (2019)
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Exsto is the world's first Cognac created by a sommelier and a master blender
Durian season in Singapore: a world beyond mao shan wang | {
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Q: Classic ASP stuck in an infinite loop, makes no sense I could try and explain what this code is for and why the output is the way it is, but it will take forever.
function Get90nums(G90_TC)
if NOT isNumeric(G90_TC) then exit function : G90_TC=int(G90_TC) : if G90_TC>6 OR G90_TC<1 then exit function
dim G90_BA : G90_BA="" : dim G90_CC : G90_CC=false : dim G90_NC : G90_NC=0 : dim G90_RC : G90_RC=0 : dim G90_TBL : G90_TBL=0
do until G90_CC
randomize : G90_RN=int((90)*rnd+1)
if inStr(G90_BA,"["&G90_RN&"]")=0 then
if G90_NC=5 then
G90_BA=G90_BA&"[91][91][91][91]NL" : G90_RC=G90_RC+1 : G90_NC=0
if G90_RC=3 then
G90_TBL=G90_TBL+1
G90_RC=0
end if
else
G90_BA=G90_BA&"["&G90_RN&"]" : G90_NC=G90_NC+1
end if
end if
if G90_TBL=G90_TC then G90_CC=true : Get90nums=G90_BA
loop
end function
response.write Get90nums(1)
If you run the function as Get90nums(1) it will return something along the lines of:
[22][15][85][31][14][91][91][91][91]NL[40][10][9][77][54][91][91][91][91]NL[49][71][6][64][4][91][91][91][91]NL
I say "something along the lines" because the output is random. Ignore the fact the numbers are in brackets, ignore all the "91's" and the "NL's", and what you get is:
[22][15][85][31][14][40][10][9][77][54][49][71][6][64][4]
That's 15 unique random numbers between 1 and 90, no number is repeated.
Run the function as Get90nums(5) and you will get 75 unique numbers between 1 and 90.
However, if you run Get90nums(6), rather than returning all 90 numbers, it just get's stuck in an infinite loop, and I have no idea why.
Can anybody please shine some light on this, it's driving me crazy!
Thanks
A: I fixed it.
Instead of generating a random number each time I put all 90 numbers into an array beforehand and shuffled it.
UK bingo differs a lot from US bingo. You get six 3*9 cards. Each row has 5 numbers spread out and in ascending order. There are 90 numbers overall and no duplicates.
It's clearly a very early draft, but if you want to play 6 card UK bingo (it works with 1 to 6 cards) here's the script:
<%@LANGUAGE="VBSCRIPT" CODEPAGE="65001"%>
<%
'----------------------------------------------------------------------------------
Dim array90(89)
for x = 0 to 89
array90(x)=x+1
next
Arr90=ArrReOrder(array90)
'----------------------------------------------------------------------------------
function Get90nums(G90_TC)
if NOT isNumeric(G90_TC) then exit function:G90_TC=G90_TC:if G90_TC>6 OR G90_TC<1 then exit function
dim G90_BA:G90_BA="":dim G90_CC:G90_CC=false:dim G90_NC:G90_NC=0:dim G90_RC:G90_RC=0:dim G90_TBL:G90_TBL=0
for x=0 to (G90_TC*15)-1
G90_RN=Arr90(x)
G90_BA=G90_BA&"["&G90_RN&"]":G90_NC=G90_NC+1
if G90_NC=5 then
G90_BA=G90_BA&"[91][91][91][91]NL":G90_RC=G90_RC+1:G90_NC=0
if G90_RC=3 then
G90_TBL=G90_TBL+1
G90_RC=0
end if
end if
if G90_TBL=G90_TC then G90_CC=true:Get90nums=G90_BA
next
end function
'----------------------------------------------------------------------------------
function ArrReOrder(aArray)
Dim iUpper,iLower,iLoop,iSwapPos,varTmp
iUpper=UBound(aArray):iLower=LBound(aArray)
randomize Timer
for iLoop=iLower to iUpper
iSwapPos=Int(Rnd*(iUpper+1))
varTmp=aArray(iLoop)
aArray(iLoop)=aArray(iSwapPos)
aArray(iSwapPos)=varTmp
next
ArrReOrder=aArray
end function
'----------------------------------------------------------------------------------
sub arrNumericalAsc(arrArray)
Dim row,j,StartingKeyValue,NewKeyValue,swap_pos
for row=0 to uBound(arrArray)-1
if NOT arrArray(row)=91 then
StartingKeyValue=int(arrArray(row))
NewKeyValue=int(arrArray(row))
swap_pos=row
for j=row+1 to uBound(arrArray)
if int(arrArray(j)) < int(NewKeyValue) then
swap_pos=j
NewKeyValue=arrArray(j)
end if
next
if int(swap_pos) <> row then
arrArray(swap_pos)=int(StartingKeyValue)
arrArray(row)=int(NewKeyValue)
end if
end if
next
end sub
'
'----------------------------------------------------------------------------------
Dim RC,NLarray,CommaArray,lineNums:lineNums=Get90nums(6):RC=0
lineNums=replace(lineNums,"[","")
lineNums=replace(lineNums,"]",",")
NLarray=split(lineNums,",NL")
for y=0 to uBound(NLarray)
RC=RC+1
if RC=1 AND NOT y=uBound(NLarray) then Response.Write("<table width=""0%"" border=""1"" cellspacing=""0"" cellpadding=""10"">")&VBcrlf
CommaArray=split(NLarray(y),",")
CommaArray=ArrReOrder(CommaArray)
arrNumericalAsc CommaArray
for z=0 to uBound(CommaArray)
if z=0 then Response.Write(" <tr>")&VBcrlf
if CommaArray(z)=91 then CA_val=" " else CA_val=CommaArray(z)
Response.Write(" <td width=""30"" align=""center"">" & CA_val & "</td>")&VBcrlf
if z=uBound(CommaArray) then Response.Write(" </tr>")&VBcrlf
next
if RC=3 then Response.Write("</table>"&VBcrlf&VBcrlf&"<br><br>")&VBcrlf&VBcrlf:RC=0
next
%>
A: Once G90_BA has filled up with 90 unique random number in the range 1..90, the expression inStr(G90_BA,"["&G90_RN&"]")=0 will consistently fail. At that point, G90_TBL will no longer be increased. Possibly it is then stuck at a value less than 6, so it will never become equal to G90_TC, G90_CC is never set to true, and the loop never ends. An alternative explanation would be that it is taking increasingly long to find unused numbers to fill the last few slots in the list. In other words, it is just slow. Try it in a debugger and see.
BTW, it is OK if takes forever to explain the code's purpose, but it is not OK if it takes forever to explain what this code is doing. Please refactor.
A: Move the randomize out of the loop and notice the change to the G90_RN line, use CInt over Int:
randomize
do until G90_CC
G90_RN=CInt((90*Rnd())+1)
if inStr(G90_BA,"["&G90_RN&"]")=0 then
if G90_NC=5 then
G90_BA=G90_BA&"[91][91][91][91]NL" : G90_RC=G90_RC+1 : G90_NC=0
if G90_RC=3 then
G90_TBL=G90_TBL+1
G90_RC=0
end if
else
G90_BA=G90_BA&"["&G90_RN&"]" : G90_NC=G90_NC+1
end if
end if
if G90_TBL=G90_TC then G90_CC=true : Get90nums=G90_BA
loop
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,733 |
Q: CSS correto Div > li > span Tem uma parte no meu código que não estou conseguindo alterar o css.
<div class="AlterarCSS">
<ul>
<li>Nome - <span>Endereço Site<span></li>
<li>Nome - <span>Endereço Site<span></li>
<li>Nome - <span>Endereço Site<span></li>
</ul>
</div>
Tentei alterar assim:
.AlterarCSS > li > span{
background: yellow;
}
Agradeço ajuda
A: O simbolo > em um seletor CSS garante que o próximo elemento será um filho imediato, e, no seu caso, o filho imediato é o ul.
Para obter o resultado esperado, utilize o seguinte seletor:
.AlterarCSS > ul > li > span {
background: yellow;
}
<div class="AlterarCSS">
<ul>
<li>Nome - <span>Endereço Site<span></li>
<li>Nome - <span>Endereço Site<span></li>
<li>Nome - <span>Endereço Site<span></li>
</ul>
</div>
Para mais informações sobre seletores CSS, utilize o guia de referência CSS
A: Seletor filho
Um seletor filho tem como alvo um filho imediato de um elemento. O seletor filho consiste de um ou mais seletores simples separados por um sinal de maior ">". O elemento pai fica à esquerda do sinal ">", e é permitido deixar espaço em branco entre o elemento de combinação e os seletores.
A regra a seguir aplica-se a todos os elementos strong que sejam filhos de um elemento div:
div > strong { color:#f00; }
Somente elementos strong que sejam descendentes diretos do elemento div serão afetados por esta regra. Se houver qualquer outro elemento entre o elemento div e o elemento strong na árvore do documento, o seletor não se aplicará. No exemplo a seguir, somente "Texto um " será afetado pela regra:
<div>
<strong>Texto um</strong>
<p><strong>Texto dois</strong></p>
</div>
Veja Mais sobre Seletores
Logo, seu código deverá ficar assim:
.AlterarCSS > ul > li > span{
background: yellow;
}
<div class="AlterarCSS">
<ul>
<li>Nome - <span>Endereço Site<span></li>
<li>Nome - <span>Endereço Site<span></li>
<li>Nome - <span>Endereço Site<span></li>
</ul>
</div>
A: Retire o ">" do CSS tente assim:
.AlterarCSS li span{
background: yellow;
}
Ou faça dessa forma que define os filhos corretamente:
.AlterarCSS > ul > li > span{
background: yellow;
}
A: Experimente assim:
.AlterarCSS ul li span {
background: yellow;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,978 |
\section{Introduction}
The problem of simultaneous localization and mapping (SLAM) has a rich history over the past two decades, which is too broad to cover here, see e.g. \cite{dissanayake-2001,durrant2006simultaneous}. The extended Kalman filter (EKF) based SLAM (the EKF-SLAM) has played an important historical role, and is still used, notably for its ability to close loops thanks to the maintenance of correlations between remote landmarks.
The fact that the EKF-SLAM is inconsistent (that is, it returns a covariance matrix that is too optimistic, see e.g., \cite{Bar-Shalom}, leading to inaccurate estimates) was early noticed \cite{julier2001counter} and has since been explained in various papers \cite{castellanos2004limits,
bailey2006consistency,
huang2008analysis,
huang2010observability,
huang2007convergence,huang2011observability}. In the present paper we consider the inconsistency issues that stem from the fact that, as only relative measurements are available, the origin and orientation of the earth-fixed frame can never be correctly estimated, but the EKF-SLAM tends to ``think" it can estimate them as its output covariance matrix reflects an information gain in those directions of the state space. This lack of observability, and the poor ability of the EKF to handle it, is notably regarded as the root cause of inconsistency in \cite{huang2010observability,huang2011observability} (see also references therein). In the present paper we advocate the use of the Invariant (I)-EKF to prevent covariance reduction in directions of the state space where no information is available.
The Invariant extended Kalman filter (IEKF) is a novel methodology introduced in \cite{bonnabel-cdc,bonnabel-martin-salaun-cdc09} that consists in slightly modifying the EKF equations to have them respect the geometrical structure of the problem. Reserved to systems defined on Lie groups, it has been mainly driven by applications to localization and guidance, where it appears as a slight modification of the multiplicative EKF (MEKF), widely known and used in the world of aeronautics. It has been
proved to possess theoretical local convergence properties the EKF lacks in \cite{barrau2014invariant}, to be an improvement over the EKF in practice (see e.g., \cite{barczyk2013invariant,barczyk2015invariant,diemer2014invariant,martin2010generalized} and more recently \cite{barrau2014invariant} where the EKF is outperformed), and has been successfully implemented in industrial applications to navigation (see the patent \cite{brevet_alignement}).
In the present paper, we slightly generalize the IEKF framework, to make it capable to handle very general observations (such as range and bearing or bearing only observations), and we show how the derived IEKF-SLAM, a simple variant of the EKF-SLAM, allows remedying the inconsistency of EKF-SLAM stemming from the non-observability of the orientation and origin of the global frame.
\subsection{Links and differences with previous literature}
The issue of EKF-SLAM inconsistency has been the object of many papers, see
\cite{julier2001counter, castellanos2004limits,
bailey2006consistency,
huang2007convergence} to cite a few, where empirical evidence (through Monte-Carlo simulations) and theoretical explanations in various particular situations have been accumulated. In particular, the insights of \cite{bailey2006consistency,
huang2007convergence} have been that the orientation uncertainty is a key feature in the inconsistency. The article \cite{huang2007convergence}, in line with \cite{julier2001counter, castellanos2004limits,martinel,
bailey2006consistency}, also underlines the importance of the linearization process, as linearizing about the true trajectory solves the inconsistency issues, but is impossible to implement in practice as the true state is unknown. It derives a relationship that should hold between various Jacobians appearing in the EKF equations when they are evaluated at the current state estimate to ensure consistency.
A little later, the works of G.P. Huang, A.I. Mourikis, and S. I. Roumeliotis \cite{huang2008analysis,
huang2010observability,huang2011observability} have provided a sound theoretical analysis of the EKF-SLAM inconsistency as caused by the EKF inability to correctly reflect the three unobservable degrees of freedom (as an overall rotation and translation of the global reference frame leave all the measurements unchanged). Indeed, the filter tends to erroneously acquire information along the directions spanned by those unobservable transformations. To remedy this problem, the above mentioned authors have proposed various solutions, the most advanced being the Observability Constrained (OC)-EKF. The idea is to pick a linearization point that is such that the unobservable subspace ``seen" by the EKF system model is of appropriate dimension, while minimizing the expected errors of the linearization points.
Our approach, that relies on the IEKF, provides an interesting alternative to the OC-EKF, based on a quite different route. Indeed, the rationale is to apply the EKF methodology, but using alternative estimation errors to the standard linear difference between the estimate and the true state. Any non-linear error that reflects a discrepancy between the true state and the estimate, necessarily defines a local frame around any point, and the idea underlying the IEKF amounts to write the Kalman Jacobians and covariances in this frame. We notice and prove here that an alternative nonlinear error defines a local frame where the unobservable subspace is \emph{everywhere} spanned by the same vectors. Using this local frame at the current estimate to express Kalman's covariance matrix will be shown to ensure the unobservable subspace ``seen" by the EKF system model is \emph{automatically} of appropriate dimension.
We thus obtain an EKF variant which automatically comes with consistency properties. Moreover, we relate unobservability to the inverse of the covariance matrix (called information matrix) rather than on the covariance matrix itself, and we derive guarantees of information decrease over unobservable directions. Contrarily to the OC-EKF, and as in the standard EKF, we use here the latest, and thus best, state estimate as the linearization point to compute the filter Jacobians.
In a nutshell, whereas the key fact for the analysis of \cite{huang2010observability} is that the choice of the linearization point affects the observability properties of the linearized state error system of the EKF, the key fact for our analysis is that the choice of the error variable has similar consequences. Theoretical results and simulations underline the relevance of the proposed approach.
Robot-centric formulations such as \cite{castellanos}, and later \cite{Guerreiro,Lourenco} are promising attempts to tackle unobservability, but they unfortunately lack convenience as the position of all the landmarks must be revised during the propagation step, so that the landmarks' estimated position becomes in turn sensitive to the motion sensor's noise. They do not provably solve the observability issues considered in the present paper, and it can be noted the OC-EKF has demonstrated better experimental performance than the robocentric mapping filter, in \cite{huang2010observability}. In particular, the very recent papers \cite{Guerreiro,Lourenco} propose to write the equations of the SLAM in the robot's frame under a constant velocity assumption. Using an output injection technique, those equations become linear, allowing to prove global asymptotic convergence of any linear observer for the corresponding deterministic linear model. This is fundamentally a deterministic approach and property, and as the matrices appearing in the obtained linear model are functions of the observations, the behavior of the filter is not easy to anticipate in a noisy context: The observation noise thus corrupts the very propagation step of the filter.
Some recent papers also propose to improve consistency through local map joining, see \cite{Zhao} and references therein. Although appealing, this approach is rather oriented towards large-scale maps, and requires the existence of local submaps. But when using submap joining algorithm, ``inconsistency in even one
of the submaps, leads to an inconsistent global map" \cite{huang2008submap}. This approach may thus prove complementary, if the IEKF SLAM proposed in the present paper is used to build consistent submaps. Note that, the IEKF SLAM can also be readily combined with other measurements such as the GPS, whereas the submap approach is tailored for pure SLAM.
From a methodology viewpoint, it is worth noting our approach does {not} bring to bear estimation errors written in a robot frame, as \cite{castellanos,Guerreiro,Lourenco,Zhao}. Although based on symmetries as well, the estimation errors we use are slightly more complicated.
Finally, nonlinear optimization techniques have become popular for SLAM recently, see e.g., \cite{Dellaert} as one of the first papers. Links between our approach, and those novel methods are discussed in the paper's conclusion.
\subsection{Paper's organization}
The paper is organized as follows. In Section \ref{Sec:1}, the standard EKF equations and EKF-SLAM algorithm are reviewed. In Section \ref{Sec:2} we recall the problem that neither the origin nor the orientation of the global frame are observable, but the EKF-SLAM systematically tends to ``think" it observes them, which leads to inconsistency. In Section \ref{Sec:22} we introduce the IEKF-SLAM algorithm. In Section \ref{Sec:3} we show how the linearized model of the IEKF always correctly captures the considered unobservable directions. In Section \ref{sect::tools} we derive a property of the covariance matrix output by the filter that can be interpreted in terms of Fisher information. In Section \ref{Sec:4} simulations support the theoretical results and illustrate the benefits of the proposed algorithm. Finally, the IEKF theory of \cite{barrau2014invariant} is briefly recapped in the appendix, and the IEKF SLAM shown to be an application of this theory indeed. The equations of the IEKF SLAM in 3D are then also derived applying the general theory.
\section{The EKF-SLAM algorithm}
\label{Sec:1}
\subsection{Statement of the general standard EKF equations}\label{gene:sec}
Consider a general dynamical system in discrete time with state $X_n \in \mathbb{R}^N$ associated to a sequence of observations $(Y_n)_{n \geqslant 0}\in{\mathbb R}^p$. The equations are as follows:
\begin{equation}
\label{eq::general_dynamical_system}
X_{n} = f(X_{n-1},u_n,w_n),
\end{equation}
\begin{equation}\label{eq::general_dynamical_system_output}
Y_n = h(X_n)+V_n,
\end{equation}
where $f$ is the function encoding the evolution of the system, $w_n$ is the process noise, $u_n$ an input, $h$ the observation function and $V_n$ the measurement noise.
The EKF propagates the estimate $\hat X_{n-1|n-1}$ obtained after the observation $Y_{n-1}$, through the deterministic part of \eqref{eq::general_dynamical_system}:
\begin{equation}\label{eq::obs_gen}
\hat X_{n|n-1} = f(\hat X_{n-1|n-1},u_n,0)\end{equation}
The update of $\hat X_{n |n-1}$ using the new observation $Y_n$ is based on the first-order approximation of the non-linear system \eqref{eq::general_dynamical_system}, \eqref{eq::general_dynamical_system_output} around the estimate $\hat X_n $, with respect to the estimation errors $e_{n-1|n-1},e_{n|n-1}$ defined as:
\begin{align}\label{err:elf}
e_{n-1|n-1} = X_{n-1} - \hat X_{n|n-1},\quad e_{n|n-1} = X_n - \hat X_{n|n-1}
\end{align}
Using the Jacobians $F_n=\dv{f}{X}(\hat X_{n-1|n-1},u_n,0)$, $G_n=\dv{f}{w}(\hat X_{n-1|n-1},u_n,0)$, and $H_n=\dv{h}{X}(\hat X_{n|n-1})$, the combination of equations \eqref{eq::general_dynamical_system}, \eqref{eq::general_dynamical_system_output} and \eqref{eq::obs_gen} yields the following first-order expansion of the error system
\begin{align}
e_{n|n-1} &= F_n e_{n-1|n-1} + G_n w_n,\label{def:A_linear}\\
Y_n - h(\hat X_{n|n-1}) &= H_n e_{n|n-1} + V_n,\label{def:H_linear}
\end{align}
where the second order terms, that is, terms of order $O\left(\norm{ e}^2,\norm{w}^2, \norm{e} \norm{w}\right) $ have been removed according to the standard way the EKF handles non-additive noises in the model (see e.g., \cite{Stengel} p. 386). Using the linear Kalman equations with $F_n,G_n,H_n,$ the gain $K_n$ is computed, and letting $z_n=Y_n - h(\hat X_{n|n-1})$, an estimate $e_{n|n}=K_nz_n$ of the error $X_n-\hat X_{n|n-1}$ accounting for the observation $Y_n$ is computed, along with its covariance matrix $P_{n|n}$. The state is updated accordingly:
\begin{equation}
\label{eq::update_linear}
\hat X_{n|n} = \hat X_{n|n-1} +K_nz_n
\end{equation}
The detailed equations are recalled in Algorithm \ref{algo::EKF}. The assumption underlying the EKF is that through first-order approximations \emph{of the state error} evolution, the linear Kalman equations allow computing a Gaussian approximation of the error $e_{n|n}\sim\mathcal N(0,P_{n|n})$ after each measurement, yielding an approximation of the sought density $\mathbb P(X_{n}|Y_1,\cdots,Y_n)\approx \mathcal N( \hat X_{n|n},P_{n|n})$. However, the linearizations involved induce inevitable approximations that may lead the filter to inconsistencies and sometimes even divergence.
\begin{algorithmic}
\begin{algorithm}
\caption{Extended Kalman Filter (EKF)}
\label{algo::EKF}
\STATE{Choose an initial uncertainty matrix $P_0$ and estimate $\hat X_0$}
\LOOP
\STATE Define $F_n, G_n$ and $H_n$ through \eqref{def:A_linear} and \eqref{def:H_linear}.
\STATE Define $Q_n$ as $\Cov(w_n)$ and $R_n$ as $\Cov(V_n)$.
\STATE \textbf{Propagation}
\STATE $\hat X_{n|n-1} = f \left( \hat X_{n-1|n-1},u_n,0 \right)$
\STATE $P_{n|n-1} = F_n P_{n-1|n-1} F_n^T + G_n Q_n G_n^T$
\STATE \textbf{Update}
\STATE $ z_n=Y_n- h \left( \hat X_{n|n-1} \right)$
\STATE $ S_n = H_n P_{n|n-1} H_n^T + R_n $,
\STATE $
K_n = P_{n|n-1} H_n^T S_n^{-1}$
\STATE $P_{n|n} = [I-K_n H_n]P_{n|n-1} $
\STATE{ $ \hat X_{n|n} = \hat X_{n|n-1} +K_n z_n$}
\ENDLOOP
\end{algorithm}
\end{algorithmic}
\subsection{The considered SLAM problem}
\label{sect::SLAM_problem}
For simplicity's sake let us focus on the standard ``steered" bicycle (or unicycle) model \cite{durrant1996autonomous}. The state is defined as:
\begin{equation}\label{state:def}
X_n = \left( \theta_n, x_n, p_n^1, \ldots, p_n^K \right),
\end{equation}
where $\theta_n \in \mathbb{R}$ denotes the heading, $x_n \in {\mathbb R}^2$ the 2D position of the robot/vehicle, $p_n^j \in {\mathbb R}^2$ the position of unknown landmark $j$ (landmarks or synonymously features, constitute the map). The equations of the model are:
\begin{equation}
\label{eq::bicycle_dyn}
\begin{aligned}
\theta_n & = \theta_{n-1} + \omega_n+w_n^{\omega}, \\
x_n & = x_{n-1} + R(\theta_{n-1}) (v_n+ w_n^v), \\
p_n^j & = p_{n-1}^j,\quad 1\leq j\leq K
\end{aligned}
\end{equation}
where $\omega_n \in \mathbb{R}$ denotes the odometry-based estimate of the heading variation of the vehicle, $v_n \in \mathbb{R}^2$ the odometry-based indication of relative shift, $w_n^{\omega}$ and $w_n^v$ their associated noises, and $R(\theta)$ is the matrix encoding a rotation of angle $\theta$:
$$
R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}.
$$Note that a forward Euler discretization of the continuous time well-known unicycle equations leads to $v_n \in \mathbb{R}^2$ having its second entry null. More sophisticated integration methods or models including side slip may yet lead to non-zero values of both entries of $v_n$ so we opt for a more general model with $v_n \in \mathbb{R}^2$.
The covariance matrix of the noises will be denoted by
\begin{equation}
\label{eq::Qn}
Q_n = \Cov \begin{pmatrix} w_n^{\omega} \\ w_n^v\\0_{2K\times 1} \end{pmatrix} = \mathbb{E} \left(\begin{pmatrix} w_n^{\omega} \\ w_n^v \\0_{2K\times 1}\end{pmatrix} \begin{pmatrix} w_n^{\omega} \\ w_n^v \\0_{2K\times 1}\end{pmatrix}^T \right)\in{\mathbb R}^{l\times l}
\end{equation}
with $l=3+2K$. A general landmark observation in the robot's frame reads:
\begin{equation}
\label{eq::bicycle_obs}
Y_n = \begin{pmatrix}
\tilde h \left[ R(\theta_n)^T \left( p^1-x_n \right) \right]+V_n^1 \\
\vdots \\
\tilde h \left[ R(\theta_n)^T \left( p^K-x_n \right) \right]+V_n^K
\end{pmatrix}
\end{equation}
where $Y_n \in {\mathbb R}^{2 K}$ (or $ {\mathbb R}^{ K}$ for monocular visual SLAM) is the observation of the features at time step $n$, and $V_n$ the observation noise, and $\tilde h$ is any function.
\begin{rem}Only a subset of the features is actually observed at time $n$. However, to simplify the exposure of the filters' equations, we systematically assume in the sequel that all features are observed.\end{rem}
We let the output noise covariance matrix be
\begin{equation}
\label{eq::Rn}
R_n = \Cov \begin{pmatrix} V_n^1 \\ \vdots \\ V_n^K \end{pmatrix}.
\end{equation}
\begin{rem}
Note that, the observation model \eqref{eq::bicycle_obs} encompasses the usual range and bearing observations used in the SLAM problem
by letting $\tilde h \begin{pmatrix}
y_1 \\ y_2 \end{pmatrix} = \left(
\sqrt{y_1^2 + y_2^2} ,
\arctan 2 \left( y_2, y_1 \right)
\right)
$. If we choose instead the one dimensional observation $\tilde h \begin{pmatrix}
y_1 \\ y_2
\end{pmatrix} =
\arctan 2 \left( y_2, y_1 \right)
$ we recover the 2D monocular SLAM measurement. Note also we do not provide any specific form for the noise in the output: this is because the properties we are about to prove are related to the observability and thus only depend on the deterministic part of the system, so they are in fact totally insensitive to the way the noise enters the system.
\end{rem}
\subsection{The EKF-SLAM algorithm}
We merely apply here the methodology of EKF to the SLAM problem described in Section \ref{sect::SLAM_problem}. The first-order expansions \eqref{def:A_linear}, \eqref{def:H_linear} applied to equations \eqref{eq::bicycle_dyn}, \eqref{eq::bicycle_obs} yield:
\begin{equation}
\begin{gathered}
\label{eq::F_H_linear_SLAM}
F_n = \begin{pmatrix} 1 & 0_{1,2} & 0_{1,2K} \\
R \left(\hat \theta_{n-1|n-1} \right) J v_n^T & I_2 & 0_{2,2K} \\
0_{2K,1} & 0_{2K,2} & I_{2K}
\end{pmatrix},\\
G_n = \begin{pmatrix}
1 & 0_{1,2} & 0_{1,2K} \\
0_{2,1} & R \left(\hat \theta_{n-1|n-1} \right) & 0_{2,2K} \\
0_{2K,1} & 0_{2K,2} & 0_{2K,2K}
\end{pmatrix}, ~
H_n =\begin{pmatrix} \nabla h^1 \cdot H_n^1 \\ \vdots \\ \nabla h^K \cdot H_n^K \end{pmatrix} \\H_n^k =\begin{pmatrix} - J R \left( \tilde\theta \right)^T \left( \hat{p}_{n|n-1}^k-\hat{x}_{n|n-1} \right) & -R \left(\tilde\theta \right)^T & R \left(\tilde\theta \right)^T \end{pmatrix},
\end{gathered}
\end{equation}
with $\tilde\theta=\hat{\theta}_{n|n-1}$, $J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and $\nabla h^k$ denotes the Jacobian of $\tilde h$ computed at $ R \left( \hat \theta_{n|n-1} \right)^T \left[ \hat p_{n|n-1}^k - \hat x_{n|n-1} \right] \in {\mathbb R}^2$. The obtained EKF-SLAM algorithm is recaped in Algorithm \ref{algo::EKF_SLAM_linear}.
\begin{algorithmic}
\begin{algorithm}
\caption{EKF SLAM}
\label{algo::EKF_SLAM_linear}
\STATE{Choose an initial uncertainty matrix $P_0$ and estimate $\hat X_0$}
\LOOP
\STATE Define $F_n, G_n$ and $H_n$ as in \eqref{eq::F_H_linear_SLAM}.
\STATE Define $Q_n$, $R_n$ as in \eqref{eq::Qn} and \eqref{eq::Rn}.
\STATE \textbf{Propagation}
\STATE{ $\hat \theta_{n|n-1} = \hat \theta_{n-1|n-1} + \omega_n$}
\STATE{ $\hat x_{n|n-1} = \hat x_{n-1|n-1} + R \left( \hat \theta_{n-1|n-1} \right) v_n$}
\STATE{ $\hat p_{n|n-1}^j = \hat p_{n-1|n-1}^j$} for all $1\leq j\leq K$
\STATE $P_{n|n-1} = F_n P_{n-1|n-1} F_n^T + G_n Q_n G_n^T$
\STATE \textbf{Update}
\STATE $ z_n=Y_n- \begin{pmatrix}
\tilde h \left[ R(\hat \theta_{n|n-1})^T \left( \hat p^1_{n|n-1}- \hat x_{n|n-1} \right) \right] \\
\vdots \\
\tilde h \left[ R(\hat \theta_{n|n-1})^T \left( \hat p^K_{n|n-1}- \hat x_{n|n-1} \right) \right]
\end{pmatrix}$
\STATE $ S_n = H_n P_{n|n-1} H_n^T + R_n $,
\STATE $
K_n = P_{n|n-1} H_n^T S_n^{-1}$
\STATE $P_{n|n} = [I-K_n H_n]P_{n|n-1} $
\STATE{$ \hat X_{n|n} = \hat X_{n|n-1} + K_n z_n$ }
\ENDLOOP
\end{algorithm}
\end{algorithmic}
\section{Observability issues and consistency of the EKF}\label{Sec:2}
In this section we come back to the general framework \eqref{eq::general_dynamical_system}, \eqref{eq::general_dynamical_system_output}.
The standard issue of observability \cite{gauthier1994observability} is fundamentally a deterministic notion so the noise is systematically turned off.
\begin{dfn}[Unobservable transformation]
\label{def::non_obs}
We say a transformation $\phi: \mathbb{R}^N \rightarrow {\mathbb R}^N$ of the system \eqref{eq::general_dynamical_system}-\eqref{eq::general_dynamical_system_output} is unobservable if for any initial conditions $X_0^1\in{\mathbb R}^N$ and $X_0^2 = \phi \left(X_0^1\right)$ the induced solutions of the dynamics \eqref{eq::general_dynamical_system} with noise turned off, i.e., $ X_n=f(X_{n-1},u_n,0)$ yield the same output at each time step $n \geqslant 0$, that is:
$$h(X_n^1)=h(X_n^2).$$
\end{dfn}
It concretely means that (with all noises turned off) if the transformation is applied to the initial state then none of the observations $Y_n$ are going to be affected. As a consequence, there is no way to know this transformation has been applied. In line with \cite{lee2006observability,huang2008analysis,
huang2010observability} we will focus here on the observability properties of the linearized system. To that end we define the notion of non-observable (or unobservable) shift which is an infinitesimal counterpart to Definition \ref{def::non_obs}, and is strongly related to the infinitesimal observability \cite{gauthier1994observability}:
\begin{dfn}[Unobservable shift]
\label{def::non_obs_first_order}
Let $(X_n)_{n \geqslant 0}$ denote a solution of \eqref{eq::general_dynamical_system} with noise turned off. A vector $\delta X_0 \in \mathbb{R}^N$ is said to be an unobservable shift of \eqref{eq::general_dynamical_system}-\eqref{eq::general_dynamical_system_output} around $X_0$ if:
$$
\forall n \geqslant 0, \quad H_n \delta X_n = 0,
$$
where $H_n$ is the linearization of $h$ at $X_n$ and where $\delta X_n$ is the solution at $n$ of the linearized system $\delta X_n = F_n \delta X_{n-1}$ initialized at $\delta X_0$, with $F_n $ denoting the Jacobian matrix of $f(\cdot, u_n,0)$ computed at $ X_{n-1}$.
In other words (see e.g. \cite{huang2010observability}), for all $n\geq 0$, $\delta X_n$ lies in the kernel of the observability matrix between steps $0$ and $n$ associated to the linearized error-state system model, i.e., $\delta X_0^T[H_0^T;(H_{1}F_1)^T;\cdots ;(H_{n}F_{n}\cdots F_1)^T]=0$.
\end{dfn}
The interpretation is as follows: consider another initial state shifted from $X_0$ to $X_0 + \delta X_0$. Saying that $\delta X_0$ is unobservable means no difference on the sequence of observations up to the first order could be detected between both trajectories. Formally, this condition reads: $h(X_n + \delta X_n)=h(X_n)+\circ \left( \delta X_n \right)$, i.e., $H_n \delta X_n=0$. An estimation method conveying its own estimation uncertainty as the EKF, albeit based on linearizations, should be able to detect such directions and to reflect that accurate estimates along such directions are beyond reach.
\subsection{Considered unobservable shifts}\label{direc:sec}
In the present paper we consider unobservability corresponding to the impossibility to observe the position and orientation of the global frame \cite{lee2006observability,huang2008analysis}. The corresponding shifts have already been derived in the literature.
\begin{prop}
\label{prop::first_order_rotations_prelim}\cite{huang2010observability}
Let $\hat X = \begin{pmatrix}
\hat \theta , \hat x , \hat p
\end{pmatrix} $ be an estimate of the state. Only one feature is considered, the generalization of the proposition to several features is trivial. The first-order perturbation of the estimate corresponding to an infinitesimal rotation of angle $\delta \alpha$ of the global frame consists of the shift $ \begin{pmatrix}
1 \\ J \hat x \\ J \hat p
\end{pmatrix} \delta \alpha,$ with $J=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. In the same way, the first-order perturbation of the estimate corresponding to an infinitesimal translation of the global frame of vector $\delta u\in{\mathbb R}^2$ consists of the shift $\begin{pmatrix} 0 \\ \delta u \\ \delta u \end{pmatrix}$.
\end{prop}
\begin{proof}
When rotating the global frame the heading becomes:
$$
\hat \theta \rightarrow \hat \theta + \delta \alpha.
$$
The position of the robot becomes:
$$
\hat x \rightarrow
\begin{pmatrix}
\cos(\delta \alpha) & -\sin(\delta \alpha) \\
\sin(\delta \alpha) & \cos(\delta \alpha)
\end{pmatrix} \hat x \approx \hat x+\delta \alpha \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \hat x.
$$
The position of the feature becomes:
$$
\hat p \rightarrow
\begin{pmatrix}
\cos(\delta \alpha) & -\sin(\delta \alpha) \\
\sin(\delta \alpha) & \cos(\delta \alpha)
\end{pmatrix} \hat p \approx \hat p +\delta \alpha \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \hat p.
$$
Stacking these results we obtain the first-order variation of the full state vector (regarding the rotation only, the effect of infinitesimal translation being trivial to derive):
\begin{align}\label{stacking}
\begin{pmatrix} \hat \theta \\ \hat x \\ \hat p \end{pmatrix}
\rightarrow \begin{pmatrix} \hat \theta \\ \hat x \\ \hat p \end{pmatrix} +
\begin{pmatrix} 1 \\ J \hat x \\ J \hat p \end{pmatrix} \delta \alpha.
\end{align}
\end{proof}
\begin{prop}
\label{prop::first_order_rotations}\cite{huang2010observability}
The shifts of Proposition \ref{prop::first_order_rotations} that correspond to infinitesimal rotations, are unobservable shifts of \eqref{eq::bicycle_dyn}-\eqref{eq::bicycle_obs} in the sense of Definition \ref{def::non_obs_first_order}.
\end{prop}
The intuitive explanation is clear \cite{huang2010observability}: ``if the robot and landmark positions are shifted equally along those vectors, it will not be possible to distinguish the shifted position from the original one through the measurements.''
\subsection{Inconsistency of the EKF}
\label{sect::EKS_SLAM8inconsistency}
This section recalls using the notations of the present paper, a result of \cite{huang2008analysis}. It shows the infinitesimal rotations defined in Proposition \ref{prop::first_order_rotations} are not, in general, unobservable shifts of the system linearized about the trajectory estimated by the EKF. Indeed, applying Definition \ref{def::non_obs_first_order} to \eqref{eq::F_H_linear_SLAM} in the case of a single feature (the generalization being straightforward) with $ \delta X_0 = ( 1 , J \hat x_{0|0},J \hat p_{0|0} )^T$ and $\delta X_n =F_n\cdots F_1\delta X_0 $ yields the condition for an infinitesimal rotation of the initial state to be unobservable for the linearized system. This condition writes $H_n\delta X_n\equiv0$ and boils down to have for any $n >0$ (see \cite{huang2008analysis}):
$$
\nabla h_n \cdot J \cdot R(\theta_{n|n-1})^T \left[ - \left( \hat p_{n|n-1} - \hat p_{0|0} \right) + \sum_{i=1}^{n-1} \left( \hat x_{i|i} - \hat x_{i|i-1} \right) \right]=0
$$
where $ \nabla h_n$ is the Jacobian of $\tilde h$ computed at $R \left( \hat \theta_{n|n-1} \right)^T \left( \hat p_{n|n-1} - \hat x_{n|n-1} \right)$. For example, if $\tilde h$ is invertible the condition boils down to
\begin{align}
\forall n>0,~~ \left[ - \left( \hat p_{n|n-1} - \hat p_{0|0} \right) + \sum_{i=1}^{n-1} \left( \hat x_{i|i} - \hat x_{i|i-1} \right) \right]=0.\label{never_verified}
\end{align}
We see the quantities involved are the updates of the state. As they depend on the noise, there is a null probability for the condition to be respected, and it is always violated in practice.
But the point of the present paper is to show that the problem is related to the (arbitrary in a non-linear context) choice to represent the estimation error as the linear difference $X - \hat X$, not to an inconsistency issue inherent to EKF-like methods applied to SLAM. By devising an EKF-SLAM based on another estimation error variable, which in some sense amounts to change coordinates, the false observability problem can be corrected.
The qualitative reason why this is sufficient is related to the basic cause of false observability: a given fixed shift may or may not be observable depending on the linearization point $\hat X$, as proved by Proposition \ref{prop::first_order_rotations}. It turns out that the latter property is not inherently related to the SLAM problem: it is in fact a mere consequence of the errors' definition \eqref{err:elf}. Defining those errors otherwise can dramatically modify the condition \eqref{never_verified}. This is the object of the remainder of this article.
\section{A novel EKF-SLAM algorithm}\label{Sec:22}
Building upon the theory of the Invariant (I)EKF on matrix Lie groups, as described and studied in \cite{barrau2014invariant}, we introduce in this section a novel IEKF for SLAM. In Appendix \ref{primer}-\ref{gen:IEKF} the general theory of the IEKF is recalled and slightly extended to account for the very general form of output \eqref{eq::bicycle_obs}, and the algorithm derived herein is shown to be a direct application of the theory. To spare the reader a study of the Lie group based theory, we attempt to explain in simple terms the IEKF methodology on the particular SLAM example throughout the present section.
Consider the model equations \eqref{eq::bicycle_dyn} with state $X_n$ given by \eqref{state:def}. Exactly as the EKF, the IEKF propagates the estimated state obtained after the observation $Y_{n-1}$ of \eqref{eq::bicycle_obs} through the deterministic part of \eqref{eq::bicycle_dyn} i.e., $\hat \theta_{n|n-1} = \hat \theta_{n-1|n-1} + \omega_n$, $\hat x_{n|n-1} = \hat x_{n-1|n-1} + R \left( \hat \theta_{n-1|n-1} \right) v_n$, $\hat p_{n|n-1}^j = \hat p_{n-1|n-1}^j$ for all $1\leq j\leq K$.
To update the predicted state $\hat X_{n|n-1}$ using the observation $Y_n$ we use a first order Taylor expansion of the error system. But, \emph{instead of considering the usual state error $X-\hat X$}, we rather use the (linearized) estimation error defined as follows
\begin{equation}
\label{xi::non_linear_error}
\xi_{n|n-1} = \begin{pmatrix}
\theta_n - \hat \theta_{n|n-1} \\
x_n - \hat x_{n|n-1} -\left( \theta_n -{\hat \theta}_{n|n-1} \right)J \hat x_n\\
p_n^1- \hat p^1_{n|n-1}-\left( \theta_n -{\hat \theta}_{n|n-1} \right)J \hat p^1_{n|n-1}
\\
\vdots \\
p^K_n - \hat p^K_{n|n-1}-\left( \theta_n -{\hat \theta}_{n|n-1} \right)J \hat p^K_{n|n-1}
\end{pmatrix}
\end{equation}and $\xi_{n|n}$ is analogously defined. For close-by $X,\hat X$, this represents an error variable in the usual sense indeed, as $\xi=0$ if and only if $\hat X=X$. As in the standard EKF methodology, let us see how this \emph{alternative} estimation error is propagated through a first-order approximation of the error system. Using the propagation equations of the filter, and \eqref{eq::bicycle_dyn}, we find
\begin{equation}\begin{aligned}\label{first:linearized}\xi_{n|n-1}=\xi_{n-1|n-1}+\begin{pmatrix}w_n^\omega \\
- w_n^\omega J \hat x_{n-1|n-1}+R({\hat \theta}_{n-1|n-1}) w_n^v \\
- w_n^\omega J \hat p^1_{n-1|n-1}
\\
\vdots\\
- w_n^\omega J \hat p^K_{n-1|n-1} \end{pmatrix}
\end{aligned}\end{equation}
where terms of order $O(\norm{\xi_{n-1|n-1}}^2)$, $O( \norm{w_n^\omega}\norm{\xi_{n-1|n-1}})$, and $O( \norm{w_n^\omega}^2)$ have been neglected as in the standard the EKF handles non-additive noises \cite{Stengel}. To derive \eqref{first:linearized} we have used the equalities $\forall \theta,\hat\theta\in{\mathbb R}$, $w\in{\mathbb R}^2$:
\begin{align}
& R(\theta)w=R(\hat\theta)w+O(|\hat\theta-\theta|~\norm{w})\label{K1}
\\&R(\theta)-R(\hat\theta)-(\theta-\hat\theta)JR(\hat\theta)=O(|\theta-\hat\theta|^2)\label{K2}
\end{align}
Note that, the odometer outputs $\omega_n,v_n$ have miraculously vanished. This is in fact a characteristics - and a key feature - of the IEKF approach.
Let us now compute the first-order approximation of the observation error, using the alternative state error \eqref{xi::non_linear_error}. Define $H_n$ as the matrix, depending on $\hat X_{n|n-1}$ only, such that for all $\xi_{n|n-1} \in{\mathbb R}^{2K+3}$ defined by \eqref{xi::non_linear_error}, the innovation term
\begin{equation*}
\begin{pmatrix}
\tilde h \left[ R(\theta_n)^T \left( p^1-x_n \right) \right] \\
\vdots \\
\tilde h \left[ R(\theta_n)^T \left( p^K-x_n \right) \right]
\end{pmatrix}-\begin{pmatrix}
\tilde h \left[ R(\hat \theta_{n|n-1} )^T \left(\hat p^1_{n|n-1}-\hat x_{n|n-1} \right) \right] \\
\vdots \\
\tilde h \left[ R(\hat \theta_{n|n-1} )^T \left( \hat p^K_{n|n-1}-\hat x_{n|n-1} \right) \right]
\end{pmatrix}
\end{equation*}is equal to $H_n\xi_{n|n-1} +O(\norm{\xi_{n|n-1} }^2)$. Using that $R(\theta)^T(p-x)-R(\hat \theta)^T(\hat p-\hat x)=R(\theta)^T[(p-x)-R(\theta-\hat \theta)(\hat p-\hat x)]$, and $R(\theta)^T {\xi}=R(\hat \theta)^T {\xi}+O(\norm{\xi}^2)$ from \eqref{K1}, we see that $H_n$ is defined as in \eqref{eq::A_H_G_IEKF} below. Thus the linearized (first-order) system model with respect to alternative error \eqref{xi::non_linear_error} writes
\begin{equation}
\begin{aligned}\label{def:A_non_linear_Lie_2}
\xi_{n|n-1} &=F_n\xi_{n-1|n-1} +G_n w_n,\\
Y_n-h(\hat X_{n|n-1})&=H_n\xi_{n|n-1} +V_n
\end{aligned}
\end{equation}
with $w_n^T=(w_n^\omega,(w_n^v)^T,0_{1\times 2K})^T$, and
\begin{equation}
\label{eq::A_H_G_IEKF}
\begin{gathered}
F_n = I_{2K+3}, ~
G_n = \begin{pmatrix} 1 & 0_{1,2} & 0_{1,2K} \\
- J \hat x_{\tiny{n-1|n-1}} & R \left(\hat \theta_{n-1|n-1}\right) & 0_{2,2K} \\
-J \hat p^1_{n-1|n-1} & 0_2 & 0_{2,2K} \\
\vdots & \vdots & \vdots \\
-J \hat p^K_{n-1|n-1} & 0_2 & 0_{2,2K}
\end{pmatrix},\\
H_n = \begin{pmatrix}
\nabla \tilde h^1 \cdot R \left( \hat \theta_{n|n-1} \right)^T \begin{pmatrix} 0_{2,1} & -I_2 & I_2 & 0_{2,2(K-1)} \end{pmatrix} \\
\nabla \tilde h^2 \cdot R \left( \hat \theta_{n|n-1} \right)^T \begin{pmatrix} 0_{2,1} & -I_2 & 0_{2,2} & I_2 & 0_{2,2(K-2)} \end{pmatrix} \\
\vdots \\
\nabla \tilde h^K \cdot R \left( \hat \theta_{n|n-1} \right)^T \begin{pmatrix} 0_{2,1} & -I_2 & 0_{2,2(K-1)} & I_2 \end{pmatrix}
\end{pmatrix},
\end{gathered}
\end{equation}
where $\nabla \tilde h^k$ is the Jacobian of $\tilde h$ computed at $R \left( \hat \theta_{n|n-1} \right)^T \left( \hat p_{n|n-1}^k - \hat x_{n|n-1} \right)$.
As in the standard EKF methodology, the matrices $F_n,G_n,H_n$ allow to compute the Kalman gain $K_n$ and covariance $P_{n}$. Letting $z_n$ be the standardly defined innovation (see Algorithm \ref{algo::IEKF_SLAM} just after ``Update"), $\xi_{n|n}=K_nz_n$ is an estimate of the linearized error $\xi_{n|n-1}$ accounting for the observation $Y_n$, and $P_{n|n}$ is supposed to encode the dispersion $\mathbb E(\xi_{n|n}\xi_{n|n}^T)$.
The final step of the standard EKF methodology is to update the estimated state $\hat X_{n|n-1}$ thanks to the estimated linearized error $\xi_{n|n}=K_nz_n$. There is a small catch, though: $\xi$ being not anymore defined as a mere difference $X-\hat X$, simply adding $\xi_{n|n}$ to $\hat X_{n|n}$ would not be appropriate. The most natural counterpart to \eqref{eq::update_linear} in our setting, would be to choose for $\hat X_{n|n}$ the values of $(\theta_n,x_n,\cdots, p_n^1,p_n^K)$ making the right member of \eqref{xi::non_linear_error} equal to the just computed $\xi_{n|n}$. However, the IEKF theory recalled in Appendix \ref{gen:IEKF}, suggests an update that amounts to the latter to the first order, but whose non-linear structure ensures better properties \cite{barrau2014invariant}. Thus, the state is updated as follows $\hat X_{n|n}=\varphi(\xi_{n|n},\hat X_{n|n-1})=\varphi(K_nz_n,\hat X_{n|n-1})$, with $\varphi$ defined by
\begin{equation}
\begin{aligned}
\varphi\bigl(\begin{pmatrix}\delta \theta \\ \delta x \\ \delta p^1 \\ \vdots \\ \delta p^K\end{pmatrix},\begin{pmatrix}\hat \theta \\ \hat x \\ \hat p^1 \\ \cdots \\ \hat p^K\end{pmatrix}\bigr)= \begin{pmatrix} \hat \theta +\delta {\theta} \\ R(\delta \theta) \hat x+ B(\delta \theta) \delta x \\ R(\delta \theta) \hat p^1 + B(\delta \theta) \delta p^1 \\ \vdots \\ R(\delta \theta) \hat p^K + B(\delta \theta) \delta p^K \end{pmatrix}\label{eq::exp_SE2}\end{aligned}
\end{equation}where $B(\alpha) = \begin{pmatrix} \frac{ \sin \left( \alpha \right) }{ \alpha } & - \frac{ 1 - \cos \left( \alpha \right) }{ \alpha } \\ \frac{1-\cos \left( \alpha \right) }{ \alpha } & \frac{ \sin \left( \alpha \right) }{ \alpha } \end{pmatrix}$. Algorithm \ref{algo::IEKF_SLAM} recaps the various steps of the IEKF SLAM.
\begin{algorithmic}
\begin{algorithm}
\caption{IEKF SLAM}
\label{algo::IEKF_SLAM}
\STATE{The state is defined by $X=(\theta,x^T,(p^1)^T,\cdots,(p^K)^T))\in{\mathbb R}^{3+2K}$. Pick an initial uncertainty matrix $P_0$ and estimate $\hat X_0$.}
\LOOP
\STATE Define $F_n, G_n$ and $H_n$ as in \eqref{eq::A_H_G_IEKF}.
\STATE Define $Q_n$, $R_n$ as in \eqref{eq::Qn} and \eqref{eq::Rn}.
\STATE \textbf{Propagation}
\STATE{ $\hat \theta_{n|n-1} = \hat \theta_{n-1|n-1} + \omega_n$}
\STATE{ $\hat x_{n|n-1} = \hat x_{n-1|n-1} + R \left( \hat \theta_{n-1|n-1} \right) v_n$}
\STATE{ $\hat p_{n|n-1}^j = \hat p_{n-1|n-1}^j$} for all $1\leq j\leq K$
\STATE $P_{n|n-1} = F_n P_{n-1|n-1} F_n^T + G_n Q_n G_n^T$
\STATE \textbf{Update}
\STATE $ z_n=Y_n- \begin{pmatrix}
\tilde h \left[ R(\hat \theta_{n|n-1})^T \left( \hat p^1_{n|n-1}- \hat x_{n|n-1} \right) \right] \\
\vdots \\
\tilde h \left[ R(\hat \theta_{n|n-1})^T \left( \hat p^K_{n|n-1}- \hat x_{n|n-1} \right) \right]
\end{pmatrix}$
\STATE $ S_n = H_n P_{n|n-1} H_n^T + R_n $,
\STATE $
K_n = P_{n|n-1} H_n^T S_n^{-1}$
\STATE $P_{n|n} = [I-K_n H_n]P_{n|n-1} $
\STATE{Use \eqref{eq::exp_SE2} to compute $ \hat X_{n|n}=\varphi(K_nz_n,\hat X_{n|n-1})$. }
\ENDLOOP
\end{algorithm}
\end{algorithmic}
\section{Remedying EKF SLAM consistency}\label{Sec:3}
In this section we show the infinitesimal rotations and translations of the global frame are unobservable shifts in the sense of Definition \ref{def::non_obs_first_order} regardless of the linearization points used to compute the matrices $F_n$ and $H_n$ of eq. \eqref{eq::A_H_G_IEKF}, a feature in sharp contrast with the usual restricting condition \eqref{never_verified} on the linearization points. In other words we show that infinitesimal rotations and translations of the global frame are always unobservable shifts of the system model \emph{linearized} with respect to error \eqref{xi::non_linear_error} regardless of the linearization point, a feature in sharp contrast with previous results (see Section \ref{sect::EKS_SLAM8inconsistency} and references therein).
\subsection{Main result}
We can consider only one feature ($K=1$) without loss of generality. The expression of the linearized system model has become much simpler, as the linearized error has the remarkable property to remain constant during the propagation step in the absence of noise, since $F_n = I_{3+2K}$ in \eqref{def:A_non_linear_Lie_2}-\eqref{eq::A_H_G_IEKF}.
First, let us derive the impact of first-order variations stemming from rotations and translations of the global frame on the error as defined by \eqref{xi::non_linear_error}, that is, an error of the following form
\begin{equation}
\label{eq::non_linear_error_f}
\xi=\begin{pmatrix}(\theta -\hat\theta)\\x-\hat x- (\theta-\hat \theta) J\hat x\\p-\hat p- (\theta-\hat \theta) J\hat p\end{pmatrix},
\end{equation}
\begin{prop}
\label{prop::first_order_rotations_non_linear}
Let $\hat X = (
\hat \theta ,\hat x , \hat p )^T$ be an estimate of the state. The first-order perturbation of the \emph{linearized} estimation error defined by \eqref{eq::non_linear_error_f} around 0, corresponding to an \emph{infinitesimal} rotation of angle $\delta \alpha$ of the global frame, reads $ \begin{pmatrix}
1 \\ 0_{2,1} \\ 0_{2,1}
\end{pmatrix} \delta \alpha.$ In the same way, an \emph{infinitesimal} translation of the global frame with vector $\delta u\in{\mathbb R}^2$ implies a first-order perturbation of the error system \eqref{eq::non_linear_error_f} of the form $(0,\delta u,\delta u)^T.$
\end{prop}
\begin{proof}
According to Proposition \ref{prop::first_order_rotations_prelim}, an infinitesimal rotation by an angle $\delta\alpha\ll 1$ of the true state corresponds to the transformation $\theta\to\theta+\delta\alpha$. $x\to x+\delta\alpha Jx$ and $p\to p+\delta\alpha Jp$. Regarding $\xi$ of eq \eqref{eq::non_linear_error_f} it corresponds to the variation
$$
\xi\to\xi+
\begin{pmatrix}
\delta \alpha \\
0_{2,1} \\
0_{2,1}
\end{pmatrix}+O(\delta\alpha\norm{\xi})+O(\delta\alpha^2).
$$This direction of the state space is ``seen" by the \emph{linearized} error system as the vector $(\delta\alpha,0,0,0,0)^T$. Similarly, a translation of vector $\delta u$ of the global frame yields the transformation $ \theta\to \theta,~ x\to\ x+\delta u,~ p\to p+\delta u$. The effect on the linearized error $\xi$ of \eqref{eq::non_linear_error_f} is obviously the perturbation $(0,\delta u,\delta u)^T$ neglecting terms of order $\delta u\norm{\xi}$.
\end{proof}
We can now prove the first major result of the present article: the infinitesimal transformations stemming from rotations and translations of the gobal frame are unobservable shifts for the IEKF linearized model.
\begin{thm}
\label{SLAM:thm:obs}
Consider the SLAM problem defined by equations \eqref{eq::bicycle_dyn} and \eqref{eq::bicycle_obs}, and the IEKF-SLAM algorithm \ref{algo::IEKF_SLAM}. Let $\delta X_0$ denote a linear combination of infinitesimal rotations and translations $\delta X_0^R, \delta X_0^1, \delta X_0^2$ of the whole system defined as follows
$$
\delta X_0^R = \begin{pmatrix}
1 \\
0_{2,1} \\
0_{2,1}
\end{pmatrix},
\qquad
\delta X_0^1 = \begin{pmatrix}
0 \\
1 \\
0 \\
1 \\
0
\end{pmatrix},
\qquad
\delta X_0^2 = \begin{pmatrix}
0 \\
0 \\
1 \\
0 \\
1
\end{pmatrix}.
$$
Then $\delta X_0$ is an unobservable shift of the linearized system model \eqref{def:A_non_linear_Lie_2}-\eqref{eq::A_H_G_IEKF} of the IEKF SLAM in the sense of Definition \ref{def::non_obs_first_order}, and this whatever the sequence of true states and estimates $(X_n,\hat X_{n|n},\hat X_{n|n-1})$: the very structure of the IEKF is consistent with the considered unobservability.
\end{thm}
\begin{proof}
Note that Definition \ref{def::non_obs_first_order} involves a propagated perturbation $\delta X_n$, but as here $F_n$ is $I_5$: we have $\forall n>0, \delta X_n = \delta X_0$. Thus, the only point to check is:
$
H_n \left( \delta X_0 \right) = 0,
$
i.e., $\nabla \tilde h \cdot R \left( \hat \theta_{n|n-1} \right)^T \begin{pmatrix} 0_{2,1} & -I_2 & I_2 \end{pmatrix}\delta X_0= 0$. This is straightforward replacing $\delta X_0$ with alternatively $\delta X_0^1, \delta X_0^2$ and $\delta X_0^R$.
\end{proof}
We obtained the consistency property we were pursuing: the linearized model correctly captures the unobservability of global rotations and translations. As a byproduct, the unobservable seen by the filter is automatically of appropriate dimension.
\subsection{Interpretation and discussion}\label{basis:sec}
The standard EKF is tuned to reduce the state estimation error $\hat X-X$ defined through the original state variables $X,\hat X$ of the problem. Albeit perfectly suited to the linear case, the latter state error has in fact absolutely no fundamental reason to rule the linearization process in a non-linear setting. The basic difference when analyzing the EKF and the IEKF is that
\begin{itemize}
\item In the standard EKF, there is a trivial correspondence between a small variation of the true state and a small variation of the estimation error \eqref{err:elf}. But the global rotations of the frame make the error vary in a non-trivial way as recalled in Section \ref{Sec:2}.
\item In the IEKF approach, the effect of a small rotation of the state on the variation of the estimation error \eqref{xi::non_linear_error} becomes trivial as ensured by Proposition \ref{prop::first_order_rotations_non_linear}. But the error is non-trivially related to the state, as its definition explicitly depends on the linearization point $\hat X$.
\end{itemize}
Many consistency issues of the EKF stem from the fact that the updated covariance matrix $P_{n|n}$ is computed before the update, namely at the predicted state $\hat X_{n|n-1}$, and thus does not account for the updated state's value $\hat X_{n|n}$, albeit supposed to reflect the covariance of the updated error. This is why the OC-EKF typically seeks to avoid linearizing at the latest, albeit best, state estimate, in order to find a close-by state such that the covariance matrix resulting from linearization preserves the observability subspace dimension. The IEKF approach is wholly different: the updated covariance $P_{n|n}$ is computed at the latest estimate $\hat X_{n|n-1}$, which is akin to the standard EKF methodology. But it is then indirectly adapted to the updated state, since it is \emph{interpreted} as the covariance of the error $\xi_{n|n}$. And contrarily to the standard case, the definition of this error depends on $\hat X_{n|n}$. More intuitively, we can say the confidence ellipsoids encoded in $P_{n|n}$ are attached to a basis that undergoes a transformation when moved over from $\hat X_{n|n-1}$ to $\hat X_{n|n}$, this transformation being tied to the unobservable directions. This prevents spurious reduction of the covariance over unobservable shifts, which are not identical at $\hat X_{n|n-1}$ and $\hat X_{n|n}$.
Finally, note the alternative error \eqref{eq::non_linear_error_f} is all but artificial: it naturally stems from the Lie group structure of the problem. This is logical as the considered unobservability actually pertains to an \emph{invariance} of the model \eqref{eq::bicycle_dyn}-\eqref{eq::bicycle_obs}, that is the SLAM problem, to global translations and rotations. Thus it comes as no surprise the \emph{Invariant} approach, that brings to bear {invariant} state errors that encode the very symmetries of the problem, prove fruitful (see the appendix for more details).
\section{IEKF consistency and information}
\label{sect::tools}
Our approach can be related to the previous work \cite{huang2010observability}. Indeed, according to the latter article, failing to capture the right dimension of the observability subspace in the linearized model leads to ``spurious
information gain along directions of the state space where no
information is actually available'' and results in ``unjustified reduction of the covariance estimates, a primary cause of filter inconsistency''. Theorem \ref{SLAM:thm:obs} proves that infinitesimal rotations and translations of the global frame, which are unobservable in the SLAM problem, are always ``seen'' by the IEKF linearized model as unobservable directions indeed, so this filter does not suffer from ``false observability'' issues. This is our major theoretical result.
That said, the results of the latter section concern the system with noise turned off, and pertain to an automatic control approach to the notion of observability as in \cite{huang2010observability}. The present section is rather concerned with the estimation theoretic consequences of Theorem \ref{SLAM:thm:obs}. We prove indeed, that the IEKF's output covariance matrix correctly reflects an absence of ``information gain'' along the unobservable directions, as mentioned above, but where the information is now to be understood in the sense of Fisher information.
As a by-product, this allows relating our results to a slightly different approach to SLAM consistency, that rather focuses on the Fisher information matrix than on the observability matrix, see in particular \cite{Wang,Cetto}.
\subsection{The general Bayesian Fisher Information Matrix}\label{BFIM:sec}
The exposure of the present section is based on the seminal article \cite{Tichavsky}. See also \cite{Wang,Cetto} for related ideas applied to SLAM.
Consider the system \eqref{eq::general_dynamical_system} with output \eqref{eq::general_dynamical_system_output}. Define the collection of state vectors and observations up to time $n$:
$$
\tilde X_n=(X_0^T,\cdots,X_n^T)^T,\quad \tilde Y_n=(Y_0^T,\cdots,Y_n^T)^T,
$$The joint probability distribution of the $(n+1)N $ vector $\tilde X_n$ and of the $np$ vector $\tilde Y_n$ is
$$
p(\tilde Y_n,\tilde X_n)=p(X_0)\Pi_{i=1}^np(Y_i\mid X_i) p(X_i\mid X_{i-1})
$$The Bayesian Fisher Information Matrix (BIFM) is defined as the following $Nk\times Nk$ matrix based upon the dyad of the gradient of the log-likelihood:
$$
J(\tilde X_n)=\mathbb E([\nabla_{\tilde X_n} \log p(\tilde Y_n,\tilde X_n)][\nabla_{\tilde X_n} \log p(\tilde Y_n,\tilde X_n]^T)
$$and note that, for the SLAM problem it boils down to the matrix of \cite{huang2010observability}. This matrix is of interest to us as it yields a lower bound on the accuracy achievable by any estimator used to attack the filtering problem \eqref{eq::general_dynamical_system}-\eqref{eq::general_dynamical_system_output}. Indeed let $J_n$ be defined as the \emph{inverse} of the $N\times N$ right-lower block of $[J(\tilde X_n)]^{-1}$. This matrix provides a lower bound on the mean square error of estimating $X_n$ from past and present measurements $\tilde Y_n$ and prior $p(X_0)$. Indeed, for any unbiased estimator $T:{\mathbb R}^{np}\to {\mathbb R}^N$:
$$
\mathbb E([T(\tilde Y_n)][T(\tilde Y_n)]^T)\succeq J_n^{-1}
$$
where $A\succeq B$ means $A-B$ is positive semi-definite. $J_n^{-1}$ is called the Bayesian or Posterior Cram\'er-Rao lower bound for the filtering problem \cite{Tichavsky}. Most interestingly, in the case where $f$ and $h$ are linear, the prior distribution is Gaussian, and the noises are additive and Gaussian, we have
$$
J_n=P_{n|n}^{-1}
$$where $P_{n|n}$ is the covariance matrix output by the Kalman filter. Thus, in the linear Gaussian case, $P_{n|n}^{-1}$ reflects the statistical information available at time $n$ on the state $X_n$. By extension in the SLAM literature $P_{n|n}^{-1}$ is often simply referred to as the information matrix, in non-linear contexts also, e.g. when using extended information filters \cite{Thrun}.
\subsection{Application to IEKF-SLAM consistency}
In the last section, we have recalled that in the linear Gaussian case, the inverse of the covariance matrix output by the Kalman filter {is} the Fisher information available to the filter (this is also stated in \cite{Bar-Shalom} p. 304). In the light of those results, it is natural to expect from any EKF variant, that the inverse of the output covariance matrix $P_{n|n}^{-1}$ reflect an absence of information gain along unobservable directions indeed. If the filter fails to do so, the output covariance matrix will be too optimistic, that is, inconsistent, and wrong covariances yield wrong gains \cite{Bar-Shalom}. The following theorem shows the linearized system model of the IEKF allows ensuring the desired property of the covariance matrix. It is our second major result.
\begin{thm}
\label{SLAM:big:thm}
Consider the SLAM problem defined by equations \eqref{eq::bicycle_dyn}-\eqref{eq::bicycle_obs} and the IEKF-SLAM Algorithm \ref{algo::IEKF_SLAM}. Let $\delta X_0$ denote a linear combination of infinitesimal rotations and translations $\delta X_0^R, \delta X_0^1, \delta X_0^2$ of the whole system, as defined in Theorem \ref{SLAM:thm:obs}. $\delta X_0$ is thus an unobservable shift. If the matrix $P_{n|n}$ output by the IEKF remains invertible, we have at all times:
\begin{align}\left( \delta X_0 \right)^T P_{n|n}^{-1} \left( \delta X_0 \right) \leq \left( \delta X_0 \right)^T P_{n-1|n-1}^{-1} \left( \delta X_0 \right). \label{decrease:eq}
\end{align}
\end{thm}
\begin{proof}
As $F_n=I_{2K+3}$ in \eqref{eq::A_H_G_IEKF}, the unobservable shifts remain fixed i.e. $\delta X_n=\delta X_0$. At the propagation step we have:
\begin{align*}
\delta & X_0^T P_{n|n-1}^{-1} \delta X_0 = \delta X_{0}^T \left( P_{n-1|n-1} + G_nQ_nG_n^T \right)^{-1} \delta X_{0} \\
\leqslant & \delta X_{0}^TP_{n-1|n-1}^{-1}\delta X_{0}.
\end{align*}
as $G_nQ_nG_n^T$ is positive semi-definite. And at the update step (see the Kalman Information Filter form in \cite{Bar-Shalom}) we have:
$$
\delta X_0^T P_{n|n}^{-1} \delta X_0 = \delta X_0^T \left[ P_{n|n-1}^{-1} + H_n^T R_n^{-1} H_n \right] \delta X_0
= \delta X_0^T P_{n|n-1}^{-1} \delta X_0$$
as $H_n \delta X_0=0$ as shown in the proof of Theorem \ref{SLAM:thm:obs}. Thus $(\delta X_0)^T P_{n|n}^{-1} (\delta X_0)$ is non-increasing over time $n$. Note that, the proof evidences that if $P_{n|n}$ is not invertible, the results of the theorem still hold, writing the IEKF in information form.
\end{proof}
Our result essentially means the \emph{linearized model} of the IEKF has a structure which guarantees that the covariance matrix at all times reflects an absence of ``spurious" (Bayesian Fisher) information gain over directions that correspond to the unobservable rotations and translations of the global frame.
\section{Simulation results}
\label{Sec:4}
In this section we verify in simulation the claimed properties on the one hand, and on the other we illustrate the striking consistency improvement achieved by the IEKF SLAM. To that end, we propose to consider a similar numerical experiment as in the sound work \cite{huang2010observability} dedicated to the inconsistency of EKF and the benefits of the OC-EKF. The IEKF is compared here to the standard EKF, the UKF, the OC-EKF, and the ideal EKF, which is the - impossible to implement - variant of the EKF where the state is linearized about the \emph{true} trajectory.
\begin{figure}
\begin{center}
\includegraphics[width=.60\columnwidth]{ssExperiment_map}
\end{center}
\caption{ Simulated trajectory : the displayed loop is driven 10 times by a robot able to measure the relative position of the landmarks lying in a range of 5 m around him. Velocity is constant (1m/s) as well as angular velocity (9 deg/s).}
\label{fig::map}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\columnwidth]{ssNEES_1}
\end{center}
\caption{ Evaluation of the consistency of the five filters using the NEES indicator (for the entire 3-DoF pose) over 50 runs, in the experimental setting described in Section \ref{sect::exp_setting}. We see the indicator stays around 1 for IEKF SLAM and OC-EKF SLAM over the whole time interval, as expected from a consistent estimation method. The ``ideal" EKF, where the system is linearized on the true value of the state, yields similar results. To the opposite, we see the EKF is inconsistent, and the UKF also.}
\label{fig::NEES}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{ssPos_RMS_1}
\includegraphics[width=\columnwidth]{ssHeading_RMS_1}
\end{center}
\caption{
Evaluation of the performance of the proposed IEKF SLAM algorithm, in terms of RMS of the vehicle heading and position error (in rad and m respectively). We see the results are very similar to those of the OC-EKF and "ideal" EKF, the latter being impossible to implement for real (the system is linearized at the true state). These results are much better than those of the UKF and classical EKF.
}
\label{fig::heading_RMS}
\end{figure}
\begin{figure}
\includegraphics[width=.49 \columnwidth]{soEKF}
\includegraphics[width=.49 \columnwidth]{soUKF}
\includegraphics[width=.49 \columnwidth]{soidEKF}
\includegraphics[width=.49 \columnwidth]{soOC-EKF}
\includegraphics[width=.49\columnwidth]{soIEKF}
\caption{Inconsistency illustrated on a single run. The plotted EKF and UKF heading errors (top plots) do not remain in the $99 \%$ uncertainty envelope computed by the filter. Filters whith theoretical properties regarding non-observable directions (IEKF, OC-EKF and ideal EKF) remedy this problem.
}
\label{fig::EKF_enveloppe}
\end{figure}
\begin{figure}[h]
\includegraphics[width=.49 \columnwidth]{smEKF}
\includegraphics[width=.49 \columnwidth]{smUKF}
\includegraphics[width=.49 \columnwidth]{smidEKF}
\includegraphics[width=.49 \columnwidth]{smOC-EKF}
\includegraphics[width=.49 \columnwidth]{smIEKF}
\caption{ Trajectory and final covariance ellipsoids returned by the implemented filters. We see EKF and UKF are not consistent, mainly because the elongation of ellipsoids, that is related to heading uncertainty, is underestimated. It is not the case with ideal EKF (idEKF), OC-EKF and IEKF.
}
\label{fig::EKF_enveloppe2}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\columnwidth]{Info}
\includegraphics[width=\columnwidth]{Info_zoom}
\end{center}
\caption{Illustration of Theorem \ref{SLAM:big:thm}. Bottom plot is a zoom of the first time steps. The information over an infinitesimal perturbation corresponding to a rotation of the whole system is decreasing for the IEKF SLAM, which is a consistent behavior as this perturbation is unobservable. Ideal EKF and OCEKF give similar results, but EKF and UKF do not. The plot also confirms EKF and UKF SLAM tend to acquire spurious information over this unobservable direction.}
\label{fig::info_rot}
\end{figure}
\subsection{Simulation setting}
\label{sect::exp_setting}
The simulation setting we chose is (deliberately) similar to the one used in \cite{huang2010observability} (Section 6.2). The vehicle (or robot) drives a 15m-diameter loop ten times in the 2D plane, finding on its path 20 unknown features as displayed on Figure \ref{fig::map}. The velocity and angular velocity are constant (1 m/s and 9 deg/s respectively). The relative position of the features in the reference frame of the vehicle is observed once every second (where $\tilde h$ of \eqref{eq::bicycle_obs} is the identity). The standard deviation $\sigma$ of the velocity measurement on each wheel of the vehicle is $2 \%$ of the velocity. This yields the standard deviation $\sigma_v$ of the resulting linear velocity and $\sigma_{\omega}$ of the rotational velocity: $\sigma_v=(\sqrt 2/2)\sigma$ and $\sigma_{\omega} = (\sqrt 2/a) \sigma$, where $a=0.5m$ is the distance between the drive wheels (see \cite{huang2010observability}). The features are visible if they lie within a sensing range of 5 m, in which case they are observed with an isotropic noise of standard deviation 10 cm. The initial uncertainty over position and heading is zero - which will prove a condition not sufficient to prevent failure of the EKF. Each time a landmark is seen for the first time, its position is initialized in the earth frame using the current estimated pose of the robot, the associated uncertainty is set to a very high value compared to the size of the map, then a Kalman update is performed to correlate the position of the new feature withe the other variables. Each second, all visible landmarks (i.e. those in a range of 5m) are processed simultaneously in a stacked observation vector.
Five algorithms are compared:
\begin{enumerate}
\item The classical EKF, described in Algorithm \ref{algo::EKF_SLAM_linear}.
\item The proposed IEKF SLAM algorithm described in Algorithm \ref{algo::IEKF_SLAM}.
\item The ideal EKF as defined in \cite{huang2010observability}, i.e., a classical EKF where the Riccati equation is computed at the true trajectory of the system instead of the estimated trajectory. Although not usable in practice, the latter is a good reference to compare with, as it is supposed to be an EKF with consistent behavior.
\item The OC-EKF described in \cite{huang2010observability}, which is so far the the only method that guarantees the non-observable subspace has appropriate dimension.
\item The Unscented Kalman Filter (UKF), known to better deal with the non-linearities than the EKF.
\end{enumerate}
Before going further, the next subsection introduces the NEES indicator used in the simulations to measure the consistency of these methods.
\subsection{The NEES indicator}
Classical criteria used to evaluate the performance of an estimation method, like Root Mean Squared (RMS) error do not inform about consistency as they do not take into account the uncertainty returned by the filter. This point is addressed by the Normalized Estimation Error Squared (NEES), which computes the average squared value of the error, normalized by the covariance matrix of the EKF. For a sample $(X_i)_{i=1,p}$ of error values having dimension $d$, each of them with a covariance matrix $P_i$ of size $d\times d$, the NEES is defined by:
$$
\text{NEES} = \frac{1}{p\times d} \sum_{i=1}^p X_i^T P_i^{-1} X_i.
$$
If each $X_i$ is a zero mean Gaussian with covariance matrix $P_i$, then for large $p$ we have NEES $\approx 1$. The case NEES $>1$ reveals an inconsistency issue: the actual uncertainty is higher than the computed uncertainty. This situation typically occurs when the filter is optimistic as it believes to have gained information over a non-observable direction. The NEES indicator will be used, along with the usual RMS, to illustrate our solution to SLAM inconsistency in the sequel.
\subsection{Numerical results}
\label{sect::results}
Figure \ref{fig::NEES} displays the NEES indicator of the vehicle pose estimate (heading and position) over time, computed for 50 Monte-Carlo runs of the experiment described in Section \ref{sect::exp_setting}. As expected, the profile of the NEES for classical EKF, ideal EKF and OC-EKF is the same as in the previous paper \cite{huang2010observability} which inspired this experimental section. Note that we used here a normalized version of the NEES, making its swing value equal to 1. We see also that the result is similar for OC-EKF SLAM and ideal EKF SLAM: the NEES varies between 1 and 1.7, in contrast to the EKF SLAM and UKF SLAM which exhibit large inconsistencies over the robot pose We see here that the IEKF remedies inconsistency, with a NEES value that remains close to 1. Note that, it performs here even better than OC-EKF and ideal EKF (whose results are very close to each other), in terms of consistency. The basic difference between IEKF and these filters lies in using or not the current estimate as a linearization point. Uncertainty directions being very dependent from the estimate, what Figure \ref{fig::NEES} suggests is that they may not be correctly captured if computed on a different point.
The other aspect of the evaluation of an EKF-like method is performance: regardless of the relevance of the covariance matrix returned by the filter (i.e. consistency), pure performance can be evaluated through RMS of the heading and position error, whose values over time are displayed in Figure \ref{fig::heading_RMS}. They confirm an expectable result: solving consistency issues improves the accuracy of the estimate as a byproduct, as wrong covariances yield wrong gains \cite{Bar-Shalom}.
Selecting a \emph{single run}, we can also illustrate the inconsistency issue in terms of covariance and information. Figure \ref{fig::EKF_enveloppe} displays the heading error for EKF, UKF, IEKF, OC-EKF and ideal EKF SLAM, and the $99 \%$ envelope returned by each filter. This illustrates both the false observability issue and the resulting inconsistency of EKF and UKF: the heading uncertainty is reduced over time while the estimation error goes outside the $99 \%$ envelope. To the opposite, the behavior of IEKF, OC-EKF and ideal EKF is sound. Figure \ref{fig::EKF_enveloppe2} shows the map and the landmarks $99 \%$ uncertainty ellipsoids: Similarly, both EKF and UKF fail capturing the true landmarks 'positions within the $99 \%$ ellipsoids whereas the three over filters succeed to do so. Finally, Figure \ref{fig::info_rot} displays the evolution of the information over a shift corresponding to infinitesimal rotations as defined in Theorem \ref{SLAM:big:thm}, that is the evolution over time of the quantity $\delta X_n^TP_{n|n}^{-1}\delta X_n$. The theorem is successfully illustrated: the latter quantity is always decreasing for the IEKF, ideal EKF, OC-EKF but not for the EKF and the (slightly better to this respect) UKF.
\section{Conclusion}
This work evidences that the EKF algorithm for SLAM is not inherently inconsistent - at least regarding inconsistency related to unobservable transformations of the global frame - but the choice of the right coordinates for the linearization process is pivotal. We showed that applying the recent theory of the IEKF - an EKF (slight) variant - leads to provable properties regarding observability and consistency. Extensive Monte-Carlo simulations have illustrated the consistency of the new method and the striking improvement over EKF, UKF, OC-EKF, and more remarkably over the ideal EKF also, which is the - impossible to implement - variant of the EKF where the system is linearized about the true trajectory.
Note that, the IEKF approach may prove relevant beyond SLAM to some other problems in robotics as well, such as autonomous navigation (see \cite{barczyk2015invariant}), and in combination with controllers, notably for motion planning purposes, see \cite{diemer2014invariant}. In \cite{barraunolcos} the IEKF has proved to possess \emph{global} asymptotic convergence properties on a simple localization problem of a wheeled robot, which is a strong property. The IEKF has also been patented for navigation with inertial sensors \cite{brevet_alignement}.
Nowadays nonlinear optimization based SLAM algorithms are becoming
popular as compared with EKF SLAM, see e.g. \cite{Dellaert} for one of the first papers on the subject. We yet anticipate a simple EKF SLAM with consistency properties will prove useful to the research community, the EKF SLAM having been abandoned in part due to its inconsistency. The general EKF has proved useful in numerous industrial applications, especially in the field of guidance and navigation. It has the benefits of being 1-recursive, avoiding to store the whole trajectory and 2-suited to on-line real-time applications. Moreover the aerospace and defense industry has developed a corpus of experience for its industrial implementation and validation. And the IEKF is a variant that, being in every respect similar to EKF, retains all its advantages, but which possesses additional guaranteed properties. Note also that, all the improvements of the EKF for SLAM such as e.g., the SLAM of \cite{Paz} and sparse extended information filters \cite{Thrun}, can virtually be turned into their invariant counterpart.
The high dimensional optimization formulation of the SLAM problem being prone to local minima, having an accurate initial value (i.e. a small initial estimation error) is very critical \cite{Zhao}. The IEKF SLAM algorithm proposed in the present paper may thus be advantageously used to initialize those methods in challenging situations.
Besides, we anticipate our approach based on symmetries could help improve (at least first order) optimization techniques for SLAM. To understand why, assume by simplicity the sensors to be noise free. Then, moving a candidate trajectory along unobservable directions will not change the cost function, and an efficient optimization algorithm should account for this. And when a gradient descent algorithm is used, only a first-order expansion of the cost function is considered. Our Lie group approach will allow defining steepest descent directions in a alternative geometric way, that will ``stick'' to the unobservable directions, and the corresponding update will move along the (Lie group) state space in a non-linear yet relevant way. This issue is left for future work, but a thorough understanding of the interest of the invariant approach for the EKF, is a first step in this direction.
\subsection*{Acknowledgements}The authors would like to thank Cyril Joly for his advice.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,125 |
<?php
namespace ZendTest\Form\TestAsset;
use Zend\Form\Form;
use Zend\Form\Element;
use Zend\Validator;
use Zend\Stdlib\Hydrator\ClassMethods as ClassMethodsHydrator;
class CustomForm extends Form
{
public function __construct()
{
parent::__construct('test_form');
$this->setAttribute('method', 'post')
->setHydrator(new ClassMethodsHydrator());
$field1 = new Element('name', array('label' => 'Name'));
$field1->setAttribute('type', 'text');
$this->add($field1);
$field2 = new Element('email', array('label' => 'Email'));
$field2->setAttribute('type', 'text');
$this->add($field2);
$this->add(array(
'name' => 'csrf',
'type' => 'Zend\Form\Element\Csrf',
'attributes' => array(
),
));
$this->add(array(
'name' => 'submit',
'attributes' => array(
'type' => 'submit'
)
));
}
public function getInputFilterSpecification()
{
return array(
'name' => array(
'required' => true,
'filters' => array(
array('name' => 'Zend\Filter\StringTrim'),
),
),
'email' => array(
'required' => true,
'filters' => array(
array('name' => 'Zend\Filter\StringTrim'),
),
'validators' => array(
new Validator\EmailAddress(),
),
),
);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,264 |
\section{Particle optimization}
\subsection{Introduction}
\subsubsection{Pseudo-Boolean optimization}
We call $f\colon\mathbb{B}^d:=\set{0,1}^d\to\mathbb{R}$ a pseudo-Boolean function. The present work discusses approaches to obtain heuristics for the program
\begin{equation}
\label{eq:pb program}
\begin{tabular}{ll}
\text{maximize } & $f(\v x)$ \\[0.2em]
\text{subject to} & $ \v x\in\mathbb{B}^d$
\end{tabular}
\end{equation}
using sequential Monte Carlo techniques. In the sequel, we refer to $f$ as the \emph{objective function}. For an excellent overview of applications of binary programming and equivalent problems we refer to the survey paper by \citeN{boros2002pseudo} and references therein.
The idea to use particle filters for global optimization is not new [\citeNP{del2006sequential}, Section 2.3.1.c], but novel sequential Monte Carlo methodology making use of suitable parametric families on binary spaces \cite{schaefer2011sequential} may allow to construct more efficient samplers for the special case of pseudo-Boolean optimization. We particularly discuss how this methodology connects with the cross-entropy method \cite{Rub:CE1} which is another particle driven optimization algorithm based on parametric families.
The sequential Monte Carlo algorithm as developed by \citeN{schaefer2011sequential} is rather complex compared to local search algorithms such as simulated annealing \cite{kirkpatrick1983optimization} or $k$-opt local search \cite{merz2002greedy} which can be implemented in a few lines. The aim of this paper is to motivate the use of advanced particle methods and sophisticated parametric families in the context of pseudo-Boolean optimization and to provide conclusive numerical evidence that these complicated algorithms can indeed outperform simple heuristics if the objective function has poorly connected strong local maxima. This is not at all clear, since, in terms of computational time, multiple randomized restarts of fast local search heuristics might very well be more efficient than comparatively complex particle approaches.
\subsubsection{Outline}
The article is structured as follows. We first introduce some notation and review how to model an optimization problem \eqref{eq:pb program} as a filtering problem on an auxiliary sequence of probability distributions. Section \ref{sec:smc} describes a sequential Monte Carlo sampler \cite{del2006sequential} designed for global optimization on binary spaces \cite{schaefer2011sequential}. Section \ref{sec:pfbs} reviews three parametric families for sampling multivariate binary data which can be incorporated in the proposed class of particle algorithms. Section \ref{sec:algorithms} discusses how the cross-entropy method \cite{Rub:CE1} and simulated annealing \cite{kirkpatrick1983optimization} can be interpreted as special cases of the sequential Monte Carlo sampler. In Section \ref{sec:applications} we carry out numerical experiments on instances of the unconstrained quadratic binary optimization problem. First, we investigate the performance of the proposed parametric families in particle-driven optimization algorithms. Secondly, we compare variants of the sequential Monte Carlo algorithm, the cross-entropy method, simulated annealing and simple multiple-restart local search to analyze their respective efficiency in the presence or absence of strong local maxima.
\subsubsection{Notation}
We briefly introduce some notation that might be non-standard. We denote scalars in italic type, vectors in italic bold type and matrices in straight bold type. Given a set $M$, we write $\card M$ for the number of its elements and $\mathds{1}_M$ for its indicator function. For $a,b\in\mathbb{Z}$ we denote by $\dset{a,b}=\set{a,\dots,b}$ the discrete interval from $a$ to $b$. Given a vector $\v x\in\mathbb{B}^d$ and an index set $I\subseteq\dset{1,d}$, we write $\v x_I\in\mathbb{B}^{\card I}$ for the sub-vector indexed by $I$ and $\v x_{-I}\in\mathbb{B}^{d-\card I}$ for its complement. We occasionally use the norms $\norm{\v x}_{\infty}:=\max_ix_i$ and $\abs{\v x}:=\sum_{i=1}^{d}\abs{x_i}$.
\subsection{Statistical modeling}
\label{sec:stat model}
\subsubsection{Associated probability measures}
For particle optimization, the common approach is defining a family of probability measures $\set{\pi_\varrho\colon\varrho\geq0}$ associated to the optimization problem $\max_{\v x\in\mathbb{B}^{d}} f(\v x)$ in the sense that
\begin{align*}
\pi_0=\uni_{\mathbb{B}^d}, \quad \lim_{\varrho\to\infty}\pi_\varrho=\uni_{M_f},
\end{align*}
where $\uni_S$ denotes the uniform distribution on the set $S$ and $M_f=\mathop{\mathrm{argmax}}_{\v x\in\mathbb{B}^{d}} f(\v x)$ the set of maximizers. The idea behind this approach is to first sample from a simple distribution, potentially learn about the characteristics of the associated family and smoothly move towards distributions with more mass concentrated in the maxima. We review two well-known techniques to explicitly construct such a family $\pi_\varrho$.
\begin{definition}[Tempered family]
We call $\set{\pi_\varrho\colon\varrho\geq0}$ a tempered family, if it has probability mass functions of the form
\begin{equation}
\label{eq:tempered}
\pi_{\varrho}(\v \gamma):=\nu_{\varrho}\exp({\varrho}\,f(\v \gamma)),
\end{equation}
where $\nu_{\varrho}^{-1}:=\textstyle\sum_{\v\gamma\in\mathbb{B}^d} \exp({\varrho}\,f(\v \gamma))$.
\end{definition}
As ${\varrho}$ increases, the modes of $\pi_{\varrho}$ become more accentuated until, in the limit, all mass is concentrated on the set of maximizers. The name reflects the physical interpretation of $\pi_{\varrho}(\v x)$ as the probability of a configuration $\v x\in\mathbb{B}^d$ for an inverse temperature ${\varrho}$ and energy function $-f$. This is the sequence used in simulated annealing \cite{kirkpatrick1983optimization}.
\begin{definition}[Level set family]
We call $\set{\pi_\varrho\colon\varrho\geq0}$ a level set family, if it has probability mass functions of the form
\begin{equation}
\label{eq:level set}
\pi_{\varrho}(\v \gamma):=\card{L^{+}_{\varrho}}^{-1}\mathds{1}_{L^{+}_{\varrho}}(\v \gamma),
\end{equation}
where $L^{+}_{\varrho}:=\set{\v \gamma\in\mathbb{B}^d\colon \varrho[f(\v x^{*})-f(\v \gamma)]\leq 1}$ for $\v x^{*}\in M_f$.
\end{definition}
Indeed, $L^{+}_{\varrho}$ is the super-level set of $f$ with respect to the level $c=f(\v x^{*})-1/{\varrho}$, for $\varrho>0$, and $\pi_{\varrho}(\v \gamma)$ is the uniform distribution on $L^{+}_{\varrho}$. As ${\varrho}$ increases, the support of $\pi_{\varrho}$ becomes restricted to the points that have an objective value sufficiently close to the maximum of the $f$. In the limit, the support is reduced to the set of global maximizers.
\begin{figure}[ht]
\caption{Associated sequences $\pi_{\varrho_t}$ for a toy example $f\colon\mathbb{B}^3\to[-20,20]$. The colors indicate the advance of the sequences from yellow to red. For simplicity, we choose $\varrho_t=t$ for $t\in\dset{0,16}$.}
\label{fig:sequ}
\subfigure[objective function $f(\v x)$]{
\includegraphics[width=0.3\textwidth]{distr}
}
\subfigure[tempered sequence \eqref{eq:tempered}]{
\includegraphics[width=0.3\textwidth]{tempered}
}
\subfigure[level set sequence \eqref{eq:level set}]{
\includegraphics[width=0.3\textwidth]{rare_event}
}
\end{figure}
The particle-driven optimization algorithms are computationally more involved than local search heuristics since we need to construct a sequence of distributions instead of a sequence of states. We shall see that this effort pays off in strongly multi-modal scenarios, where even sophisticated local search heuristics can get trapped in a subset of the state space.
\subsubsection{Rare event simulation}
While the tempered sequence is based on a physical intuition, the level set sequence has an immediate interpretation as a sequence of rare events since, as $\varrho$ increases, the super-level set becomes a `rare event' with respect to the uniform measure. Rare event simulation and global optimization are therefore closely related concepts and methods for rare event estimation can often be adapted to serve as optimization algorithms.
Particle algorithms for rare event simulation include the cross-entropy method \cite{Rub:CE1} and the sequential Monte Carlo sampler \cite{johansen2006sequential}. The former uses the level set sequence, the latter uses a \emph{logistic potential family}
\begin{align*}
\pi_{\varrho}(\v \gamma):=\nu_{\varrho}\,\ell(\varrho[f(\v \gamma)-f(\v x^{*})]),
\end{align*}
where $\nu_{\varrho}^{-1}:=\textstyle\sum_{\v\gamma\in\mathbb{B}^d} \ell(\varrho[f(\v \gamma)-f(\v x^{*})])$ and $\ell\colon\mathbb{R}\to(0,1),\ \ell(x)=[1+\exp(-x)]^{-1}$ denotes the logistic function. \citeN{johansen2006sequential} did not specifically design their algorithm for optimization but their approach to static rare event simulation is closely related to the particle optimization framework.
\section{Sequential Monte Carlo}
\label{sec:smc}
We discuss a static sequential Monte Carlo sampler that uses a transition kernel with independent proposals in the move step based on suitable parametric families. This methodology has been demonstrated to reliably estimate the mean of the posterior distribution in Bayesian variable selection problems for linear normal models \cite{schaefer2011sequential}. In this section, we provide a self-contained description of this framework. For a more general overview of sequential Monte Carlo methods we refer to \citeN{del2006sequential}.
\subsection{Sequential Importance Sampling}
For convenience of notation, we index the sequence of distributions $(\pi_{\varrho_t})_t$ directly by $t$ rather that by the parameter of the family $\varrho_t$. We refer to $\m X=(\v x_1,\dots,\v x_n)^\intercal\in\mathbb{B}^{n\times d}$ and $\v w\in[0,1]^n$ with $\abs{\v w}=1$ as a \emph{particle system} with $n$ particles. We say the particle system $(\v w,\m X)$ \emph{targets} the probability distribution $\pi$ if the empirical distribution $\sum_{k=1}^n w_k\,\delta_{\v x_{k}}$ converges to $\pi$ for $n\to\infty$.
\subsubsection{Importance weights}
We sample $(\v w_0, \m X_0)$ with $\v x_{1,0},\dots,\v x_{n,0} \sim\uni_{\mathbb{B}^d}=\pi_0$ and set $\v w_0=\v 1/n$ to initialize the system. Suppose we are given a particle approximation $(\v w_{t},\m X_{t})$ of $\pi_{t}$ and want to target the subsequent distribution $\pi_{t+1}$. For all $k\in\dset{1,n}$ and $\alpha>0$, we update the weights to
\begin{equation}
\label{eq:imp weights}
w_{k,t,\alpha}:=\frac{u_{k,t,\alpha}}{\sum_{i=1}^n u_{i,t,\alpha}},\text{ where }
u_{k,t,\alpha}:= w_{k,t}\frac{\tilde\pi_{\varrho_{t}+\alpha}(\v x_{k,t})}{\tilde\pi_{\varrho_{t}}(\v x_{k,t})},
\end{equation}
and $\tilde\pi_{\varrho}\propto\pi_{\varrho}$ denotes the unnormalized version of $\pi_{\varrho}$. The normalizing constants $\nu_{\varrho}$ and $\card{L^{+}_{\varrho}}$ defined in equations \eqref{eq:tempered} and \eqref{eq:level set} are unknown but the algorithm only requires ratios of unnormalized probability mass functions. We refer to $\alpha$ as the \emph{step length} at time $t$. After updating, the particle system targets the distribution
\begin{equation}
\textstyle
\label{eq:emp approx}
\pi_{\varrho_t+\alpha}\approx\sum_{k=1}^n w_{k,t,\alpha}\,\delta_{\v x_{t,k}}.
\end{equation}
As we choose $\alpha$ larger, that is $\pi_{\varrho_t+\alpha}$ further from $\pi_{\varrho_t}$, the weights become more uneven and the accuracy of the importance approximation deteriorates. If we repeat the weighting step, we just increase $\alpha$ and finally obtain an importance sampling estimate of $\pi_\infty=\uni_{M_f}$ with instrumental distribution $\pi_0=\uni_{\mathbb{B}^d}$. This yields a poor estimator since the probability to hit the set $M_f$ with $n$ uniform draws is $1-(1-2^{-d}\card{M_f})^{n}$ and decreases rapidly as the dimension $d$ grows. The pivotal idea behind sequential Monte Carlo is to alternate moderate updates of the importance weights and improvements of the particle system via resampling and Markov transitions.
\begin{definition}[Effective sample size]
The importance weight degeneracy is often measured through the \emph{effective sample size} criterion \cite{KongLiuWong} defined as
\begin{equation*}
\textstyle\eta_n(\v w) := \left\lbrack n\sum_{k=1}^n w_{k}^2\right\rbrack^{-1}\in[1/n,1].
\end{equation*}
The effective sample size is $1$ if the weights are uniform, that is equal to $1/n$; the effective sample size is $1/n$ if all mass is concentrated in a single particle.
\end{definition}
\subsubsection{Finding the step length}
Given any increasing sequence $(\pi_{\varrho_t})_t$, we could repeatedly reweight and monitor whether the effective sample size falls below a critical threshold. For the special case of annealing via sequential Monte Carlo, however, the effective sample size after weighting $\eta_n(\v w_{t,\alpha})$ is merely a function of $\alpha$. For a particle system $(\m X_{t},\v w_t)$ at time $t$, we pick a step length $\alpha$ such that
\begin{equation}
\label{eq:ess}
\eta_n(\v w_t)\,\beta=\eta_n(\v w_{t,\alpha}),
\end{equation}
that is we lower the effective sample with respect to the current particle approximation by some fixed ratio $\beta\in(0,1)$ [\citeNP{jasra2011inference}, \citeNP{del2011adaptive}]. This ensures a `smooth' transition between two auxiliary distributions, in the sense that consecutive distributions are close enough to approximate each other reasonably well using importance weights; in our numerical experiments, we took $\beta=9/10$. We obtain the associated sequence $(\varrho_t)_t$ by setting $\varrho_{t+1}=\varrho_{t}+\alpha_t$ where $\alpha_t$ is a unique solution of \eqref{eq:ess}. Since $\eta_n(\v w_{t,\alpha})$ is continuous and monotonously decreasing in $\alpha$ we can use bi-sectional search to solve \eqref{eq:ess}.
\subsection{Conditioning the particle system}
In this section, we discuss how to condition the weighted system and improve its quality before we proceed with the next weighting step.
\subsubsection{Resampling}
We replace the system $(\v w_{t+1}, \m X_t)$ targeting $\pi_{t+1}$ by a selection of particles $\hat{\v x}_1,\dots,\hat{\v x}_n$ drawn from the current particle reservoir $\v x_{1,t},\dots,\v x_{n,t}$ such that
\begin{equation*}
\ev{n(\v x_{k})}=n\,w_k,
\end{equation*}
where $n(\v x)$ denotes the number of particles identical with $\v x$. Thus, in the resampled system particles with small weights have vanished while particles with large weights have been multiplied. For the implementation of the resampling step, there exist several recipes. We could apply a multinomial resampling \cite{Gordon} which is straightforward. There are, however, more efficient ways like residual \cite{LiuChen}, stratified \cite{kitagawa1996monte} and systematic resampling \cite{CarClifFearn}. We use the latest in our simulations, see Procedure \ref{algo:resample}.
\begin{algorithm}[ht]
\KwIn{$\v w=(w_1,\dots,w_n),\,\m X=(\v x_1,\dots,\v x_n)^\intercal$}
$\v v\gets n\,\v w,\ i\gets1,\ c\gets v_1$ \\
\textbf{sample} $u\sim\uni_{[0,1]}$ \\
\For{$k\in\dset{1,n}$}{
\lWhile{$c < u$}{$i\gets i+1,\ c \gets c+v_i$}\\
$\hat{\v x}_k\gets \v x_i,\ u\gets u+1$
}
\Return $\m{\widehat{X}}=(\hat{\v x}_1\dots,\hat{\v x}_n)^\intercal$
\caption{Resampling (systematic)}
\label{algo:resample}
\end{algorithm}
\subsubsection{Moving the system}
\label{sec:move}
If we repeated the weighting and resampling steps several times, we would rapidly reduce the number of different particles to a very few. The key to fighting the depletion of the particle reservoir is moving the particles according to a Markov transition kernel $\kappa_{t+1}$ with invariant measure $\pi_{t+1}$. The particle $\hat{\v x}_{k,t+1}^{(0)}$ is by construction approximately distributed according to $\pi_{t+1}$, and a draw
\begin{equation*}
\hat{\v x}_{k,t+1}^{(1)}\sim\kappa_{t+1}(\bullet\mid\hat{\v x}_{k,t+1}^{(0)})
\end{equation*}
is therefore again approximately distributed according to $\pi_{t+1}$. The last sample of the generated Markov chain
$
(\hat{\v x}_{k,t+1}^{(0)},\dots,\hat{\v x}_{k,t+1}^{(s)})
$
is, for sufficiently many move steps $s\in\mathbb{N}$, almost exactly distributed according to the invariant measure $\pi_{t+1}$ and independent of its starting point.
\subsubsection{Stopping rule}
While we could always apply a fixed number of move steps, we rather use an adaptive stopping criterion based on the number of distinct particles.
\begin{definition}[Particle diversity]
We define the \emph{particle diversity} as
\begin{equation*}
\label{eqn:pd}
\zeta_n(\m X) := n^{-1}\card{\set{\v x_k\colon k\in\dset{1,n}}}\in[1/n,1].
\end{equation*}
\end{definition}
Ideally, the sample diversity $\zeta_n(\m X)$ should correspond to the expected diversity
\begin{equation*}
\textstyle
\zeta_n(\pi) := 1\wedge n^{-1}
\sum_{\v\gamma\in\mathbb{B}^d}\mathds{1}_{\set{\v x\in\mathbb{B}^d\colon c_n\pi(\v x)\geq 1}}(\v\gamma),
\end{equation*}
where $c_n$ is the smallest value that solves $\sum_{\v\gamma\in\mathbb{B}^d}\lfloor c_n\pi(\v\gamma)\rfloor \geq n$. This is the particle diversity we would expect if we had an independent sample from $\pi(\v x)$. Therefore, if $\kappa_{t+1}$ is fast-mixing, we want to move the system until
\begin{equation*}
\zeta_n(\widehat{\m{X}}\, ^{(s)}_{t+1})\approx\zeta_n(\pi_{t+1}).
\end{equation*}
Since the quantity on the right hand side is unknown, we stop moving the system as soon as the particle diversity reaches a steady state we cannot push it beyond \cite{schaefer2011sequential}.
More precisely, we stop if the absolute diversity is above a certain threshold $\zeta^{*}\approx0.95$ or the last improvement of the diversity is below a certain threshold $\zeta_{\Delta}^{*}>0$. We always stop after a finite number of steps but the thresholds $\zeta^{*}$ and $\zeta_{\Delta}^{*}$ need to be calibrated to the efficiency of the transition kernel. For slow-mixing kernels, we recommend to perform batches of consecutive move steps instead of single move steps.
If the average acceptance rate $\overline\lambda$ of the kernel (see Section \ref{sec:kernels}) is smaller than $\zeta_{\Delta}^{*}$, it is likely that the algorithm stops after the first iteration although further moves would have been necessary. We could adaptively adjust the threshold $\zeta_{\Delta}^{*}$ to be proportional to an estimate of the average acceptance rate; for our numerical experiments, however, we kept it fixed to $\zeta_{\Delta}^{*}\approx 10^{-2}$.
\begin{algorithm}
\DontPrintSemicolon
\KwIn{
\parbox{0.6\textwidth}{
$\m X=(\v x_1^{(0)},\dots,\v x_n^{(0)})^\intercal$ \textbf{ targeting } $\pi$ \\[0.1em]
$\kappa(\v\gamma\mid\bullet)$ \text{ with } $\pi(\v\gamma)=\sum_{\v x\in\mathbb{B}} \pi(\v x) \kappa(\v\gamma\mid\v x)$
}
}
$s\gets 1$ \\
\Repeat{$\zeta(\m X^{(s)})-\zeta(\m X^{(s-1)})<\zeta_{\Delta}^{*}$ \textnormal{\bf or} $\zeta(\m X^{(s)})>\zeta^{*}$}{
\textbf{sample} ${\v x}_k^{(s)}\sim \kappa (\bullet\mid{\v x}_k^{(s-1)})$ \textbf{for all} $k\in\dset{1,n}$ \\
}
\Return $\m X^{(s)}=(\v x_1^{(s)}\dots,\v x_n^{(s)})^\intercal$
\caption{Move}
\label{algo:move}
\end{algorithm}
\subsubsection{Transition kernels}
\label{sec:kernels}
Most transition kernels in Monte Carlo simulations are some variant of the Metropolis-Hastings kernel (see e.g. \citeN{RobCas}),
\begin{equation*}
\kappa_{t+1}\left(\v\gamma\mid\v x\right)
:=\lambda_{q_{t+1}}(\v\gamma,\v x)q_{t+1}(\v\gamma\mid\v x)+
\textstyle
\delta_{\v x}(\v\gamma)\left\lbrack 1-\sum_{\v y\in\mathbb{B}^d}\lambda_{q_{t+1}}(\v y,\v x)q_{t+1}(\v y\mid\v x) \right\rbrack,
\end{equation*}
where we sample from the kernel by proposing a new state $\v\gamma\sim q_{t+1}(\v\gamma\mid \v x)$ and accepting the proposal with probability
\begin{equation}
\label{eq:acc prob}
\lambda_{q_{t+1}}(\v\gamma,\v x)
:= 1\wedge\frac{\tilde\pi_{t+1}(\v\gamma)q_{t+1}(\v x\mid \v\gamma)}{\tilde\pi_{t+1}(\v x)q_{t+1}(\v\gamma\mid \v x)}
\end{equation}
or returning $\v x$ otherwise. Again, we denote by $\tilde\pi_{t+1}\propto\pi_{t+1}$ the unnormalized version of $\pi_{t+1}$ since the kernel only requires the ratio of the unnormalized probability mass functions.
\begin{definition}[Symmetric kernel]
On binary spaces, a common choice for the proposal distribution is
\begin{equation}
\label{eq:sym kernel}
\textstyle
q(\v\gamma\mid\v x)=\sum_{k=1}^d p_k\delta_k(\abs{\v x-\v\gamma})\,k!(d-k)!/d!,
\end{equation}
with weight vector $\v p\in[0,1]^d$ normalized such that $\abs{\v p}=1$.
\end{definition}
With probability $p_k$, the kernel proposes a uniform draw from the $k$-neighborhood of $\v x$,
\begin{equation}
\label{eq:k neighborhood}
N_k(\v x):=\set{\v\gamma\in\mathbb{B}^{d}\colon \abs{\v x- \v \gamma}=k}.
\end{equation}
We refer to this type of kernel as \emph{symmetric kernel} since $q(\v\gamma\mid\v x)=q(\v x\mid\v \gamma)$ and equation \eqref{eq:acc prob} simplifies. This class of kernels provide a higher mutation rate than the random-scan Gibbs kernel (see \citeN{schaefer2011sequential} for adiscussion).
Locally operating transition kernels of the symmetric type are known to be slowly mixing. If we put most weight on small values of $k$, the kernel only changes one or a few entries in each step. If we put more weight on larger values of $k$, the proposals will hardly ever be accepted if the invariant distribution $\pi$ is multi-modal. Ideally, we want the particles sampled from the transition kernel to be nearly independent after a few move steps which is often hard to achieve using local transition kernels.
\begin{definition}[Adaptive independent kernel]
For the sequential Monte Carlo algorithm, we use \emph{adaptive independent kernels} which have proposal distributions of the kind
\begin{equation*}
q(\v\gamma\mid\v x)=q_{\theta}(\v\gamma),\quad \theta\in\Theta,
\end{equation*}
which do not depend on the current state $\v x$ but have a parameter $\theta$ which we adapt during the course of the algorithm.
\end{definition}
The adaptive independent kernel is rapidly mixing if we can fit the \emph{parametric family} $q_\theta$ such that the proposal distribution $q_{t+1}=q_{\theta_{t+1}}$ is sufficiently close to the target distribution $\pi_{t+1}$, yielding thus, on average, high acceptance rates $\lambda_{q_{t+1}}$. The general idea behind this approach is to take the information gathered in the current particle approximation into account (see e.g. \citeN{chopin2002sequential}). The usefulness of this strategy for sampling on binary spaces has been illustrated by \citeN{schaefer2011sequential}.
We fit a parameter $\theta_{t+1}$ to the particle approximation of $\pi_{t+1}$ according to some suitable criterion. Precisely, $\theta_{t+1}$ is taken to be the maximum likelihood or method of moments estimator applied to the weighted sample $(\v w_{t+1},\m X_t)$. The choice of the parametric family $q_\theta$ is crucial to the implementation of a sequential Monte Carlo sampler with adaptive independent kernel. We discuss this issue in detail in Section \ref{sec:pfbs}.
Adaptation could, to a certain extent, also be done for local transition kernels. \citeN{nott2005adaptive} propose an adaptive kernel which replaces the full conditional distribution of the Gibbs sampler by an easy to compute linear approximation which is estimated from the sampled particles. This method accelerates Gibbs sampling if the target distribution $\pi$ is hard to evaluate but does not provide fast mixing like the adaptive independent kernel (see \citeN{schaefer2011sequential} for a comparison).
Still, the use of local kernels in the context of the proposed sequential Monte Carlo algorithm might be favorable if, for instance, the structure of the problem allows to rapidly compute the acceptance probabilities of local moves. Further, batches of local moves can be alternated with independent proposals to ensure that the algorithm explores the neighborhood of local modes sufficiently well.
\subsection{Remark on discrete state spaces}
Since the sample space $\mathbb{B}^d$ is discrete, a given particle is not necessarily unique. This raises the question whether it is sensible to store multiple copies of the same weighted particle in our system. In the sequel, we discuss some more details concerning this issue which has only been touched upon briefly by \citeN{schaefer2011sequential}.
Let $n(\v x)$ denote the number of copies of the particle $\v x$ in the system $(\v w,\m X)$. Indeed, for parsimonious reasons, we could just keep a single representative of $\v x$ and aggregate the associated weights to $w_*(\v x)=n(\v x)\,w(\v x)$.
\subsubsection{Impact on the effective sample size}
Shifting weights between identical particles does not affect the nature of the particle approximation but it obviously changes the effective sample size $\eta_n(\v w)$ which is undesirable since we introduced the effective sample size as a criterion to measure the goodness of a particle approximation. From an aggregated particle system, we cannot distinguish the weight disparity induced by reweighting according to the importance function \eqref{eq:imp weights} and the weight disparity induced by multiple sampling of the same states which occurs if the mass of the target distribution is concentrated. More precisely, we cannot tell whether the effective sample size is actually due to the gap between $\pi_t$ and $\pi_{t+1}$ or the presence of particle copies due to the mass of $\pi_t$ concentrating on a small proportion of the state space which occurs by construction of the auxiliary distribution in Section \ref{sec:stat model}.
\subsubsection{Impact on the resample-move step}
Aggregating the weights means that the number of particles is not fixed at runtime. In this case, the straightforward way to implement the move step presented in Section \ref{sec:move} is breaking up the particles into multiple copies corresponding to their weights and moving them separately. But instead of permanently splitting and pooling the weights it seems more efficient to just keep the multiple copies.
We could, however, design a different kind of resample-move algorithm which first augments the number of particles in the move step and then resamples exactly $n$ weighted particles from this extended system using a variant of the resampling procedure proposed by \citeN{fearnhead2003line}. A simple way to augment the number of particles is sampling and reweighting via
\begin{equation*}
\v x_k^{(1)}\sim q_{t+1}(\bullet\mid \v x_k^{(0)}), \quad w_k^{(1)}=w_k\lambda,\ w_k^{(0)}=w_k(1-\lambda),
\end{equation*}
where $\lambda=\lambda_{q_{t+1}}(\v x_{k}^{(1)},\v x_{k}^{(0)})$ denotes the acceptance probability \eqref{eq:acc prob} of the Metropolis-Hastings kernel. We tested this variant but could not see any advantage over the standard sampler presented in the preceding sections. For the augment-resample type algorithm the implementation is more involved and the computational burden significantly higher. In particular, the Rao-Blackwellization effect one might achieve when replacing the accept-reject steps of the transition kernel by a single resampling step does not seem to justify the extra computational effort.
Indeed, aggregating the weights does not only prevent us from using the effective sample size criterion, but also requires extra computational time of $\mathcal O(n\log n)$ in each iteration of the move step since pooling the weights is as complex as sorting. With our application in mind, however, computational time is more critical than memory, and we therefore recommend to refrain from aggregating the weights.
\section{Parametric families on binary spaces}
\renewcommand{\algorithmcfname}{Procedure}
\label{sec:pfbs}
We review three parametric families on $\mathbb{B}^d$. In contrast to the similar discussion in \cite{schaefer2011sequential}, we also consider a parametric family which cannot be used in sequential Monte Carlo samplers but in the context of the cross-entropy method. For more details on parametric families on binary spaces we refer to \citeN{schaefer2012logistic}.
\subsection{Suitable parametric families}
\label{sec:properties}
We frame some properties making a parametric family suitable as proposal distribution in sequential Monte Carlo algorithms.
\begin{enumerate}[(a)]
\item For reasons of parsimony, we prefer a family of distributions with at most $d(d+1)/2$ parameters like the multivariate normal.
\item Given a sample $\m X=(\v x_1,\dots,\v x_n)^\intercal$ from the target distribution $\pi$, we need to estimate $\theta^*$ in a reasonable amount of computational time.
\item We need to generate samples $\m Y=(\v y_1,\dots, \v y_m)^\intercal$ from the family $q_\theta$. We need the rows of $\m Y$ to be independent.
\item For the sequential Monte Carlo algorithm, we need to evaluate $q_\theta(\v y)$ point-wise. However, the cross-entropy method still works without this requirement.
\item We want the calibrated family $q_{\theta^*}$ to reproduce e.g. the marginals and covariance structure of $\pi$ to ensure that the parametric family $q_{\theta^*}$ is sufficiently close to $\pi$.
\end{enumerate}
\subsection{Product family}
The simplest non-trivial distributions on $\mathbb{B}^d$ are certainly those having independent components.
\begin{definition}[Product family]
For a vector $\m m\in(0,1)^d$ of marginal probabilities, we define the \emph{product family}
\begin{equation}
\label{eq:product family}
\textstyle
q^{{\raisemath{-1.1pt}{\sqcap}}}_{\v m}(\v \gamma):= \prod_{i=1}^d m_i^{\gamma_i}(1-m_i)^{1-{\gamma_i}}.
\end{equation}
\end{definition}
\subsubsection{Properties}
We check the requirement list from Section \ref{sec:properties}:
(a) The product family is parsimonious with $\mathrm{dim}(\theta)=d$.
(b) The maximum likelihood estimator $\hat{\v m}$ is the weighted sample mean.
(c) We can easily sample $\v y\sim q^{{\raisemath{-1.1pt}{\sqcap}}}_{\v m}$.
(d) We can easily evaluate the mass function $q^{{\raisemath{-1.1pt}{\sqcap}}}_{\v m}(\v y)$.
(e) However, the product family does not reproduce any dependencies we might observe in $(\v w,\m X)$.
The last point is the crucial weakness which makes the product family impractical for particle optimization algorithms on strongly multi-modal problems. Consequently, the rest of this section deals with ideas on how to sample binary vectors with a given dependence structure. There are, to our knowledge, two major strategies to this end.
\begin{enumerate}[(1)]
\item We construct a generalized linear model which permits to compute the conditional distributions. We apply the chain rule and write $q_\theta$ as
\begin{equation}
\label{eq:chain rule factorization}
\textstyle
q_{\v\theta}(\v\gamma)=q_{\v\theta}(\v\gamma_1)\prod_{i=2}^d q_{\v\theta}(\v\gamma_i\mid\v\gamma_{1:i-1}),
\end{equation}
which allows to sample the entries of a random vector component-wise.
\item We sample from an auxiliary distribution $\varphi_\theta$ and map the samples into $\mathbb{B}^d$. We call
\begin{equation}
\label{eq:aux distr}
\textstyle
q_{\theta}(\v\gamma)=\int_{\tau^{-1}(\v \gamma)} \varphi_\theta(\v v) d\v v
\end{equation}
a copula family, although we refrain from working with explicit uniform marginals.
\end{enumerate}
We first present a generalized linear model and then review a copula approach.
\subsection{Logistic conditionals family}
\label{sec:logistic family}
Even for rather simple non-linear models we usually cannot derive closed-form expressions for the marginal probabilities required for sampling according to \eqref{eq:chain rule factorization}. Therefore, we might directly construct a parametric family from its conditional probabilities.
\begin{definition}[Logistic conditionals family]
We define, for a lower triangular matrix $\m A\in\mathbb{R}^{d\times d}$, the \emph{logistic conditionals family} as
\begin{align*}
\label{eq:lb}
q^{{\raisemath{-1.1pt}{\,\logistic}}}_{\m A}(\v \gamma)
&
:=
\textstyle
\prod_{i\in \dset{1,d}}\ell\left(a_{ii}+\sum_{j=1}^{i-1} a_{ij}\gamma_j\right)^{\gamma_i}
\left[1-\ell\left(a_{ii}+\sum_{j=1}^{i-1} a_{ij}\gamma_j\right)\right]^{1-\gamma_i}
\end{align*}
where $\ell\colon\mathbb{R}\to(0,1),\ \ell(x)=[1+\exp(-x)]^{-1}$ is the logistic function. We readily identify the product family $q^{{\raisemath{-1.1pt}{\sqcap}}}_{\v m}$ as the special case $\m A=\mathrm{diag}{\ell^{-1}(\v m)}$.
\end{definition}
The virtue of the logistic conditionals family is that, by construction, we can sample a random vector component-wise while the full probability $q_{\m A}^{{\raisemath{-1.1pt}{\,\logistic}}}(\v y)$ of the sample $\v y$ is computed as a by-product of Procedure \ref{algo:sampling}. We refer to the Online Supplement for instructions on how to fit the parameter $\m A$.
\begin{algorithm}
$\v y=(0,\dots,0),\ p\gets 1$ \\
\For{$i\in\dset{1,d}$}{
$r\gets q_{\m A}^{{\raisemath{-1.1pt}{\,\logistic}}}(y_i=1\mid\v y_{1:i-1})=\ell({a_{ii}+\sum_{j=1}^{i-1}a_{ij}y_j})$ \\
$u\sim\mathcal{U}[0,1]$ \\
\bf{ if } $u<r$ \bf{ then }$y_i\gets1$ \\
$p\gets\begin{cases}
p\cdot r & \textbf{if }\ \ y_i=1 \\
p\cdot (1-r) & \textbf{if }\ \ y_i=0
\end{cases}$ \\
}
\Return $\v y,\ p$
\caption{Sampling via chain rule factorization}
\label{algo:sampling}
\end{algorithm}
\subsubsection{Properties}
We check the requirement list from Section \ref{sec:properties}:
(a) The logistic conditionals family is sufficiently parsimonious with $\mathrm{dim}(\theta)=d(d+1)/2$.
(b) We can fit the parameter $\m A$ via likelihood maximization. The fitting is computationally intensive but feasible.
(c) We can sample $\v y\sim q^{{\raisemath{-1.1pt}{\,\logistic}}}_{\m A}$ using the chain rule factorization \eqref{eq:chain rule factorization}.
(d) We can exactly evaluate $q^{{\raisemath{-1.1pt}{\,\logistic}}}_{\m A}(\v y)$.
(e) The family $q^{{\raisemath{-1.1pt}{\,\logistic}}}_{\m A}$ reproduces the dependency structure of the data $\m X$ although we cannot explicitly compute the marginal probabilities.
\subsection{Gaussian copula family}
\label{sec:Gaussian copula}
Let $\varphi_\theta$ be a family of multivariate auxiliary distributions on $\mathbb X$ and $\tau\colon\mathbb X \to \mathbb{B}^d$ a mapping into the binary space. We can sample from the copula family \eqref{eq:aux distr} by setting $\v x=\tau(\v v)$ for a draw $\v v\sim \varphi_\theta$ from the auxiliary distribution. Most multivariate parametric families with at most $d(d+1)/2$ parameters appear to either have a rather limited dependency range or they do not scale to higher dimensions \cite{joe1996families}. Therefore, the natural and seemingly only viable option for $\varphi_\theta$ is the multivariate normal distribution \cite{emrich1991method}.
\begin{definition}[Gaussian copula family]
For a vector $\v a\in\mathbb{R}^{d}$ and a correlation matrix $\m \Sigma\in\mathbb{R}^{d\times d}$, we introduce the mapping
\begin{equation*}
\tau_{\v a}\colon\mathbb{R}^{d}\to\mathbb{B}^{d},\
\tau_{\v a}(\v v):=(\mathds{1}_{(-\infty,a_i]}(v_1),\dots,\mathds{1}_{(-\infty,a_d]}(v_d)),
\end{equation*}
and define the \emph{Gaussian copula family} as
\begin{equation*}
\textstyle
q^{{\raisemath{-1.1pt}{gc}}}_{\v a,\m \Sigma}(\v\gamma)
:=(2\pi)^{-\frac{d}{2}}\mathrm{det}{\m \Sigma}^{-\frac{1}{2}}\textstyle\int_{\tau_{\v a}^{-1}(\v\gamma)}\exp\left(-\frac{1}{2}\,\v v^\intercal\m \Sigma^{-1}\v v\right)\,d\v v.
\end{equation*}
\end{definition}
For index sets $I\subseteq \dset{1,d}$, the cross-moments
\begin{equation*}
m_I=\textstyle\sum_{\v\gamma\in\mathbb{B}^{d}}q^{{\raisemath{-1.1pt}{gc}}}_{\v a,\m \Sigma}(\v\gamma)\prod_{i\in I}\gamma_i
\end{equation*}
are equal the cumulative distribution function of the multivariate normal with respect to the entries indexed by $I$ (see \citeN{schaefer2012logistic} for a more detailed discussion). In particular, the first and second moments are
\begin{equation*}
m_i=\varPhi_1(a_i),\quad m_{ij}=\varPhi_2(a_i,a_j;\sigma_{ij}),\quad i,j\in\dset{1,d},
\end{equation*}
where $\varPhi_1(\cdot)$ and $\varPhi_2(\cdot,\cdot;\sigma_{ij})$ denote the cumulative distribution functions of the univariate and bivariate normal distributions with zero mean, unit variance and correlation coefficient $\sigma_{ij}\in[-1,1]$. We refer to the Online Supplement for instructions on how to fit the parameters $\v a$ and $\m\Sigma$.
\subsubsection{Properties}
We check the requirement list from Section \ref{sec:properties}:
(a) The Gaussian copula family is sufficiently parsimonious with $\mathrm{dim}(\theta)=d(d+1)/2$.
(b) We can fit the parameters $\v a$ and $\m \Sigma$ via method of moments. However, the parameter $\m \Sigma$ is not always positive definite.
(c) We can sample $\v y\sim q^{{\raisemath{-1.1pt}{gc}}}_{\v a,\m \Sigma}$ using $\v y=\tau_{\v a}(\v v)$ with $\v v\sim \varphi_{\m \Sigma}$.
(d) We cannot easily evaluate $q^{{\raisemath{-1.1pt}{gc}}}_{\v a,\m \Sigma}(\v y)$ since this requires computing high-dimensional integral expressions which is a computationally challenging problem in itself (see e.g. \citeN{genz2009computation}). The Gaussian copula family is therefore less useful for sequential Monte Carlo samplers but can be incorporated into the cross-entropy method reviewed in Section \ref{sec:cross entropy}.
(e) The family $q^{{\raisemath{-1.1pt}{gc}}}_{\v a,\m \Sigma}$ reproduces the exact mean and, possibly scaled, correlation structure.
\subsection{Toy example}
We briefly discuss a toy example to illustrate the usefulness of the parametric families. For the quadratic function
\begin{equation}
\label{eq:toy exa}
f(\v x)=\v x^\intercal\m F\v x,\quad \m F:=
\begin{pmatrix}
1& 2& 1& 0 \\
2& 1& -3& -2 \\
1& -3& 1& 2 \\
0& -2& 2& -2
\end{pmatrix},
\end{equation}
the associated probability mass function $\pi(\v\gamma)\propto\exp(\v\gamma^\intercal\m F\v\gamma)$ has a correlation matrix
\begin{equation*}
\m R\approx\begin{pmatrix}
1& 0.127& -0.106& -0.101 \\
0.127& 1& -0.941& -0.866 \\
-0.106& -0.941& 1& 0.84 \\
-0.101& -0.866& 0.84& 1
\end{pmatrix},
\end{equation*}
which indicates that this distribution has considerable dependencies and its mass function is therefore strongly multi-modal. We generate pseudo-random data from $\pi$, adjust the parametric families to the data and plot the mass functions of the fitted parametric families.
Figure \ref{fig:toy exa} shows how the three parametric families cope with reproducing the true mass function. Clearly, the product family is not close enough to the true mass function to yield a suitable instrumental distribution while the logistic conditional family almost copies the characteristics of $\pi$ and the Gaussian copula family allows for an intermediate goodness of fit.
\begin{figure}[ht]
\caption{Toy example showing how well the parametric families replicate the mass function of the distribution $\pi(\v\gamma)\propto\exp(\v\gamma^\intercal\m F\v\gamma)$ as defined in \eqref{eq:toy exa}.}
\label{fig:toy exa}
\begin{center}
\subfigure[True mass function $\pi(\v\gamma)$]{
\includegraphics[width=0.46\textwidth]{function}
}
\subfigure[Product family $q_{\v m}(\v\gamma)$]{
\includegraphics[width=0.46\textwidth]{product}
}
\subfigure[Logistic conditionals family $q_{\m A}(\v\gamma)$]{
\includegraphics[width=0.46\textwidth]{logregr}
}
\subfigure[Gaussian copula family $q_{\v a, \m\Sigma}(\v\gamma)$]{
\includegraphics[width=0.46\textwidth]{gaussian}
}
\end{center}
\end{figure}
\section{Optimization algorithms}
\label{sec:algorithms}
In this section, we provide a synopsis of all steps involved in the sequential Monte Carlo algorithm and connect this framework to the cross-entropy method and simulated annealing. In Table \ref{tab:seq}, we state the necessary formulas for the tempered and the level set sequence introduced in Section \ref{sec:stat model}.
\subsection{Sequential Monte Carlo}
For convenience, we summarize the complete sequential Monte Carlo sampler in Algorithm \ref{algo:smc}. Note that, in practice, the sequence $\pi_{\varrho_t}$ is not indexed by $t$ but rather by $\varrho_t$, which means that the counter $t$ is only given implicitly.
The algorithm terminates if the particle diversity sharply drops below some threshold $\delta>0$ which indicates that the mass has concentrated in a single mode. If we use a kernel with proposals from a parametric family $q_{\theta_t}$, we might already stop if the family degenerates in the sense that only a few components of $q_{\theta_t}$, say less than $d^{*}=12$, are random while the others are constant ones or zeros. In this situation, additional moves using a parametric family are a pointless effort. We either return the maximizer within the particle system or we solve the subproblem of dimension $d^*$ by brute force enumeration. We might also perform some final local moves in order to further explore the regions of the state space the particles concentrated on.
\renewcommand{\algorithmcfname}{Algorithm}
\setcounter{algocf}{0}
\begin{algorithm}
\KwIn{$f\colon \mathbb{B}^d\to\mathbb{R}$}
\textbf{sample} $\v x_k\stackrel{\mathrm{iid}}{\sim}\uni_{\mathbb{B}^d}$ \textbf{for all} $k\in\dset{1,n}$. \\[0.3em]
$\alpha\gets\textbf{find step length}(0,\m X)$, $\v w\gets\textbf{importance weights}(\alpha,\pi,\m X)$ \\[0.3em]
\While{$\zeta_{n}(\m X)>\delta$}{$ $\\[0.3em]
\begin{tabular}{lll}
$q_\theta $&\hspace{-3mm}$\gets\textbf{fit parametric family}(\v w, \m X)$ &\hspace{-3mm} (see Section \ref{sec:pfbs}) \\[0.3em]
$\m{\widehat{X}} $&\hspace{-3mm}$\gets\textbf{resample}(\v w,\m X)$ &\hspace{-3mm} (Procedure \ref{algo:resample}) \\[0.3em]
$\m X $&\hspace{-3mm}$\gets\textbf{move}(\kappa_{\pi, q_\theta}, \m{\widehat{X}})$&\hspace{-3mm} (Procedure \ref{algo:move}) \\[0.3em]
$\alpha $&\hspace{-3mm}$\gets\textbf{find step length}(\varrho, \m X)$ & \\[0.3em]
$\v w $&\hspace{-3mm}$\gets\textbf{importance weights}(\alpha,\pi_{\varrho},\m X)$ & \\[0.3em]
$\varrho $&\hspace{-3mm}$\gets\varrho+\alpha$&
\end{tabular}
}
\Return $\mathop{\mathrm{argmax}}_{\v x\in\set{\v x_1,\dots,\v x_n}} f(\v x)$
\caption{Sequential Monte Carlo optimization}
\label{algo:smc}
\end{algorithm}
\subsection{Cross-entropy method}
\label{sec:cross entropy}
For the level set sequence, the effective sample size is the fraction of the particles which have an objective function value greater than $\max_{\v x\in\mathbb{B}^d}f(\v x)-1/(\varrho_t+\alpha)$; see Table \ref{tab:seq} and equation \eqref{eq:level set}. The remaining particles are discarded since their weights equal zero. Consequently, there is no need to explicitly compute $\alpha_t$ as a solution of \eqref{eq:ess}. We simply order the particles $\v x_k$ according to their objective values $f(\v x_k)$ and only keep the $n(1-\beta)$ particles with the highest objective values.
Rubinstein \citet{Rub:CE1}, who popularizes the use of level set sequences in the context of the cross-entropy method, refers to $n(1-\beta)$ as the size of the \emph{elite sample}. The cross-entropy method has been applied successfully to a variety of combinatorial optimization problems, some of which are equivalent to pseudo-Boolean optimization \cite{Rub:bookCE}, and is closely related to the proposed sequential Monte Carlo framework.
However, the central difference between the cross-entropy method and the sequential Monte Carlo algorithm outlined above is the use of the invariant transition kernel in the latter. We obtain the cross-entropy method as a special case if we replace the kernel $\kappa_t$ by its proposal distribution $q_{\theta_t}$. The sequential Monte Carlo approach uses a smooth family of distributions $\set{\pi_\varrho\colon\varrho\geq0}$ and explicitly schedules the evolution $\pi_{\varrho_t}$ which in turn leads to the proposal distributions $q_{\theta_t}$. The cross-entropy method, in contrast, defines the subsequent proposal distribution
\begin{equation*}
q_{\theta_{t+1}}\approx q_{\theta_{t}}\mathds{1}_{L^{+}_{\varrho_{t+1}}}
\end{equation*}
without any reference sequence $\pi_t$ to balance the speed of the particle evolution.
In order to decelerate the advancement of the cross-entropy method, we introduce a lag parameter $\tau\in[0,1)$ and use a convex combination of the previous parameter $\theta_{t-1}$ and the parameter $\hat\theta_t$ fit to the current particle system, setting
\begin{equation*}
\theta_{t}:=(1-\tau)\hat\theta_{t}+\tau\theta_{t-1}.
\end{equation*}
However, there are no guidelines on how to adjust the lag parameter during the run of the algorithm. Therefore, the sequential Monte Carlo algorithm is easier to calibrate since the reference sequence $\pi_t$ controls the stride and automatically prevents the system from overshooting.
On the upside, the cross-entropy method allows for a broader class of auxiliary distributions $q_{\theta_t}$ since we do not need to evaluate $q_{\theta_t}(\v x)$ point-wise which is necessary in the computation of the acceptance probability of the Hastings kernel; see Section \ref{sec:Gaussian copula}.
\begin{table}
\caption{Formulas for optimization sequences}
\label{tab:seq}
\begin{center}
\begin{tabular}{l|c|c}
& $\exp(\varrho f)$
& $\mathds{1}_{L^{+}_{\varrho}}$ \\[1mm]
\hline
&&\\[-3mm]
$u_{t,\alpha}(\v x_{k,t})$
& $\displaystyle e^{\alpha f(\v x_{k,t})}$
& $\displaystyle \mathds{1}_{L^{+}_{\varrho_{t}+\alpha}}(\v x_{k,t})$ \\[5mm]
$\eta_n(\v w_{t,\alpha})$
& $\displaystyle\frac{\left\lbrack\sum_{k=1}^n e^{\alpha f(\v x_{k,t})}\right\rbrack^2}{n\sum_{k=1}^n e^{2\alpha f(\v x_{k,t})}}$
& $\displaystyle\frac{\card{\set{\v x_{k,t}\mid k\in\dset{1,n}} \cap L^{+}_{\varrho_{t}+\alpha}}}{n}$ \\[5mm]
$\lambda_{q_{t+1}}(\v\gamma\mid\v x_{k,t})$
& $\displaystyle 1\wedge \frac{e^{\alpha(f(\v\gamma)-f(\v x_{k,t}))}}{e^{\log q_t(\v \gamma)-\log q_t(\v x_{k,t})}}$
& $\displaystyle 1\wedge\frac{\mathds{1}_{L^{+}_{\varrho_{t+1}}}(\v \gamma)}{e^{\log q_t(\v \gamma)-\log q_t(\v x_{k,t})}}$
\end{tabular}
\end{center}
\end{table}
\subsection{Simulated annealing}
\label{sec:sim ann}
A well-studied approach to pseudo-Boolean optimization is simulated annealing \cite{kirkpatrick1983optimization}.
While the name stems from the analogy to the annealing process in metallurgy, there is a pure statistical meaning to this setup. We can picture simulated annealing as approximating the mode of a tempered sequence \eqref{eq:tempered} using a single particle. Since a single observation does not allow for fitting a parametric family, we have to rely on symmetric transition kernels \eqref{eq:sym kernel} in the move step.
A crucial choice is the sequence $\varrho_t$ which in this context is often referred to as the \emph{cooling schedule}. There is a vast literature advising on how to calibrate $\varrho_t$ where a typical guideline is the expected acceptance rate of the Hastings kernel. We calibrate $\varrho_t$ such that the empirical acceptance rate
\begin{equation*}
\textstyle
\overline{\lambda}_{t-s:t}:=\sum_{r=t-s}^{t}\lambda_r,\quad s>0
\end{equation*}
follows approximately $(t+1)^{-5}$ for $t\in[0,1]$. There are variants of simulated annealing which use more complex cooling schedules, tabu lists and multiple restarts, but we stick to this simple version for the sake of simplicity. Algorithm \ref{algo:sim ann} describes the version we use in our numerical experiments in Section \ref{sec:perf algo}.
\begin{algorithm}[ht]
\KwIn{$f\colon \mathbb{B}^d\to\mathbb{R},\,T^{*}\in\mathbb{R}$}
$\v x\sim\uni_{\mathbb{B}^d},\, \v x^*\gets\v x,\,t\gets0,\, T_{\Delta}\gets0$ (time elapsed) \\
\While{$T_{\Delta}<T^{*}$}{
\textbf{sample }$\v\gamma\sim \uni_{N_1(\v x)},\ u\sim\uni_{[0,1]},\ \lambda_t\gets1\wedge \exp\left[\varrho_t\,(f(\v \gamma)-f(\v x))\right]$ \\
\lIf{$u<\lambda_t$}{$\v x \gets \v\gamma$} \\
\lIf{$f(\v x) > f(\v x^*)$}{$\v x^* \gets \v x$} \\
\textbf{adjust }$\varrho_t$ \textbf{ such that } $\overline{\lambda}_{t-s:t}\approx(1+T_{\Delta}/T^{*})^{-5}$ \\
$t\gets t+1$
}
\Return $\v x^*$
\caption{Simulated annealing optimization}
\label{algo:sim ann}
\end{algorithm}
\subsection{Randomized local search}
\label{sec:local search}
We describe a greedy local search algorithm which works on any state space that allows for defining a neighborhood structure. The typical neighborhood on binary spaces is the $k$-neighborhood defined in \eqref{eq:k neighborhood}. A greedy local search algorithm computes the objective value of all states in the current neighborhood and moves to the best state found until a local optimum is reached. The local search algorithm is called $k$-opt if it searches the neighborhood $\cup_{i=1}^{k}N_i(\cdot)$ (see e.g. \citeN{merz2002greedy} for a discussion).
The algorithm can be randomized by repeatedly restarting the procedure from randomly drawn starting points. There are more sophisticated versions of local search algorithms exploit the properties of the objective function but even a simple local search procedure can produce good results \cite{alidaee2010theorems}. Algorithm \ref{algo:local search} describes the $1$-opt local search procedure we use in our numerical experiments in Section \ref{sec:perf algo}.
\begin{algorithm}[ht]
\KwIn{$f\colon \mathbb{B}^d\to\mathbb{R},\,T^{*}\in\mathbb{R}$}
$\v x^{*}\sim\uni_{\mathbb{B}^d},\, T_{\Delta}\gets0$ (time elapsed) \\
\While{$T_{\Delta}<T^{*}$}{
$\v x\sim\uni_{\mathbb{B}^d}$ \\
\While{$\v x$\textnormal{\textbf{ is not a local optimum}}}{
$\v x\gets\mathop{\mathrm{argmax}}_{\v \gamma\in N_1(\v x)} f(\v \gamma)$\\
}
\lIf{$f(\v x) > f(\v x^*)$}{$\v x^* \gets \v x$}\\
}
\Return $\v x^*$
\caption{Randomized local search}
\label{algo:local search}
\end{algorithm}
\section{Applications}
\label{sec:applications}
\subsection{Unconstrained Quadratic Binary Optimization}
\subsubsection{Introduction}
It is well-known that any pseudo-Boolean function $f\colon\mathbb{B}^{d}\to\mathbb{R}$ can be written as a multi-linear function
\begin{align}
\label{eq:multi-linear}
f(\v x)
=\sum_{I\subseteq \dset{1,d}} f\left(\mathds{1}_I(1), \dots,\mathds{1}_I(d)\right)\prod_{i\in I}x_i\prod_{i\in \dset{1,d}\setminus I}(1-x_i)
=\sum_{I\subseteq \dset{1,d}} a_I \prod_{i\in I}x_i,
\end{align}
where $a_I\in\mathbb{R}$ are real-valued coefficients. We say the function $f$ is of order $k$ if the coefficients $a_I$ are zero for all $I\subseteq \dset{1,d}$ with $\card I > k$. While optimizing a first order function is trivial, optimizing a non-convex second order function is already an NP-hard problem \cite{garey1979guide}.
In the sequel, we focus on optimization of second order pseudo-Boolean functions to exemplify the stochastic optimization schemes discussed in the preceding sections. If $f$ is a second order function, we restate program \eqref{eq:pb program} as
\begin{equation}
\label{eq:ubqo program}
\begin{tabular}{ll}
\text{maximize } & $\v x^\intercal \m F \v x$ \\[0.2em]
\text{subject to} & $ \v x\in\mathbb{B}^d$,
\end{tabular}
\end{equation}
where $\m F\in\mathbb{R}^{d\times d}$ is a symmetric matrix. We call \eqref{eq:ubqo program} an unconstrained quadratic binary optimization problem (\textsc{uqbo}); we refer to \citeN{boros2007local} for a list of applications and equivalent problems. In the literature it is also referred to as unconstrained quadratic Boolean or bivalent or zero-one programming \cite{beasley1998heuristic}.
\subsubsection{Particle optimization and meta-heuristics}
\label{sec:meta}
Meta-heuristics are a class of algorithms that optimize a problem by improving a set of candidate solutions without systematically enumerating the state space; typically they deliver solutions in polynomial time while an exact solution has exponential worst case running time. The outcome is neither guaranteed to be optimal nor deterministic since most meta-heuristics are randomized algorithms. We briefly discuss the connection to particle optimization against the backdrop of the unconstrained quadratic binary optimization problem where we roughly separate them into two classes: local search algorithms and particle-driven meta-heuristics.
Local search algorithms iteratively improve the current candidate solution through local search heuristics and judicious exploration of the current neighborhood; examples are local search \cite{boros2007local,merz2002greedy}, tabu search \cite{glover1998adaptive,palubeckis2004multi}, simulated annealing \cite{katayama2001performance}. Particle driven meta-heuristics propagate a set of candidate solutions and improve it through recombination and local moves of the particles; examples are genetic algorithms \cite{merz1999genetic}, memetic algorithms \cite{merz2004memetic}, scatter search \cite{amini1999scatter}. For comparisons of these methods we refer to \citeN{hasan2000comparison} or \citeN{beasley1998heuristic}.
The sequential Monte Carlo algorithm and the cross-entropy method are clearly in the latter class of particle-driven meta-heuristics. The idea behind sequential Monte Carlo is closely related to the intuition behind population (or swarm) optimization and genetic (or evolutionary) algorithms. However, the mathematical framework used in sequential Monte Carlo allows for a general formulation of the statistical properties of the particle evolution while genetic algorithms are often problem-specific and empirically motivated.
\subsubsection{Particle optimization and exact solvers}
If we can explicitly derive the multi-linear representation \eqref{eq:multi-linear} of the objective function, there are techniques to turn program \eqref{eq:pb program} into a linear program. For the \textsc{uqbo} it reads
\begin{equation}
\label{eq:lin ubqo}
\begin{tabular}{ll}
\text{maximize } & $\displaystyle f(\v x)=2\sum_{i=1}^d\sum_{j=1}^{i-1} f_{ij}x_{ij}+\sum_{i=1}^df_{ii}x_{ii}$ \\[1.5em]
\text{subject to} & $\v x\in\mathbb{B}^{d(d+1)/2}$ \\
& $\left.
\begin{array}{l}
\hspace{-1ex}x_{ij}\leq x_{ii} \\
\hspace{-1ex}x_{ij}\leq x_{jj} \\
\hspace{-1ex}x_{ij}\geq x_{ii} + x_{jj} -1 \\
\end{array}
\right\}\text{ for all}\ i,j\in \dset{1,d}$.
\end{tabular}
\end{equation}
Note, however, that there are more parsimonious linearization strategies than this straightforward approach [\citeNP{hansen2009improved}, \citeNP{gueye2009linear}]. The transformed problem allows to access the tool box of linear integer programming which consist of branch-and-bound algorithms that are combined with rounding heuristics, various relaxations techniques and cutting plane methods [\citeNP{pardalos1990computational}, \citeNP{palubeckis1995heuristic}].
Naturally, the question arises whether particle-driven meta-heuristics can be incorporated into exact solvers to improve branch-and-bound algorithms. Indeed, stochastic meta-heuristics deliver lower bounds for maximization problems, but particle-driven algorithms are computationally somewhat expensive for this purpose unless the objective function is strongly multi-modal and other heuristics fail to provide good results; see the discussion in Section \ref{sec:extreme}.
However, the sequential Monte Carlo approach in combination with the level set sequence \eqref{eq:level set} might also be useful to determine a global branching strategy, since the algorithm provides an estimator for
\begin{equation*}
\textstyle
\overline{\v\gamma}_{c}:=\card{L^{+}_{c}}^{-1}\sum_{\v\gamma\in\mathbb{B}^d}\v\gamma\,\mathds{1}_{L^{+}_{c}}(\v\gamma),
\end{equation*}
which is the average of the super-level set $L^{+}_{c}:=\set{\v x\in\mathbb{B}^d\colon f(\v x)\geq c}$. These estimates given for a sequence of levels $c$ might provide branching strategies than are superior to local heuristics or branching rules based on fractional solutions. A further discussion of this topic is beyond the scope of this paper but it certainly merits consideration.
\subsection{Construction of test problems}
\label{sec:test problems}
\subsubsection{Introduction}
The meta-heuristics we want to compare do not exploit the quadratic structure of the objective function and might therefore be applied to any binary optimization program. If the objective function can be written in multi-linear form like \eqref{eq:ubqo program} there are efficient local search algorithms \cite{boros2007local,merz2002greedy} which exploit special properties of the target function and easily beat particle methods in terms of computational time.
Therefore, the use of particle methods is particularly interesting if the objective function is expensive to compute or even a black box. The posterior distribution in Bayesian variable selection for linear normal models is an example of such an objective function (see \citeN{schaefer2011sequential} and references therein). We stick to the \textsc{uqbo} for our numerical comparison since problem instances of varying difficulty are easy to generate and interpret while the results carry over to general binary optimization.
In the vast literature on \textsc{uqbo}, authors typically compare the performance of meta-heuristics on a suite of randomly generated problems with certain properties. \citeN{pardalos1991construction} proposes standardized performance tests on symmetric matrices $\m F\in\mathbb{Z}^{d\times d}$ with entries $f_{ij}$ drawn from the uniform
\begin{equation*}
q_{c}(k):=\frac{1}{2c}\mathds{1}_{\dset{-c,c}}(k), \quad c\in\mathbb{N}.
\end{equation*}
The test suites generated by \citet[\href{http://people.brunel.ac.uk/~mastjjb/jeb/orlib/bqpinfo.html}{OR-library}]{beasley1990or} and \citeN{glover1998adaptive} follow this approach have been widely used as benchmark problems in the \textsc{uqbo} literature (see \citeN{boros2007local} for an overview). In the sequel we discuss the impact of diagonal dominance, shifts, the density and extreme values of $\m F$ on the expected difficulty of the corresponding \textsc{uqbo} problem.
\subsubsection{Diagonal}
Generally, stronger \emph{diagonal dominance} in $\m F$ corresponds to easier \textsc{uqbo} problems \cite{billionnet1994minimization}. Consequently, the original problem generator presented by \citeN{pardalos1991construction} is designed to draw the off-diagonal elements from a uniform on a different support $\dset{-q,q}$ with $q\in\mathbb{N}$.
In this context, we point out that the impact of diagonal dominance carries over to the statistical properties of the tempered distributions \eqref{eq:tempered} we defined in the introductory Section \ref{sec:stat model}. Indeed, stronger diagonal dominance in $\m F$ corresponds to exponential quadratic distributions
\begin{equation*}
\pi(\v\gamma):=\frac{\exp(\v\gamma^\intercal\m F\v\gamma)}{\sum_{\v\gamma\in\mathbb{B}^d}\exp(\v\gamma^\intercal\m F\v\gamma)}
\end{equation*}
having lower dependencies between the components of $\v\gamma$. We can analytically derive a parameter $\m A\in\mathbb{R}^{d\times d}$ for a logistic conditionals family $q^{{\raisemath{-1.1pt}{\,\logistic}}}_{\m A}$ that approximates $\pi(\v\gamma)$ where the quality of the approximation increases as the diagonal of $\m F$ becomes more dominant \cite{schaefer2012logistic}. We can accelerate the sequential Monte Carlo algorithm by initializing the system from $q^{{\raisemath{-1.1pt}{\,\logistic}}}_{\m A}$ instead of $\uni_{\mathbb{B}^d}$. However, we did not exploit this option to keep the present work more concise.
For positive definite $\m F\succ0$, the optimization problem is convex and can be solved in polynomial time \cite{kozlov1979polynomial}; in exact optimization, this fact is exploited to construct upper bounds for maximization problems \cite{poljak1995convex}. We observe a corresponding complexity reduction in statistical modeling. For $\m F\succ0$, the auxiliary distribution
\begin{equation*}
\pi(\v\gamma):=\frac{\v\gamma^\intercal\m F\v\gamma}{2^{d-2}\left(\v 1^\intercal \m F \v 1 + \tr{\m F}\right)},
\end{equation*}
is a feasible mass function, and we can derive analytical expressions concerning all cross-moments and marginal distributions \cite{schaefer2011parametric} which allows to largely analyze the properties of $\pi(\v\gamma)$ without enumerating the state space.
\subsubsection{Shifts}
The global optimum of the \textsc{uqbo} problem is more difficult to detect as we shift the entries of the matrix $\m F$ but the relative gap between the optimum and any heuristic value diminishes. If we sample $f_{ij}=f^{\tau}_{ij}$ from a uniform on the \emph{shifted} support
\begin{equation*}
q_{c,\tau}(k):=\uni_{\dset{-c+\tau,c+\tau}}(k), \quad c\in\mathbb{N},\,\tau\in\dset{-c,c},
\end{equation*}
we obtain an objective function
\begin{equation*}
f_{\tau}(\v x)
=\v x^\intercal\m F^{\tau}\v x
\stackrel{d}{=}\v x^\intercal(\m F^{0}+\tau\v1\v1^\intercal)\v x
=f_{0}(\v x)+\tau\abs{\v x}^{2},
\end{equation*}
where $\stackrel{d}{=}$ means equality in distribution. Hence, with growing $\abs{\tau}$ the optimum depends less on $\m F$ and the relative gap between the optimum and a solution provided by any meta-heuristic vanishes. \citeN{boros2007local} define a related criterion
\begin{equation*}
\bar\rho:=
\frac{1}{2}+\frac{\tau+2\tau c}{2(\tau^2+c^2+c)}\in[0,1]
\end{equation*}
and report a significant impact of $\bar\rho$ on the solution quality of their local search algorithms which is not surprising.
\subsubsection{Density}
The difficulty of the optimization problem is related to the number of interactions, that is the number of non-zero elements of $\m F$. We call the proportion of non-zeros the \emph{density} of $\m F$. Drawing $f_{ij}$ from the mixture
\begin{equation*}
q_{c,\omega}(k)=\omega\,\uni_{\dset{-c,c}}(k)+(1-\omega)\delta_{0}(k), \quad c\in\mathbb{N},\,\omega\in(0,1]
\end{equation*}
we adjust the difficulty of the problem to a given expected density $\omega$.
Note that not all algorithms are equally sensitive to the density of $\m F$. Using the basic linearization \eqref{eq:lin ubqo}, each non-zero off-diagonal element requires the introduction of an auxiliary variable and three constraints. Thus, the expected total number of variables and the expected total number of constraints, which largely determine the complexity of the optimization problem, are proportional to the density $\omega$.
On the other hand, many randomized approaches, including the particle algorithms discussed in Section \ref{sec:smc}, are less sensitive to the density of the problem in the sense that replacing zero elements by small values has a minor impact on the performance of these algorithms. Rather than the zero/non-zero duality, we suggest that the presence of extreme values determines the difficulty of providing heuristic solutions.
\subsubsection{Extreme values}
\label{sec:extreme}
The uniform sampling approach advocated by \citeN{pardalos1991construction} is widely used in the literature for comparing meta-heuristics. Certainly, particle-driven methods are computationally too expensive to outperform local search heuristics on test problems with uniformly drawn entries; \cite{beasley1998heuristic} confirms this intuition with respect to genetic algorithms versus tabu search and simulated annealing. However, the uniform distribution does not produce \emph{extreme values} and it is vital to keep in mind that these have an enormous impact on the performance of local search algorithms.
Extreme values in $\m F$ lead to the existence of distinct local maxima $\v x^{*}\in\mathbb{B}^d$ of $f$ in the sense that there is no better candidate solution than $\v x^{*}$ in the neighborhood $\cup_{i=1}^{k}N_{i}(\v x^{*})$ even for relatively large $k$. Further, extreme local minima might completely prevent a local search heuristic from traversing the state space in certain directions. Consequently, local search algorithms, as discussed in Section \ref{sec:meta}, depend more heavily on their starting value, and their performance deteriorates with respect to particle-driven algorithms.
We propose to draw the matrix entries $f_{ij}$ from a discretized Cauchy distribution
\begin{equation}
\label{eq:cauchy}
\cau_{c}(k)\propto(1+(k/c)^2)^{-1}, \quad c\in\mathbb{N}
\end{equation}
that has heavy tails which cause extreme values to be frequently sampled. Figure \ref{fig:problem distr} shows the distribution of a Cauchy and a uniform to illustrate the difference. The resulting \textsc{uqbo} problems have quite distinct local maxima; in that case we also say that the function $f(\v x)$ is \emph{strongly multi-modal}.
\begin{figure}[ht]
\caption{Histograms of a Cauchy $\cau_{5}$ and a uniform $\uni_{10}$ distribution.\vspace{-5mm}}
\label{fig:problem distr}
\begin{center}
\includegraphics[width=\textwidth]{problem_distr}
\end{center}
\end{figure}
\subsection{Numerical comparison}
\label{sec:outline num ex}
In this section, we provide numerical comparisons based on instances of the \textsc{uqbo} problem. We generated two random test suites of dimension $d=250$, each having $10$ instances. For the first suite, we sampled the matrix entries from a uniform distribution $\uni_{100}$ on $\dset{-100,100}$; for the second, we sampled from a Cauchy distribution $\cau_{100}$ as defined in \eqref{eq:cauchy}. For performance evaluation, we run a specified algorithm $100$ times on the same problem and denote the outcome by $\v x_1,\dots,\v x_{100}$.
\subsubsection{Visualization}
Since the absolute values are not meaningful, we report the relative ratios
\begin{equation*}
\varrho_k:=\frac{f(\v x_k)-\text{worst solution found}}{\text{best known solution}-\text{worst solution found}}\in[0,1],
\end{equation*}
where the best known solution is the highest objective value ever found for that instance and the worst solution is the lowest objective value among the $100$ outcomes. We summarize the results in a histogram. The first $n$ bins are singletons $b_{k}:=\set{\varrho_k^{*}}$ for the highest values $\varrho_1^{*}>\cdots>\varrho_{n}^{*}\in\set{\varrho_k\colon k\in\dset{1,100}}$; the following $n$ bins are equidistant intervals $b_k^{<}:=[\frac{n-k}{n}\varrho_{n}^{*},\frac{n-k+1}{n}\varrho_{n}^{*})$. The graphs show the bins $b_{1},\dots,b_{n},b_{1}^{<},\dots,b_{n}^{<}$ in descending order from left to right on the $x$-axis. The interval bins are marked with a sign ``$<$'' and the lower bound. The $y$-axis represents the counts.
For comparison, we draw the outcome of several algorithms into the same histogram, where the worst solution found is the lowest overall objective value among the outcomes. For each algorithm, the counts are depicted in a different color and, for better readability, with diagonal stripes in a different angle. To put it plainly, an algorithm performs well if its boxes are on the left of the graph since this implies that the outcomes where often close to the best known solution.
\subsubsection{Comparison of binary parametric families}
\label{sec:perf fam}
We study how the choice of the binary parametric family affects the quality of the delivered solutions. The focus is on the cross-entropy method, since we cannot easily use the Gaussian copula family in the context of sequential Monte Carlo. We use $n=1.2\times10^4$ particles, set the speed parameter to $\beta=0.8$ (or the elite fraction to $0.2$) and the lag parameter to $\tau=0.5$.
The numerical comparisons, given in Figures \ref{fig:uni mc} and \ref{fig:cauchy mc}, clearly suggest that using more advanced binary parametric families allows the cross-entropy method to detect local maxima that are superior to those detected using the product family. Hence, the numerical experiments confirm the intuition of our toy example in Figure \ref{fig:toy exa}.
On the strongly multi-modal instance \ref{fig:cauchy mc} the numerical evidence for this conjecture is stunningly clear-cut; on the weakly multi-modal problem \ref{fig:uni mc} its validity is still unquestionable. This result seems natural since reproducing the dependencies induced by the objective function is more relevant in the former case than in the latter.
\begin{figure*}[ht]
\begin{centering}
\caption{The cross-entropy method using different binary parametric families.}
\subfigure[problem $f(\v x)=\v x^\intercal\m F\v x$ with $f_{ij}\sim\cau_{100}$ for $i,j\in\dset{1,250}$]{
\includegraphics[width=0.95\textwidth]{v_r250c_mc_01}
\label{fig:cauchy mc}
}
\subfigure[problem $f(\v x)=\v x^\intercal\m F\v x$ with $f_{ij}\sim\uni_{100}$ for $i,j\in\dset{1,250}$]{
\includegraphics[width=0.95\textwidth]{v_r250u_mc_01}
\label{fig:uni mc}
}
\end{centering}
\end{figure*}
\subsubsection{Comparison of optimization algorithms}
\label{sec:perf algo}
We compare a sequential Monte Carlo sampler with parametric family, a sequential Monte Carlo sampler with a single-flip symmetric kernel \eqref{eq:sym kernel}, the cross-entropy method, simulated annealing and $1$-opt local search as described in Section \ref{sec:algorithms}.
For the cross entropy method, we use the same parameters as in the preceding section. For the sequential Monte Carlo algorithm, we use $n=0.8\times10^4$ particles and set the speed parameter to $\beta=0.9$; we target a tempered auxiliary sequence \eqref{eq:tempered}. For both algorithms we use the logistic conditionals family as sampling distribution. With these configurations, the algorithms converge in roughly $25$ minutes. We calibrate the sequential Monte Carlo sampler with local moves to have the same average run time by processing batches of $10$ local moves before checking the particle diversity criterion. The simulated annealing and $1$-opt local search algorithms run for exactly $25$ minutes.
The results shown in Figures \ref{fig:uni ac} and \ref{fig:cauchy ac} assert the intuition that particle methods perform significantly better on strongly multi-modal problems. However, on the easy test problems, the particle methods tend to persistently converge to the same sub-optimal local modes. This effect is probably due to their poor local exploration properties. Since particle methods perform significantly less evaluations of the objective function, they are less likely to discover the highest peak in a region of rather flat local modes. The use of parametric families aggravates this effect, and it seems advisable to alternate global and local moves to make a particle algorithm more robust against this kind of behavior.
Further numerical results are shown in Figure \ref{fig:r250c} and Figure \ref{fig:r250u}.
\begin{figure*}[ht]
\begin{centering}
\caption{Comparison of stochastic optimization algorithms on two \textsc{uqbo} problems.}
\subfigure[problem $f(\v x)=\v x^\intercal\m F\v x$ with $f_{ij}\sim\cau_{100}$ for $i,j\in\dset{1,250}$]{
\includegraphics[width=0.95\textwidth]{v_r250c_01}
\label{fig:cauchy ac}
}
\subfigure[problem $f(\v x)=\v x^\intercal\m F\v x$ with $f_{ij}\sim\uni_{100}$ for $i,j\in\dset{1,250}$]{
\includegraphics[width=0.95\textwidth]{v_r250u_01}
\label{fig:uni ac}
}
\end{centering}
\end{figure*}
\begin{figure*}
\caption{Comparison of stochastic optimization algorithms. $10$ problems $f(\v x)=\v x^\intercal\m F\v x$ with $f_{ij}\sim\cau_{100}$ for $i,j\in\dset{1,250}$}
\label{fig:r250c}
\mygraph{r250c}{01}
\mygraph{r250c}{02}
\mygraph{r250c}{03}
\mygraph{r250c}{04}
\mygraph{r250c}{05}
\mygraph{r250c}{06}
\mygraph{r250c}{07}
\mygraph{r250c}{08}
\mygraph{r250c}{09}
\mygraph{r250c}{10}
\end{figure*}
\begin{figure*}
\caption{Comparison of stochastic optimization algorithms. $10$ problems $f(\v x)=\v x^\intercal\m F\v x$ with $f_{ij}\sim\uni_{100}$ for $i,j\in\dset{1,250}$}
\label{fig:r250u}
\mygraph{r250u}{01}
\mygraph{r250u}{02}
\mygraph{r250u}{03}
\mygraph{r250u}{04}
\mygraph{r250u}{05}
\mygraph{r250u}{06}
\mygraph{r250u}{07}
\mygraph{r250u}{08}
\mygraph{r250u}{09}
\mygraph{r250u}{10}
\end{figure*}
\section{Discussion and conclusion}
The numerical experiments carried out on different parametric families revealed that the use of the advanced families proposed in this paper significantly improves the performance of the particle algorithms, especially on the strongly multi-modal problems. The experiments demonstrate that local search algorithms, like simulated annealing and randomized $1$-opt local search, indeed outperform particle methods on weakly multi-modal problems but deliver inferior results on strongly multi-modal problems.
Using tabu lists, adaptive restarts and rounding heuristics, we can certainly design local search algorithms that perform better than simulated annealing and $1$-opt local search. Still, the structural problem of strong multi-modality persists for path-based algorithms. On the other hand, cleverly designed local search heuristics will clearly beat sequential Monte Carlo methods on easy to moderately difficult problems.
The results encourage the use of particle methods if the objective function is known to be potentially multi-modal and hard to analyze analytically. We have to keep in mind that multiple restarts of rather simple local search heuristics can be very efficient if they make use of the structure of the objective function. For $25$ minutes of randomized restarts, the heuristic proposed by \citeN{boros2007local}, which exploits the fact that the partial derivatives of a multi-linear function are constant, practically always returns the best known solution on all test problems treated to create Figures \ref{fig:r250c} and \ref{fig:r250u}.
The numerical work was completely done in \href{http://www.python.org/}{Python 2.6} using \href{http://www.scipy.org}{SciPy} packages and run on a cluster with $1.86$ GHz processors. The sources used in this work and the problems processed in this paper can be found at \url{http://code.google.com/p/smcdss}.
\section{Acknowledgments}
This work is part of the author's Ph.D. thesis at CREST under supervision of Nicolas Chopin whom I would like to thank for the numerous discussions on particle algorithms. I thank the editor and two anonymous referees for their detailed comments which helped to significantly improve this paper.
\section{Appendix: Fitting the parameters}
We briefly summarize how the parameters of the logistic conditionals family and the Gaussian copula family can be assessed for a given particle system $\m X=(\v x_1,\dots,\v x_n)^\intercal$ with weights $\v w=(w_{1},\dots,w_{n})$. We denote by
\begin{equation}
\label{eq:moments}
\textstyle\bar x_i:=\sum_{k=1}^n w_k x_{ki},\quad \bar x_{ij}:=\sum_{k=1}^n w_k x_{ki}x_{kj},\quad i,j\in \dset{1,d}
\end{equation}
the weighted first and second sample moments.
\subsection{Logistic conditionals family}
\subsubsection{Derivatives of the log-likelihood function}
The log-likelihood function of the weighted logistic regression of $\v y^{(i)}:= \m X_{\bullet i}$ on $\m Z^{(i)}:=(\m X_{\bullet1:i-1},\v 1)$ is
\begin{align*}
\log L(\v a)
&=\sum_{k=1}^n w_k \left[y^{(i)}_k\log[\ell(\v z_{k \bullet}^{(i)}\v a)]+(1-y^{(i)}_k)\log[1-\ell(\v z_{k \bullet}^{(i)}\v a)]\right] \\
&=\sum_{k=1}^n w_k\left[y^{(i)}_k\v z_{k \bullet}^{(i)}\v a-\log[1+\exp(\v z_{k \bullet}^{(i)}\v a)]\right],
\end{align*}
where we used that $\log[1-\ell(\v x^\intercal\v a)]=-\log[1+\exp(\v x^\intercal\v a)]=-\v x^\intercal\v a+\log[\ell(\v x^\intercal\v a)]$. Since $\partial\log[1+\exp(\v x^\intercal \v a)]/\partial\v a=\ell(\v x^\intercal\v a)\v x$, the gradient of the log-likelihood is
\begin{align*}
s(\v a)
=\sum_{k=1}^n w_k\left[y^{(i)}_k\v z_{k \bullet}^{(i)}-\ell(\v z_{k \bullet}^{(i)}\v a)\v z_{k \bullet}^{(i)}\right]
=(\m Z^{(i)})^\intercal \mathrm{diag}(\v w)[\v y^{(i)} - \v p^{(i)}_{\v a}],
\end{align*}
where $(p^{(i)}_{\v a})_k:=\ell(\v z_{k \bullet}^{(i)}\v a)$. Since $\partial\ell(\v x^\intercal \v a)/\partial\v a=-\ell(\v x^\intercal \v a)[1-\ell(\v x^\intercal \v a)]\v x$, the Hessian matrix of the log-likelihood is
\begin{align*}
s'(\v a)
=-\sum_{k=1}^n w_k\left[\ell(\v z_{k \bullet}^{(i)}\v a)[1-\ell(\v z_{k \bullet}^{(i)}\v a)]\right]\v z_{k \bullet}^{(i)}(\v z_{k \bullet}^{(i)})^\intercal
=-(\m Z^{(i)})^\intercal \mathrm{diag}(\v w) \mathrm{diag}(\v q^{(i)}_{\v a}) \m Z^{(i)},
\end{align*}
where $(q^{(i)}_{\v a})_k:=\ell(\v z_{k \bullet}^{(i)}\v a)[1-\ell(\v z_{k \bullet}^{(i)}\v a)]$.
\subsubsection{Complete separation}
The data might suffer from complete or quasi-complete separation \cite{albert_84} which causes the likelihood function $L(\v a)$ to be monotonic. In that case there is no maximizer. We can avoid the monotonicity by assigning a suitable prior distribution to the parameter $\v a$. \citeN{firth_93} recommends the Jeffreys prior for its bias reduction which can conveniently be implemented via data adjustment \cite{kosmidis2009bias}.
For the sake of simplicity, however, we only assign a simple Gaussian prior with variance $\varepsilon^{-1}>0$ such that, up to a constant, the log-posterior distribution is the log-likelihood function plus a quadratic penalty term and therefore always convex. The score function and its Jacobian matrix become
\begin{align*}
s(\v a) =(\m Z^{(i)})^\intercal \mathrm{diag}(\v w)[\v y^{(i)} - \v p^{(i)}_{\v a}] - \varepsilon\v a,\quad
s'(\v a)=-(\m Z^{(i)})^\intercal \mathrm{diag}(\v w) \mathrm{diag}(\v q^{(i)}_{\v a}) \m Z^{(i)}-\varepsilon \m I.
\end{align*}
The bias-reduced estimators are known to shrink towards $0.5$ which is an undesired property when fitting a parametric family. Therefore, we attempt to keep the shrinkage parameter $\varepsilon$ as small as possible.
\subsubsection{Newton-Raphson iterations}
The first order condition $s(\v a)=\v 0$ is solved iteratively
\begin{align*}
\v a^{(t+1)}
=\v a^{(t)}-[s'(\v a^{(t)})]^{-1}s(\v a^{(t)})
=\v a^{(t)}+\v x^{(t)}
\end{align*}
where $\v x^{(t)}$ is the vector that solves
\begin{align*}
\left[(\m Z^{(i)})^\intercal \mathrm{diag}(\v w) \mathrm{diag}(\v q^{(i)}_{\v a^{(t)}})(\m Z^{(i)})^\intercal+\varepsilon\m I\right]\v x^{(t)}=
\left[(\m Z^{(i)})^\intercal \mathrm{diag}(\v w) [\v y^{(i)} - \v p^{(i)}_{\v a^{(t)}}]-\varepsilon\v a^{(t)}\right].\end{align*}
If the Newton iteration at the $i$th component fails to converge, we can either augment the penalty term $\varepsilon$ which leads to stronger shrinkage of the mean $m_i$ towards $0.5$ or we can drop some covariates $\gamma_j$ for $j\in\set{1,\dots,i-1}$ from the iteration to improve the numerical condition of the procedure.
In practice, we drop the predictors from the regression model which are only weakly correlated with the explained variable. This step is important to speed up the algorithm and improve its numerical properties. For a proposal distribution, it is particularly important to take the strong dependencies into account but it is often sufficient to work with very sparse logistic conditionals families.
In particularly difficult cases, we might prefer to set $a_{ii}=\ell^{-1}(\bar x_{i})$ and $\v a_{i,1:i-1}=\v 0$, where $\bar x_{i}$ is defined in \eqref{eq:moments}, which guarantees that at least the mean is correct. This is an important issue since misspecification of the mean of $\gamma_i$ also affects the distribution of the components $\gamma_j$ for $j\in\set{i+1,\dots,d}$ which are sampled conditional on $\gamma_i$.
\begin{algorithm}
\DontPrintSemicolon
\KwIn{$\m X=(\v x_1,\dots,\v x_n)^\intercal,\ \v w=(w_1,\dots,w_n),\ \m A\in\mathbb{R}^{d\times d}$}\vspace{0.3em}
\For {$i\in\dset{1,d}$}{$ $\\[0.3em]
$\m Z\gets(\m X_{\bullet1:i-1},\v 1),\ \v y\gets\m X_{\bullet i}, \ \v a^{(0)}\gets\m A_{i,1:i}$\\
\Repeat{$\Vert\v a^{(t+1)}-\v a^{(t)}\Vert_{\infty}<\delta$}{
\begin{tabular}{rllr}
$p_k$ &\hspace{-3mm}$\gets$&\hspace{-3mm}$\ell(\m Z_{k\bullet} \v a^{(t)})$&\textbf{ for all }$k\in\dset{1,n}$ \\
$q_k$ &\hspace{-3mm}$\gets$&\hspace{-3mm}$p_k(1-p_k)$ &\textbf{ for all }$k\in\dset{1,n}$ \\
\end{tabular}
\begin{tabular}{ll}
$\v a^{(t+1)}\gets$&\hspace{-3mm}$\v a^{(t)}+\left[(\m Z^{(i)})^\intercal \mathrm{diag}[\v w] \mathrm{diag}[\v q]\m Z^{(i)}+\varepsilon\m I\right]^{-1}\times$\\
&$\qquad\qquad\left[(\m Z^{(i)})^\intercal \mathrm{diag}[\v w] \left[\v y - \v p \right]-\varepsilon\v a^{(t)}\right]$\\
\end{tabular}
}\vspace{0.3em}
$\m A_{i,1:i}\gets\v a$ \\
}
\Return $\m A$
\caption{Fitting the weighted logistic regressions}
\label{algo:fit logistic}
\end{algorithm}
\subsection{Gaussian copula family}
We adjust the Gaussian copula family $q^{{\raisemath{-1.1pt}{gc}}}_{\v a,\m \Sigma}$ by method of moments. We need to solve the non-linear equations
\begin{equation}
\label{eq:fitting Gaussian}
\varPhi_1(a_i)=\bar x_i,\quad \varPhi_2(a_i,a_j;\sigma_{ij})=\bar x_{ij},\quad i,j\in\dset{1,d},
\end{equation}
where $\bar x_i$ and $\bar x_{ij}$ are defined in \eqref{eq:moments} while $\varPhi_1(v_i)$ and $\varPhi_2(v_i,v_j;\sigma_{ij})$ denote the cumulative distribution functions of the univariate and bivariate normal distributions with zero mean, unit variance and correlation coefficient $\sigma_{ij}\in[-1,1]$.
While the parameter $a_i=\varPhi_1^{-1}(\bar x_i)$ is easy to assess, the challenging task is to compute the bivariate variances $\sigma_{ij}$ for all $i,j\in \dset{1,d}$. Recall the standard result [\citeNP{johnson2002continuous}, p.255]
\begin{equation}
\label{eq:der of norm pdf}
\frac{\partial \varPhi_2(x_1,x_2;\sigma)}{\partial\sigma}=\varphi(x_1,x_2;\sigma),
\end{equation}
where $\varphi(\cdot,\cdot; \sigma)$ denotes the density of the bivariate normal distribution with correlation coefficient $\sigma$. We obtain the following Newton-Raphson iteration
\begin{equation}
\sigma^{(t+1)}=\sigma^{(t)}-
\frac{\varPhi_2(a_i,a_j;\sigma_{ij}^{(t)})-\bar x_{ij}}{\varphi(a_i,a_j; \sigma_{ij}^{(t)})},
\end{equation}
starting at some initial value $\sigma_{ij}^{(0)}\in(-1, 1)$; see Procedure \ref{algo:fit gaussian}. In the sequential Monte Carlo context, good initial values are obtained from the parameters of the previous auxiliary distributions.
We use a fast series approximation \cite{drezner_98} to evaluate $\varPhi_2(a_i,a_j;\sigma_{ij})$. These approximations are critical when $\sigma_{ij}$ comes very close to either boundary of $[-1,1]$. The Newton iteration might repeatedly fail when restarted at the corresponding boundary $\sigma_{ij}^{(0)}\in\set{-1,1}$. In any event, $\varPhi_2(x_1,x_2;\sigma)$ is strictly monotonic in $\sigma$ since its derivative \eqref{eq:der of norm pdf} is positive, and we can switch to bi-sectional search if necessary.
\begin{algorithm}
\DontPrintSemicolon
\KwIn{$\bar x_i,\ \bar x_{ij}\ $\textbf{ for all }$i,j\in \dset{1,d}$}\vspace{0.3em}
$a_i\gets\varPhi^{-1}(\bar x_i)\ $\textbf{ for all }$i\in \dset{1,d}$\\[0.3em]
$\m \Sigma^{(0)}\gets\m I$ \\[0.3em]
\For {$i,j\in \dset{1,d}, \ i<j$}{\vspace{0.3em}
\Repeat{$\vert \sigma_{ij}^{(t+1)}-\sigma_{ij}^{(t)}\vert<\delta$}{
$\displaystyle \sigma_{ij}^{(t+1)}\gets
\sigma_{ij}^{(t)}-\frac{\varPhi_2(a_i,a_j;\sigma_{ij}^{(t)})-\bar x_{ij}}{\varphi(a_i,a_j; \sigma_{ij}^{(t)})}$ \\
}\vspace{0.3em}
$\sigma_{ji}\gets\sigma_{ij}^{(t+1)}$ \\[0.3em]
}
\textbf{if not }$\m\Sigma\succ 0$\textbf{ then }$\m \Sigma\gets(\m \Sigma+\abs{\lambda}\m I)/(1+\abs{\lambda})$ \\
\Return $\v a,\, \m \Sigma$
\caption{Fitting the dependency matrix}
\label{algo:fit gaussian}
\end{algorithm}
The locally fitted correlation matrices $\m \Sigma$ might not be positive definite for $d \geq 3$ because the Gaussian copula is not flexible enough to model the full range of cross-moments binary distributions might have (see \citeN{schaefer2012logistic} for an extended discussion). We obtain a feasible parameter replacing $\m \Sigma$ by
\begin{equation*}
\m \Sigma^*=(\m \Sigma+\abs{\lambda}\m I)/(1+\abs{\lambda}),
\end{equation*}
where $\lambda$ is smaller than all eigenvalues of the locally fitted matrix $\m \Sigma$. This approach evenly lowers the local correlations to a feasible level and is easy to implement on standard software. In practice, we also prefer to work with a sparse version of the Gaussian copula family that concentrates on strong dependencies and sets minor correlations to zero.
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"redpajama_set_name": "RedPajamaArXiv"
} | 330 |
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"redpajama_set_name": "RedPajamaC4"
} | 4,802 |
require File.expand_path('../../spec_helper', __FILE__)
describe "FSEventStream" do
it "watches directories for changes with a delegate" do
eventStream = FSEventStream.alloc.initWithPaths(['/tmp', '/tmp'], delegate:self)
eventStream.should.be.kind_of?(FSEventStream)
eventStream.delegate.should == self
end
it "watches directories for changes with a callback" do
b = Proc.new{}
eventStream = FSEventStream.alloc.initWithPaths(['/tmp', '/tmp'], onChanges:b)
eventStream.should.be.kind_of?(FSEventStream)
eventStream.callback.should == b
end
end
class Receiver
attr_accessor :status
def initialize
@status = :waiting
end
def handleEvents(events)
@status = :called
end
end
describe "An FSEventStream" do
it "starts and stops watching directories" do
eventStream = FSEventStream.alloc.initWithPaths([Dir.tmpdir], onChanges:Proc.new {})
eventStream.start.should == 1
eventStream.stop
end
# it "calls its callback on changes" do
# tmpdir = Dir.tmpdir
# receiver = Receiver.new
#
# eventStream = FSEventStream.alloc.initWithPaths([tmpdir], delegate:receiver)
# eventStream.start.should == 1
#
# filename = File.join(tmpdir, 'fsevents-test')
# FileUtils.rm_f(filename)
# FileUtils.touch(filename)
# File.should.exist(filename)
#
# # Busy wait for the block to run
# started = Time.now
# while (receiver.status == :waiting) and (Time.now - started) < 3
# sleep 0.1
# end
#
# receiver.status.should == :called
# eventStream.stop
# end
end | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,934 |
\section{Introduction}
Devising an expressive and intuitive encoding space to represent and manipulate 3D shapes is a long-standing goal in computer vision and graphics. A key challenge involves translating high-level directives into low-level geometric transformations, which maintain the structural validity of the shape.
Many existing techniques for editing 3D shapes strongly rely on the malleability of \emph{neural implicit representations} of geometry~\cite{im-net_2018, mescheder2019occupancy, deep-sdf_2019, hao2020dualsdf, sdf-style-gan_2022, hertz2022spaghetti}.
However, the resulting shapes often contain undesirable artifacts or even floating parts and are not directly usable in existing 3D modeling and computer-graphics pipelines. An alternative approach for achieving control over manipulations of 3D data is through \emph{procedural methods} that leverage a set of instructions to create a shape~\cite{shapeAssembly, shapeMOD}. Though interpretable, they often rely on coarse cuboid approximations of the geometry in order to enforce programmatic rules.
\input{figs/fig_overview}
In this work, we leverage procedural methods to build a program that maps human-interpretable parameters to a high-quality and intuitively editable 3D mesh. In our program, changing a high-level parameter produces a set of low-level instructions which ensure the modified shape is structurally valid. Our human-interpretable parameter space describes a wide range of geometric properties and enables producing broad variations of detailed shapes. Moreover, we design our program to additionally include semantic part labels. Thus, GeoCode{} contains consistent part segmentation across the generated instances.
To demonstrate the effectiveness of our procedural program, we generate a large dataset with a variety of detailed shapes by sweeping over the interpretable parameter space. Then, we train a neural network to infer the parameter space representation for a point cloud or a sketch as input. Once the input is mapped to the parameter space, and then processed through our program, the resulting shape can be intuitively edited, as seen in Fig.~\ref{fig:teaser}. Moreover, we show that shapes can be easily mixed and interpolated using their interpretable parameter representation. We also show that our system generalizes to inputs from different distributions than the training set, such as free-form user-created sketches, sketches generated from images in the wild, noisy point cloud data, real-world point cloud scans, and more.
In summary, we propose a technique for representing shapes using an intuitively editable parameter space. We build a neural network that learns to map a given point cloud or sketch to the parameter space, and our program translates the parameter representation and produces high-quality mesh outputs by construction. The resulting shapes are native to conventional 3D modeling pipelines. We compare our approach against existing techniques and find that GeoCode{} can more accurately infer and recover 3D shapes. Another notable advantage of GeoCode{} is its ability to generate physically-plausible objects, which we demonstrate by dropping samples from above a flat plane in a physics simulation.
Code and results available: \url{{https://github.com/threedle/GeoCode}}.
\section{Related Work}
\noindent \textbf{Shape reconstruction.} Several works used deep learning methods to reconstruct a 3D object from a sketch. \citet{3d-sketch-deep-volumetric} and \citet{sketch-based-freeform-surface-modeling} predict voxel grid and depth with normal maps, respectively, which are then converted to meshes. Another approach is to produce a CAD shape to improve the structural integrity and edit capability of the reconstructed mesh. For example, Sketch2CAD \cite{sketch2cad} and Free2CAD \cite{free2cad} trained neural networks to parse 2D sketches into sequences of CAD commands, and ComplexGen~\cite{GuoComplexGen2022} created CAD shapes using point clouds as input.
While GeoCode{} also converts sketches and point clouds into 3D editable shapes, the final shapes are fundamentally different from CAD shapes. The final shapes are intuitively controlled by a set of human-interpretable parameters that capture the geometric attributes of different parts of the shape. Changing one parameter results in local changes that can propagate to the rest of the shape to maintain its physical validity. In contrast, CAD objects are composed of low-level operations. Editing one CAD operation changes only a part of the shape, which makes keeping its structural integrity very challenging.
We also note that there is a large body of research on recovering the underlying surface mesh from a point cloud input~\cite{hoppe1992surface, kazhdan2006poisson, kazhdan2013poissonrecon, point2mesh, metzer2021orienting}. Our goal in this paper is different. We aim to produce an intuitively editable mesh version of the point cloud, while surface reconstruction works focus mainly on recovering a holistic 3D shape.
\smallskip
\noindent \textbf{Procedural program.} Procedural models are powerful representations of shapes, which have recently gained increasing popularity. Recent works tried to directly infer procedural program instructions to build a shape. For example, PLAD \cite{PLAD}, ShapeAssembly \cite{shapeAssembly}, and ShapeMOD \cite{shapeMOD} created instruction languages that can construct 3D shapes by defining cuboids and procedurally attaching them.
While these methods use bounding box primitives to abstract the shape, our program builds an expressive geometry of different parts, which can also be easily modified to manipulate the geometry. Another fundamental difference is that we do not directly infer the program instructions. Instead, we first predict the geometric attributes of the input. Then, these attributes are used by our structurally-designed procedural program to build the output shape.
Another approach for object construction is to learn the hierarchical structure of shape parts and assemble them to create the complete shape~\cite{li2017grass, mo2019structurenet, wang22machine, Jones_2022_CVPR, Paschalidou2020CVPR}. Such models rely on annotated datasets, with fine-grained part labels and given relation graphs~\cite{mo2019partnet}. On the contrary, our method operates on holistic sketches or point cloud inputs, and the semantic part information is built-in within our program.
\input{figs/fig_gallery.tex}
\input{figs/fig_structural_integrity.tex}
\section{Method} \label{sec:method}
Given an input object, represented as a 3D point cloud or a 2D sketch image, our goal is to recover an editable 3D mesh. To do that, we develop a procedural program, which enforces a set of shape rules and is parameterized by human-interpretable controls. We recover an editable 3D shape by training a neural network to infer the set of procedural program parameters and run the program to construct the output shape. An overview %
is given in \cref{fig:overview}.
\input{figs/fig_program_inside_look}
In addition to the intuitive controls, our program includes structural information, which results in a consistent semantic part segmentation by construction. \cref{fig:gallery} presents a gallery of our shapes, along with their semantic part segmentation. Moreover, the program keeps the geometrical validity after shape manipulation, as exemplified in \cref{fig:structural_integrity}.
Below, we elaborate on our procedural program in \cref{subsec:program}, its input, output, and operation. In~\cref{subsec:mapping}, we explain the structure of our network which infers the human-interpretable parameters from a point cloud or a sketch input and we explain its training process.
\subsection{Interpretable Shape Program} \label{subsec:program}
Our program supports three types of human-intuitive input parameters: discrete, binary, and continuous, as exemplified in~\cref{fig:teaser}. The program offers disentangled control over the shape, which enables modification of a specific part while keeping all others intact. However, it also models complex structural interactions, such that one part influences another, and the latter is adapted automatically to preserve part contact and retain the structural integrity of the shape. For example, in \cref{fig:teaser}, the number of backrest slats is increased while the rest of the chair remains the same. In contrast, replacing the square seat with a round one in \cref{fig:structural_integrity} makes the seat narrower, which also changes the leg's position and decreases the width of the backrest accordingly.
The program is implemented as a directed acyclic graph (DAG) comprised of \textit{operation nodes} and \textit{edges}. The operation nodes can hold anything from a single value, all the way to complex geometry. Examples of operation nodes include math operations or vector operations; mesh primitives, such as cubes or spheres; line primitives such as a Bezier curve; and rigid transformation (translation, rotation, and scale).
A collection of operations is responsible to generate a \textit{shape element} and one or several shape elements build a \textit{shape part}. For example, the armrest of a chair is considered a part that is comprised of up to three shape elements: an arm, an arm support, and an arm cushion. Parts are attached together and replicated using symmetry rules to create the final shape. We refer our reader to \cref{fig:program_inside_look} for an inside look at how our program is built.
Selected operation nodes are parameterized by the input parameters, allowing the user to interact with the program and control the resulting shape. The operations are chained together, receive, process and then pass the information along their edges, and model shape elements and inter-part influences. The program has a single final node, which outputs a mesh.
We group operation nodes into collections that make up a single semantic part in the final shape. For example, the backrest, seat, armrests, and legs of a chair, all have their own operation-node collections. This design choice results in segmented parts by construction, as shown in \cref{fig:gallery}. The operation-node collections also pass information between each other, which keeps the structural integrity of the shape during edits. This property is demonstrated in \cref{fig:structural_integrity}.
\smallskip
\noindent \textbf{Construction of shape elements.}
In most cases, we model a shape element using curves. In other cases, we use mesh primitives such as a sphere or a plane that we modify to our needs. A curve-based shape element is built out of two curves: a one-dimensional curve $\mathbf{c}_{\tt 1}$, which describes a path in the 3D space, and a two-dimensional curve $\mathbf{c}_{\tt 2}$, which describes the profile of the shape. The profile $\mathbf{c}_{\tt 2}$ is extruded along the path that $\mathbf{c}_{\tt 1}$ defines, a process that creates the shape element's 3D mesh.
Controlling the appearance of the shape element is done by setting the scale of the profile $\mathbf{c}_{\tt 2}$ at each point along the curve $\mathbf{c}_{\tt 1}$. An example of this process is presented in the second step in \cref{fig:program_inside_look}. Another way to alter the appearance of a shape element is to change the profile $\mathbf{c}_{\tt 2}$ itself. Examples of such edits are changing the roundness of the seat of a chair (\cref{fig:teaser}) or interpolating the body of a vase between the circle and square profiles (\cref{fig:interpolate}).
\smallskip
\noindent \textbf{Structural relations.}
A notable benefit of cultivating curves to build shape elements is the ease of defining attachment points on a curve, by setting points with relative distance along the curve. Meaning, given a shape element $\mathcal{A}$, which is built from a curve $\mathbf{c}_{\tt 1}$ and a profile $\mathbf{c}_{\tt 2}$, an attachment point $\mathbf{p}_{\tt 1}$ on the shape element is defined by a single float number $\mathbf{p}_{\tt 1} \in [0,1]$. A value of $0.0$ is the start point of the curve $\mathbf{c}_{\tt 1}$, while a value of $1.0$ is the end point of the same curve. We attach a shape element $\mathcal{B}$ to shape element $\mathcal{A}$ by defining an attachment point on each one. We optionally set the orientation of $\mathcal{B}$ to match the normal of $\mathbf{c}_{\tt 1}$ at the attachment point $\mathbf{p}_{\tt 1}$. Steps 5 and 6 in \cref{fig:program_inside_look} show the placement of a top rail and the cross-rails for a chair in this manner. Scaling shape elements is achieved by calculating distances between attachment points.
Other structural relations do not require attachment points and rely simply on symmetry. This is shown in steps 2 and 3 in \cref{fig:program_inside_look}, where we use symmetry to replicate the legs of the chair and the frame. We also employ rotational symmetry, for example, chairs with swivel legs and vases that have multiple handles in (\cref{fig:gallery}).
\smallskip
\noindent \textbf{Edits.}
The input to our program is a set of human-interpretable parameters. These can set the structure of the shape (\eg, the height of a chair's seat or the points where the handles will be attached to a vase) or modify shape elements' appearance (\eg, the width of a leg or the roundness of the seat of a chair). Structural edits affect the one-dimensional curves (\eg, $\mathbf{c}_{\tt 1}$) and propagate to other shape elements using the attachment points and symmetries that are enforced by our program.
Edits to the appearance of shape elements affect the two-dimensional curves (\eg, $\mathbf{c}_{\tt 2}$) and can cause structural edits as well. An example of this case is increasing the seat's roundness, which causes the chair to get narrower by imposing structural edits on the shape that bring the legs and the frame of the chair closer together. Considering the opposite direction, edits to the structure of the shape cannot affect the appearance of any of the shape elements. For example, changing the height of the seat of the chair will not affect the seat's roundness or the appearance of the legs. %
\smallskip
\noindent \textbf{Part visibility.} In our program, we include support for binary properties, \eg, a chair may or may not include armrests. As part of the probability distribution function for our human-interpretable parameter, we add a \textit{part existence label} to parameters that control a switchable part of the shape.
Continuing the armrests example, all the parameters that control the shape of the armrests may have no effect on the final chair shape if the binary parameter controlling the visibility of the armrests is set to false, and hence, the final chair shape will not include them. Consequently, these parameters will be given the option to be set to that part existence label. A pre-processing stage on our datasets assigns any such parameter to the part existence label if the part it controls is not visible in the final shape. This step is done based on a set of \textit{visibility rules} that are a part of our program.
\subsection{Mapping to the Program Space}
\label{subsec:mapping}
To map a point cloud or sketch input to the human-interpretable parameter representation, we employ an encoder-decoder network architecture. The encoder embeds the input into a global feature vector. Then, we use a set of decoders where each one translates the embedding vector to a single parameter. Together, the final interpretable representation is obtained. Finally, we run the program and recover the 3D shape. \cref{fig:overview} illustrates our system design.
\smallskip
{\noindent \bf Problem formulation.}
We formulate the shape recovery problem as predicting the human-interpretable parameters from a given point cloud or sketch. Let us denote the program parameters as $\{p_i\}$, where each parameter can take $N_i$ discrete values. Continuous program parameters, such as thickness or height, are discretized uniformly over their range. The ground-truth value of the parameter is encoded by a one-hot vector $\mathbf{y}_i \in \{0,1\}^{N_i}$. The ground-truth representation for all the parameters is the concatenation of all $\{\mathbf{y}_i\}$, which we denote as $\mathbf{y} \in \{0,1\}^{\sum_i \! N_i}$.
The prediction of the program parameters by the network is done as follows:
\begin{align}
\hat{\mathbf{y}}_{\tt pc} = \mathbf{D}(\mathbf{E}_{\tt pc}(\mathbf{c})), ~~~
\hat{\mathbf{y}}_{\tt sketch} = \mathbf{D}(\mathbf{E}_{\tt sketch}(\mathbf{s})),
\end{align}
\noindent where $\hat{\mathbf{y}}_{pc}$ and $\hat{\mathbf{y}}_{sketch}$ are the predicted parameters in one-hot representation from the point cloud $\mathbf{c}$ or sketch $\mathbf{s}$, respectively, $\mathbf{E}_{\tt pc}$ is the point cloud encoder and $\mathbf{E}_{\tt sketch}$ is the sketch encoder, and $\mathbf{D}$ denotes the shared decoders.
To train the network, we use our program and construct a dataset of point cloud, sketch, and ground-truth triplets, \ie, $\mathcal{D} = \{(\mathbf{c},\mathbf{s},\mathbf{y})\}$. Then, we train the network with the loss function:
\begin{equation}
\mathcal{L} = \frac{1}{|\mathcal{D}|}\sum_{(\mathbf{c},\mathbf{s},\mathbf{y}) \in \mathcal{D}} \text{CE}(\hat{\mathbf{y}}_{\tt pc},\mathbf{y}) + \text{CE}(\hat{\mathbf{y}}_{\tt sketch},\mathbf{y}),
\end{equation}
\noindent where \text{CE} denotes the \textit{sum of cross-entropy losses} over
each one-hot vector $\mathbf{y}_i$. %
\section{Experiments} \label{sec:exp}
We present qualitative and quantitative evaluations of our method's performance on shape recovery and editing. We demonstrate GeoCode's ability to recover 3D shapes from point clouds and sketches from the held-out test set of our dataset and form shapes in the wild and contrast it with alternative procedural shape reconstruction methods~\cite{mo2019structurenet, shapeAssembly}. Furthermore, we show that our system provides editing capabilities on the reconstructed shapes, such as modifying the geometry of a part, mixing two shapes, and interpolating between two shapes. Finally, we demonstrate GeoCode's robustness against various input perturbations.
\input{tabs/tab_stability}
\smallskip
\noindent \textbf{Dataset and implementation details.}
To train and evaluate our system, we generated train, validation, and test datasets using our shape program. For each generated 3D shape, we sample 1,500 points using Farthest Point Sampling~\cite{eldar1997FPS} and an additional 800 randomly sampled points. We render the 2D sketch images using Blender's~\cite{blender} stroke-bases rendering engine \textit{Freestyle}~\cite{blender-freestyle-engine} from three camera angles while training only on two of them. Sketches are randomly augmented using horizontal flip, stroke dilation, and stroke erosion.
We have control over the sampling granularity of each continuous parameter, as well as how many random shapes are generated for each possible value of each parameter. During the dataset generation, we make sure that no two shapes are the same. We design three programs that we built using Blender~\cite{blender} Geometry Nodes~\cite{blender-geometry-nodes}, each one handling a different shape domain of chairs, vases, and tables, and have 59, 39, and 36 human-interpretable parameters as input, respectively. In total, the train set for each domain contains 9,570 chairs, 9,330 vases, and 6,270 tables. The validation and test sets each contain 957 chairs, 933 vases, and 627 tables.
For the point cloud encoder, we use Dynamic Graph CNN (DGCNN)~\cite{wang2019dynamic} and employ the VGG architecture~\cite{simonyan2015a_vgg} for the sketch encoder. The decoder utilizes a multi-layer perceptron with three layers for each program parameter. Since each human-interpretable parameter may have a different number of values, the output layer size of each decoder varies according to the number of possible classes of the parameter the decoder is responsible for.
\input{tabs/tab_baseline}
\smallskip
\subsection{Shape Recovery} \label{subsec:exp_reconstruction}
We evaluate our method's ability to recover 3D shapes from samples in our test set. We also consider out-of-distribution examples, including shapes from %
COSEG~\cite{wang12aca}, real-world scan-based datasets such as ScanNet~\cite{dai2017scannet}, hand-drawn sketches, and sketches generated by CLIPasso~\cite{vinker2022clipasso} from images in the wild.
\smallskip
{\noindent\bf Baselines and evaluation metric.}
We compare our method against two structure-aware baselines, StructureNet~\cite{mo2019structurenet} and ShapeAssembly~\cite{shapeAssembly}. The decoders of both these models work in conjunction with a point cloud encoder to reconstruct shapes from unannotated point clouds. As instructed by the authors, we train a point cloud encoder~\cite{qi2017pointnetplusplus} on the PartNet dataset~\cite{mo2019partnet} to map point clouds to the StructureNet and to the ShapeAssembly latent spaces. During inference, we use the trained encoder with the decoders released by the authors to reconstruct shapes from point clouds sampled from COSEG shapes~\cite{wang12aca}.
For quantitative evaluations, we use the bi-directional Chamfer Distance~\cite{chamfer}, defined as the average minimum distance of each point in one shape to a point in the other shape and vice versa. The Chamfer Distance is calculated using 10,000 randomly sampled points on the ground truth and reconstructed shapes.
\smallskip
\noindent \textbf{Results on our dataset.}
We first verify qualitatively that our system can correctly reconstruct shapes given an input point cloud or sketch. In \cref{fig:gallery}, we show examples of reconstructed shapes and their corresponding inputs. The reconstructions are visually similar to the input point clouds and sketches, confirming that our system can faithfully recover shapes for inputs that are within our data distribution.
\smallskip
{\noindent\bf Structural stability.} During dataset generation we take measures to make sure our samples are structurally stable. For instance, the vases could have a base that is too narrow and will make them tip over under their own weight in real life. During inference time, we do not enforce any stability-inducing modification on our predictions. To ensure that our predictions are structurally stable, we define a binary \textit{stability} metric. We consider a shape to be \textit{stable} if it has no parts that are detached, and it remains standing in a physics simulation after a drop onto a flat plane from a height of 5\% of the shape's height. In~\cref{tab:stability}, we can see that GeoCode{}'s ground truth shapes are mostly stable. Moreover, the predictions our system generates are on par with the ground-truth data in terms of their stability.
\input{figs/fig_various_pc_reconstruction}
\smallskip
{\noindent\bf Comparison on the COSEG dataset.}
Next, we evaluate the generalization power of GeoCode{} beyond our generated data distribution and compare our reconstruction performance with previous works~\cite{mo2019structurenet, shapeAssembly}. For this experiment, we use the COSEG~\cite{wang12aca} chair and vase shape sets of 400 chairs and 300 vases. Neither we nor the baseline methods have been trained on this dataset, so we find it a fair test for the generalization ability.
From each shape, we create point clouds with 1,500 points that are sampled using Farthest Point Sampling~\cite{eldar1997FPS} and additional 800 randomly sampled points. \cref{tab:baseline} shows the average Chamfer Distance for the chair and vase sets from COSEG for different reconstruction methods. We observe that our system produces more accurate reconstructions than the baselines. Visual comparison results are presented in the supplementary.
\smallskip
{\noindent\bf Reconstruction from out-of-distribution shapes.}
To further demonstrate the generalization capability of GeoCode, we choose three known 3D shape datasets and reconstruct shapes from these datasets for point cloud inputs.
First, we consider two real-world scan-based datasets, ScanObjectNN~\cite{uy-scanobjectnn-iccv19} and ScanNet~\cite{dai2017scannet}, which include LiDAR scans of objects and scenes, respectively. For ScanObjectNN, we use the point clouds from the main dataset. For ScanNet, we extract objects from the scene using the provided segmentation masks and randomly sample point clouds from them. We also consider ShapeNet~\cite{shapenet2015} and produce point clouds by randomly sampling its shapes. For all the datasets, we end up with point clouds having 2,048 points.
In~\cref{fig:various_pc_reconstruction} we show the reconstruction for two domains, chairs, and tables. Our method is able to recover shapes that capture the main geometric features of the out-of-distribution samples. Moreover, point clouds produced from scanned objects are contaminated by noise and partiality. Still, GeoCode{} is able to produce compelling results in these cases as well.
\input{figs/fig_traced}
\input{figs/fig_clipasso}
\smallskip
{\noindent\bf Reconstruction from hand-drawn sketches.}
\cref{fig:traced} shows reconstruction examples from free-form, outlined, and in-the-wild sketches. Our system is able to generalize and capture the main features from the sketches and produce shapes that are visually similar to the input sketches. These sketches possess different styles and are drawn from different angles compared to the sketches we trained on.
\input{figs/fig_mix_shapes}
\smallskip
\noindent \textbf{Reconstruction from CLIPasso sketches.}
\cref{fig:clipasso} shows reconstruction examples from sketches generated by CLIPasso~\cite{vinker2022clipasso}. We use CLIPasso to convert images of chairs, vases, and tables found in the wild to sketches with various numbers of strokes. The sketches produced by CLIPasso are noisy and often contain artifacts, these make them visually different from sketches found in our datasets. %
We observe that, overall, the shapes reconstructed from our system are sensible and correctly recover features from the shapes in the original images. This suggests that our network is able to express shapes with human-interpretable parameters given an out-of-distribution input.
\subsection{Shape Editing} \label{subsec:exp_edits}
Beyond shape recovery, we show that our interpretable parameter space allows for easy shape editing. This property is exemplified by shape mixing and shape interpolation.
\input{figs/fig_interpolate}
\input{figs/fig_selective_interpolate}
\smallskip
{\noindent\bf Shape mixing.}
We show that our system can mix shapes by selecting a set of parameters from the human-interpretable parameter space representation of source shapes and combining them to form a representation that produces the final shape. When mixing disjointed parts, such as handles from one shape and legs from another, it is guaranteed that the mixed shape will capture the selected parts from the corresponding source shapes and transfer them to the resulting shape. \cref{fig:shape_mixing} shows mixing examples between pairs of shapes reconstructed from point clouds and sketches. The resulting shapes are based on the first shape with the addition of selected parts from a second shape. The final mixed shape is structurally valid and physically plausible.
\smallskip
{\noindent\bf Shape Interpolation.}
GeoCode{} is effective in interpolating between shapes and interpolating between selected parts of the shapes.~\cref{fig:interpolate} shows interpolation between pairs of shapes reconstructed from point clouds and sketches where we interpolate across all the parameters for $\alpha \in [0,1]$ in 0.2 increments.~\cref{fig:selective_interpolate} shows interpolation over selected geometric features for $\alpha \in [0,1]$ in 0.25 increments.
\input{tabs/tab_simplified_meshes}
\subsection{Robustness} \label{subsec:exp_robustness}
In this section, we evaluate our method's robustness to slightly deformed inputs. Specifically, we both qualitatively and quantitatively evaluate the performance of GeoCode{} on the following deformation: sketches of simplified shapes and point clouds sampled from simplified shapes. For other deformation/perturbation including sketches rendered from novel angles, random point clouds with decreasing number of points, and point clouds with added Gaussian noise, we refer the reader to the supplemental.
\smallskip
{\noindent\bf Reconstruction from simplified shapes.}
Simplification reduces the number of polygons and, therefore, the level of detail in the shape. In this experiment, we show that our system is able to reconstruct shapes from both point cloud and sketch inputs that are produced from simplified meshes. We compare the non-simplified reconstruction where we have approximately 25K faces (this number varies depending on the visible parts in the shape) to simplified versions where the average number of faces is decreased by a factor of $10\times$ ($\sim$2K faces), and a factor of $100\times$ ($\sim$120 faces).
\cref{tab:simplified_meshes} shows the average Chamfer Distance between the predictions from the point cloud or sketch inputs that were produced from the simplified shape and the non-simplified original ground-truth shape. It is evident that our method is robust to simplification with a factor of $10\times$ and performs slightly worst for a simplification factor of $100\times$. We refer the reader to the supplementary for qualitative results that visualize the simplified shapes and their reconstructions.
\section{Conclusions and Future Work}
In this paper, we presented GeoCode{}, a novel method that represents shapes using a human-interpretable parameter space. We achieved this by building a procedural program controlled by the intuitive parameter space and training a neural network to predict the parameter representation for an input point cloud or sketch. We showed that our system produces structurally valid 3D geometry and enables editing of the resulting shape easily and intuitively. We also demonstrated that GeoCode{} generalized well to out-of-distribution point clouds and sketches in the wild.
In the future, we are interested in incorporating additional attributes into our procedural program to include UV texture maps and material properties for the resulting shapes. Another venue we aim to explore is extending our method for representing 3D scenes.
\section{Acknowledgments}
We thank the University of Chicago for providing the AI cluster resources, services, and the professional support of the technical staff. This work was also supported in part by gifts from Adobe Research. Finally, we would like to thank R. Kenny Jones, Chen Dudai, and the members of 3DL for their thorough and insightful feedback on our work.
\section{Additional Results} \label{sec:sup_results}
\subsection{Complementary Videos} \label{subsec:sup_complementary_videos}
\smallskip
{\noindent\bf Demo video.} As part of the supplementary, we provide a video (\textbf{GeoCode Demo.mp4}) showing the variety of shapes that our method can produce. In addition, the video will help the reader understand the ease of controllability that the method provides, as well as show how the structural integrity is maintained continuously while editing the shapes.
\par
For a short length at the beginning of the video, we also show the ease at which shapes can be changed using human-interpretable parameters, by showing the shape along with the actual values of selected parameters of the chair:
\begin{itemize}[topsep=8pt,itemsep=1pt,partopsep=1pt, parsep=1pt]
\item Cross Rail Count (discrete) - controls the number of cross rails appearing on the backrest of the chair.
\item Armrest (boolean) - toggles the existence of the armrests, when set to \textit{True} the armrests will appear while maintaining the structural integrity of the current chair shape.
\item Armrest Height (continuous) - controls the attachment point of the base of the armrests to the backrest of the chair. Higher values, mean the armrests are connected to the frame at a higher position. The value is treated as a percentage of the height of the frame, for example, a value of 0.5 will place the armrests at an equal height between the seat and the top of the frame.
\item Seat Roundness (continuous) - controls the shape of the seat. The higher this value is, the more rounded the seat shape will be. The video helps to show that the entire chair is adjusted automatically to fit the changing seat shape and maintain the structural integrity of the chair. We show how this is achieved when we describe our method in \cref{fig:program_inside_look} of the main paper.
\item Seat Height (continuous) - controls the height of the seat. The legs, the frame, and other components of the chair, like the cross rail, are all adjusted automatically to fit the seat position in a continuous manner.
\item Backrest Curvature (continuous) - this parameter controls how rounded the cross rails and top rail are. A value of 0.0 means that the backrest is completely flat, while higher values will make the backrest appear more rounded. In reality, it also controls other components that are not shown in the video, like the vertical rails.
\item Backrest Slant (continuous) - allows adjusting how slanted the backrest is. A value of 0.0 means the backrest is perpendicular to the ground (or equivalently, to the seat), while higher values make the backrest lean backward in a rounded fashion.
\end{itemize}
\smallskip
{\noindent\bf Simulation demostration.} As explained in \cref{sec:exp} of the main paper, our \textit{structural stability} metric checks that a given shape is structurally valid by testing two things, checking for the existence of loose parts, and dropping the shape from a height onto a flat plane and testing that the height of the shape has not changed after the drop. In the attached video (\textbf{GeoCode Simulation.mp4}) we show 15 samples tested in that simulation, from all three domains, chair, vase, and table. The drop height was increased for the purpose of visual demonstration, from the 5\% that we used when creating \cref{tab:stability} to 20\% of the shape's longest dimension.
\subsection{Visual Comparison on the COSEG Dataset} \label{subsec:sup_results_shape_recovery}
We present qualitative results that complement \cref{tab:baseline} shown in the main paper. In this experiment, we evaluated GeoCode{} on the shape recovery task while comparing to the works~\cite{mo2019structurenet, shapeAssembly} using COSEG~\cite{wang12aca} chair and vase shape sets.
\par
The qualitative results in \cref{fig:other_methods} show the generalization power of GeoCode{} while also explaining the drastic decrease in average Chamfer Distance compared to the other two methods. GeoCode{} can more accurately capture geometric features of the ground-truth shapes when compared to both StructureNet~\cite{mo2019structurenet} and ShapeAssembly~\cite{shapeAssembly}.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth]{figures/other_methods.pdf}
\caption{\textbf{Visual Comparison on the COSEG Dataset.} We show the reconstruction results of GeoCode{}, StructureNet~\cite{mo2019structurenet}, and ShapeAssembly~\cite{shapeAssembly} on point cloud inputs generated from COSEG~\cite{wang12aca} samples. The compared methods yield crude approximations of the input geometry and may output a shape with detached parts. In contrast, GeoCode{}'s predictions are much closer to the original shape while also being structurally sound.}
\label{fig:other_methods}
\end{figure}
\section{Additional Experiments} \label{sec:sup_experiments}
\subsection{Shape Recovery} \label{subsec:sup_exp_shape_recovery}
\smallskip
{\noindent\bf Reconstruction from point cloud \vs sketch.} We study the reconstruction performance of both point cloud and sketch inputs on our test sets for all three domains. Note that the calculation of the average Chamfer Distance for the sketch input type was done using only the camera angles that were seen during training.
\par
In \cref{tab:pc_sketch} we see that, overall, reconstruction from point clouds generally performs better compared to reconstruction from sketch inputs. For example, ~\cref{tab:pc_sketch} shows that for the chair domain, the average Chamfer Distance when reconstructing from point clouds is 2.5 times better compared to reconstruction from sketches. The vase and table domains show similar results. We attribute this, in part, to the fact that the scale of the shape cannot be effectively captured from a single image, while a point cloud inherently encodes the scale of the shape. The qualitative results for this experiment are shown in \cref{fig:pc_sketch}.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth]{figures/pc_sketch_comparison.pdf}
\caption{\textbf{Reconstruction from point cloud \vs sketch.} We compare the reconstruction results of GeoCode{} from point clouds and sketches. The Chamfer Distance for each prediction is noted near the reconstruction result. In most cases, point cloud reconstruction yields a better result. However, sketch reconstruction may sometimes perform better (first vase and first table).}
\label{fig:pc_sketch}
\end{figure}
\input{tabs/tab_pc_sketch.tex}
\subsection{Robustness} \label{subsec:sup_exp_robustness}
\smallskip
{\noindent\bf Reconstruction from simplified shapes.}
We complement our quantitative results shown in \cref{tab:simplified_meshes} of the main paper with qualitative results. Additionally, we provide the quantitative results for the table domain.
\cref{tab:simplified_meshes_table} shows the average Chamfer Distance for the table domain. Similarly to the other domains, both point cloud and sketch reconstruction experience only a slight increase in the Chamfer Distance for the lower simplification factor of $\times 10$ ($\sim$2K faces). However, a drastic increase in Chamfer Distance is registered for both input types at the higher simplification factor $\times 100$ ($\sim$120 faces). We show the qualitative results for this experiment in \cref{fig:simplified_meshes}.
\input{tabs/tab_simplified_meshes_table.tex}
\input{figs/fig_simplified_meshes}
\smallskip
{\noindent\bf Reconstruction from randomly sampled point clouds.}
Our point clouds during training and testing contain 1,500 points sampled using Farthest Point Sampling~\cite{eldar1997FPS} and an additional 800 randomly sampled points. In this experiment, we test the robustness of reconstruction performance when the system is given a decreasing number of exclusively randomly sampled points. For each shape in our test set, we randomly sample point clouds with 2,048, 1,024, and 512 points. We then perform reconstructions from these point clouds and compute the average Chamfer Distance between the reconstructed and ground-truth shapes.
\cref{tab:random_pc} shows that reconstruction from a decreasing number of randomly sampled points increases the average Chamfer Distance between the reconstruction and the ground truth. However, the recon results are still visually similar to ground truth, as evident in~\cref{fig:random_pc}. In the last decrement, with only 512 points, our system produces shapes that resemble the overall structure of the ground truth, however, some details are captured with degraded accuracy.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{figures/random_pc.pdf}
\caption{\textbf{Reconstruction from randomly sampled point clouds.} We use random sampling with a progressively decreasing number of points. At 2048 points, GeoCode{} produces accurate reconstructions. A minor degradation in the level of detail appears for 1024 points, \eg, the second vase's body is slightly different compared to the ground truth. Lastly, at only 512 points, more considerable reconstruction inaccuracies start to appear. An example of this is seen in the first and last vases, where the body and handles are less accurate. Another example is the second chair, where the seat height, top rail, and armrests are inaccurate.
}
\label{fig:random_pc}
\end{figure}
\input{tabs/tab_random_pc}
\smallskip
{\noindent\bf Reconstruction from point clouds with Gaussian noise.} To further examine the robustness of our method we add varying levels of Gaussian noise to the point clouds in our test set and evaluate the reconstruction both qualitatively and quantitatively. We consider Gaussian noise with 0 mean and a progressively increasing standard deviation of 0.5\%, 1.0\%, and 2.5\% of the overall size of the shape, corresponding to $\sigma = 0.01, 0.02$, and $0.05$.
\par
~\cref{tab:gaussian} shows that our method remains somewhat resilient up to $\sigma = 0.02$, and begins to struggle when the noise level increases to $\sigma = 0.05$. The qualitative analysis in \cref{fig:gaussian} is consistent with this result.
\input{figs/fig_gaussian.tex}
\input{tabs/tab_gaussian}
\smallskip
{\noindent\bf Reconstruction from sketches of various camera angles.} We compare the reconstruction from sketches rendered from three different angles. During training, our system sees two of the angles while we leave the third \textit{novel angle} unseen. For each shape in our test set, we render sketches from all three angles and perform reconstructions from these sketch images. ~\cref{tab:sketch_angles} shows that the average Chamfer Distance between reconstructed and ground-truth shapes is similar when we reconstruct from the two sketch angles that we trained on. However, reconstructing from the novel angle results in a larger Chamfer Distance. ~\cref{fig:sketch_angles} shows that the shapes reconstructed from the novel angle are still visually close to the ground truth.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth]{figures/sketch_camera_angles.pdf}
\caption{\textbf{Reconstruction from sketches of various camera angles.} Comparison between reconstruction from sketches rendered at different camera angles. The novel angle is an angle that was never seen during training. The two first angles perform similarly with accurate results, while the novel angle produces less accurate results. For example, in the second chair, the backrest is slanted backward for the reconstruction from the novel angle. However, the ground-truth shape and the predictions from the first two angles show a forward-slanted backrest. Another example is the second vase, where the body is noticeably thinner.}
\label{fig:sketch_angles}
\end{figure}
\input{tabs/tab_sketch_angles}
\subsection{Ablation Study} \label{subsec:sup_exp_ablation}
\smallskip
\noindent \textbf{Classification vs. regression.} In our work we supported discrete, boolean, and continuous parameters. We explained in \cref{sec:method} that continuous parameters are discretized and that the loss function is cross-entropy. In this ablation experiment, we test the performance of reconstruction in terms of average Chamfer Distance while comparing the use of classification \vs regression for the continuous parameters.
\par
To make a fair comparison we also incorporate the \textit{existence lable} that we discuss in the main paper in \cref{sec:method}. To achieve that, each decoder of a continuous parameter that controls a part of the shape that can be invisible will output two elements instead of one. The first element will be the predicted value of the parameter, while the second element predicts the existence label. Whenever the prediction of the second element exceeds 0.5 we say that the actual predicted value is the existence label. During loss calculation, we accumulate the Mean Square Error of both elements. If the existence label was predicted (the second element is greater than 0.5) then we avoid collecting the regression loss of the first element (which predicts the value of the parameter).
In \cref{tab:regression} we can see that using classification for all the parameters yields better average Chamfer Distance results.
\input{tabs/tab_regression.tex}
\smallskip
\noindent \textbf{Joint training \vs separate training.} In this experiment we wish to find out whether training separately improves the average Chamfer Distance for recovered shapes. We train the network as two separate networks, one for point cloud inputs and another for sketch inputs. To begin with, the encoders (point cloud encoder and sketch encoder) are already separated, but we have to duplicate the decoders in such a way that both networks now have their own decoding network. We note that this effectively doubles the number of weights in the decoding part of each separate network.
\input{tabs/tab_jointly.tex}
In \cref{tab:jointly}, we show the average Chamfer Distances for point cloud and sketch reconstruction for both of these methods. Training separately yielded a minor improvement in both point cloud and sketch reconstructions but at the cost of additional complexity.
\section{Additional Implementation Details} \label{sec:sup_impl}
\subsection{Dataset} \label{subsec:sup_dataset}
\smallskip
\noindent \textbf{Dataset preprocessing.} In the data generation step, we create for each sample an OBJ file containing the shape, the sketches rendered from three camera views, and a YAML file containing the labels of each parameter. Prior to training, we execute a preprocessing step where we prepare for each generated sample, two point clouds, and one YAML file which holds the normalized values of the parameters.
We begin by sampling 1500 points of the generated object mesh using Farthest Point Sampling~\cite{eldar1997FPS}, then another point cloud using random sampling which will be used for augmentation. When we retrieve a sample we take all the Farthest Point Sampled points and randomly pick 800 points from the 1,500 randomly sampled points.
Finally, we convert the shape's YAML file to its normalized form, according to the following conditions: 1) integer parameters will start from 0, 2) float (and vector) parameters will be in the range of [0.0, 1.0] mapping the original values to that range is done linearly, 3) any parameter which is not visible in the final shape will be set to the \textit{existence label} which is implemented with the value -1.0. The idea here is to avoid forcing the network to decide on a label for parameters that are not visible.
\smallskip
\noindent \textbf{Recipe file.}
To allow for an easy dataset generation according to the needs of the user, we created the \textit{recipe} configuration file which contains the instructions to create the dataset.
\par
The recipe file describes the minimum and maximum allowed values for each parameter. Continuous parameters also include the sampling granularity which is used during the data generation step to split the allowed range uniformly with a number of samples matching the user-specified sampling value.
\subsection{Network Architecture} \label{subsec:sup_network}
\smallskip
\noindent \textbf{Point cloud encoder.} We base our point cloud encoder on the four-layer classification architecture of DGCNN~\cite{wang2019dynamic}. Each layer is comprised of two parts, k-NN and then \textit{EdgeConv}. k-NN finds the closest k points of each point. In the beginning, the closeness is determined in terms of physical distances between the points, but in the next layers, the closeness is determined in the graph-feature space. In our testing, we use $k=20$.
\par
For each point $x_{i}$, we take all k closest points $x_{i,j}$ to it, and EdgeConv will have $k$ inputs, where each one is the concatenation of the features of $x_{i}$ and the features of $x_{i}-x_{i,j}$. The concatenated features are fed to a multi-layer perceptron with a matching input size, a chosen channel size, and no hidden layers. The output is then followed by a batch normalization and then a Leaky ReLU with a negative slope of 0.2. For EdgeConv will take the maximum between all the outputs. After EdgeConv is completed for all the points we continue to the next k-NN and EdgeConv pair.
\par
We choose the channel size in each of the four EdgeConv layers to be 16, 16, 32, and 64. The outputs from the four layers are aggregated together and enter another multi-layer perception, with no hidden layer and an output size of 64. Finally, max and average pooling are applied to that result and aggregated to form our embedding vector of size 128.
\smallskip
\noindent \textbf{Sketch encoder.} For the sketch encoder we base our architecture on VGG11~\cite{simonyan2015a_vgg} encoder. Our encoder assumes the following numbers of channels $[8, M, 16, M, 32, 32, M, 64, 64, M, 64, 64, M]$ where $M$ is a 2D max-pooling layer with kernel size 2 and stride 2. The numbers specify the number of channels in each 2D convolution layer, all have a kernel size of 3 and padding of 1, and each one is followed by a batch normalization and then ReLU. The result goes through max pooling and an additional linear layer which outputs an embedding vector of size 128.
\par
The input size to the encoder remains unchanged at 224x224 square images. In our tests we only use grayscale images of sketches, so we further optimize to only have a single channel for the input images.
\smallskip
\noindent \textbf{Decoding network.} As already stated, each decoder in the decoding phase is a multi-layer perception made out of three layers. The first and second layers are each followed by batch normalization, Leaky ReLU with a negative slope of 0.2 then dropout with a probability of 0.5. As shown in the System Overview~\cref{fig:overview}, the input to the multi-layer perceptron is the embedding vector which, as stated, has a size of 128, the next two hidden layer sizes are 128 and 64. Finally, the output size is dictated by the number of labels of that parameter, in addition to the part existence label when required. We discuss the part existence label in \cref{sec:method}.
\subsection{Training Scheme} \label{subsec:sup_training}
We train our network for 600 epochs, with an ADAM optimizer, an initial learning rate of 1e-2, and a scheduler with a step size of 20 and a gamma value of 0.9. Our batch size was 33.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,664 |
\section{Introduction}
\label{sec1}
Large-scale genome-wide association studies (GWAS) provide opportunities for developing
genetic risk prediction models that have the potential to improve disease prevention, intervention, and treatment.
In epidemiology and genetics, there is a growing interest in utilizing the published summary statistics, especially those from GWAS, for disease risk prediction. The abundant summary data can enhance the power in signal detection using the framework of meta-analysis. Comparing with the individual-level data, the summary data are less privacy-sensitive and are more communication efficient for data sharing. However, statistical properties of learning based on the summary data remain largely unknown.
\subsection{Problem formulation}
Let $\bm X\in\mathbb{R}^{n\times p}$ denote the standardized genetic variants measurements in $n$ independent individuals whose $i$-th row is $\bm x_i^{\intercal}$, where $p$ is the dimension of genetic variants. Let $y\in\mathbb{R}^n$ denote the mean-adjusted response vector in this sample.
In the linear model for the association between the outcome and the covariates,
\begin{align}
\label{lm1}
y_i=\bm x_i^{\intercal}\bm \beta+\epsilon_i,
\end{align}
where $\mathbb{E}[\epsilon_i|\bm x_i]=0$ and $\mathbb{E}[\epsilon_i^2|\bm x_i]=\sigma^2$, the goal is to estimate and infer the effect size vector $\bm \beta\in\mathbb{R}^p$ and its functionals using only the summary data but not the individual-level data.
GWAS reports the marginal statistics
\[
\widehat{S}_j=\bm X_{.,j}^{\intercal}\bm y/n,~j=1,\dots, p
\]
and their estimated standard errors.
Besides the marginal statistics $\widehat{\bm S}$, an estimator of the covariance matrix of $\bm x_i$ is often needed for estimation and inference. One challenge is that the empirical covariance matrix for the samples involved in $\widehat{\bm S}$ is often not available because this genomic data set is too large or privacy-sensitive to share. That is, we do not observe $\widehat{\bm \Sigma}=\bm X^{\intercal}\bm X/n$. Let $\bm \Sigma$ denote the oracle $\mathbb{E}[\bm X^{\intercal}\bm X/n]$. A common practice is to obtain an estimate of covariance matrix $\bm \Sigma$ from some external genome panel, such as the 1000 genome project (\url{https://www.internationalgenome.org}).
Let $\tilde{\bx}_i\in\mathbb{R}^p, i=1,\dots,\tilde{n}$, denote the genotype of the observations in the external data and define
\[
\widetilde{\Sig}=\frac{1}{\tilde{n}}\sum_{i=1}^{\tilde{n}}\tilde{\bx}_i\tilde{\bx}_i^{\intercal}.
\]
We call $\tilde{\bx}_i, ~i=1,\dots,\tilde{n}$, the proxy data and $\widetilde{\Sig}$ the proxy covariance matrix.
In this work, we assume that $\mathbb{E}[\tilde{\bx}_i\tilde{\bx}_i^{\intercal}]=\bm \Sigma$. That is, the proxy covariates have the same covariance structure as the covariates in computing the marginal statistics $\widehat{\bm S}$.
In genetic applications, the number of SNPs can be much larger than the sample sizes. Hence, we develop methods for the regime that $p$ is larger or much larger than $\max\{n,\tilde{n}\}$.
\subsection{Motivating applications with the proxy data}
Learning the linear model (\ref{lm1}) with summary statistics has important applications in genomic studies.
Polygenic risk score (PRS) regression concerns predicting a certain health-related outcome using the associated single nucleotide polymorphisms (SNPs). In fact, PRS can be formulated as a high-dimensional regression problem \citep{Chen20}. It is crucial in PRS prediction to provide confidence intervals for $\bm x_*^{\intercal}\bm \beta$ given a new individual's genomic information $\bm x_*\in\mathbb{R}^p$.
Besides, summary data provide the opportunity to combine multiple studies (e.g., cohorts) into one large study to increase the sample size \citep{albinana2020leveraging}, which is also the goal of meta-analysis \citep{deelen2019meta}. Hence, it is also of statistical interest to estimate and make inference of $\bm \beta$ with the proxy data.
Another application of the proxy-data based inference is distributed inference, where the whole data set contains \textit{i.i.d.} observations but the data are distributed at multiple remote machines. Distributed algorithms estimate the target parameter by communicating some summary information across machines. To reduce the communication costs, the gradient vectors are communicated but not the high-dimensional Hessian matrix. The overall Hessian matrix can be approximated by a local matrix or by subsampling. See, for example, \citet{jordan2018communication} and \citet{wang2019utilizing}. This type of distributed inference also falls in the category of proxy-data based inference. In Section \ref{sec-diss}, we discuss some other modern applications of the proxy-data based inference in causal inference and genetics.
To summarize, estimation and inference for $\bm \beta$ and $\bm x_*^{\intercal}\bm \beta$ based on the proxy data $\widehat{\bm S}$ and $\widetilde{\Sig}$ have significant practical values. The statistics $\widehat{\bm S}$ and $\widetilde{\Sig}$ are also known as two-sample summary data.
In the existing literature, the ``two-sample'' setting has been largely used to refer to having two samples from different distributions, such as two-sample testing. To avoid confusion, the term ``proxy data'' is adopted.
Motivated by aforementioned applications, we study proxy-data based statistical inference in high-dimensional linear models. The key challenge is that the sample covariance matrix observed $\widetilde{\Sig}$ is exclusive of the samples for computing $\widehat{\bm S}$. This distinguishes the current problems with, say, the semi-supervised problems and transfer learning problems. We highlight some of our key contributions.
\subsection{Main results and our contribution}
\label{sec1-3}
Methodology-wise, we consider a proxy-data based $\ell_1$-penalized regression estimator and prove that it is minimax optimal under typical regularity conditions. We further propose debiased estimators to make inference for $\beta_j$ for some fixed $1\leq j\leq p$ and $\bm x_*^{\intercal}\bm \beta$ with the proxy data, respectively. The debiased estimators are asymptotically normal and can be used to construct confidence intervals and for multiple testing under certain conditions. We also demonstrate that confidence interval length given by the debiased estimator of $\beta_j$ has minimax optimal length under certain conditions.
Theoretically, we discover some interesting and new phenomena with the proxy-data based learning. First, the minimax rates for estimation are slower than the corresponding rates with one-sample individual data, even if $\tilde{n}\rightarrow \infty$. The relative loss gets larger when the signal-to-noise ratio gets larger.
Second, comparing with the debiased Lasso based on individual data, the debiased Lasso estimator of $\beta_j$ based on proxy data has larger bias and variance and its asymptotic normality requires a different sample size conditions (Theorem \ref{thm-db}). In addition, the results also imply that simply treating the two-sample summary data as one-sample data can lead to invalid inference.
Third, the power of the proxy data-based inference is always no larger than the power in the conventional setting even if $\tilde{n}\rightarrow \infty$. In fact, the power function based on the proxy data is upper truncated by a function related to the magnitude of $\min\{n, \tilde{n}\}/s$. This demonstrates a curse for the proxy data-based inference with dense signals: for finite $\min\{n, \tilde{n}\}/s$, the power for strong signals is largely deminished.
\subsection{Related literature}
Estimation and inference for the regression coefficients have been extensively studied in high-dimensional linear models based on individual data from one sample. Many penalized methods have been proposed for prediction, estimation, and variable selection in high-dimensional linear models. To name a few, \citet{Lasso, FL01, Zou06, CT07, MB10, Zhang10}.
Statistical inference for each regression coefficient has been studied in the conventional setting. One stream of methods is inference based on consistent variable selection. Under the assumption that the minimal signal strength is sufficiently large \citep{zhao2006model,wainwright2009sharp}, all the true signals can be consistently selected based on regularization \citep{Lasso, FL01, Zhang10}. Hence, the high-dimensional model is reduced to the low-dimensional problem and classical fixed-dimensional inference tools can be applied.
The second stream of methods does not rely on the minimal signal strength condition.
\citet{ZZ14}, \citet{van14}, and \citet{JM14} consider debiased estimators in linear models and generalized linear models. The minimaxity and adaptivity of confidence intervals have been studied in \citet{CG17}. \citet{NL17} proposes a general framework to de-bias regularized estimators in different models.
Weaker sample size conditions or sparsity conditions have been studied in \citet{JM19} and \citet{ZB18} under certain assumptions. \citet{cai2019individualized} and \citet{javanmard2020flexible} propose methods for inference for a linear functional of the regression coefficients in linear models.
The proxy data-based estimation and prediction methods have appeared in genetic applications. \citet{LDpred} introduces an Bayesian approach for PRS based on summary data. \citet{Lassosum} and \citet{Chen20} both consider shrinkage methods as extensions of the Lasso. The method in \citet{Lassosum} is for linear models and the method in \citet{Chen20} can deal with binary outcomes based on approximations. However, the statistical guarantees and the choice of tuning parameters are largely unknown. In addition,
inference for the linear functionals of high-dimensional regression coefficients based on proxy data has not been studied in literature.
\subsection{Organization and notation}
In Section \ref{sec-method}, we describe the $\ell_1$-regularized method for estimating $\bm \beta$ with high-dimensional proxy data and study its convergence rate and minimax optimality. In Section \ref{sec3}, we construct the debiased estimator of $\beta_j$ based on proxy data and study its limiting distribution
In Section \ref{sec3-3}, we construct confidence interval for $\bm x_*^{\intercal}\bm \beta$ and provide theoretical guarantees. In Section \ref{sec-simu}, we study the empirical performance of our proposals via extensive numerical experiments. In Section \ref{sec-data}, we apply the one-sample and two-sample methods to a GWAS in an outbred mice population. In Section \ref{sec-diss}, we discuss some other summary data motivated problems for future research. The proofs and other supplementary information are provided in the supplements \citep{Supp}.
\textbf{Notation}. For real-valued sequences $\{a_n\}, \{b_n\}$, we write $a_n \lesssim b_n$ if $a_n \leq cb_n$ for some universal constant $c \in (0, \infty)$, and $a_n \gtrsim b_n$ if $a_n \geq c'b_n$ for some universal constant $c' \in (0, \infty)$. We say $a_n \asymp b_n$ if $a_n \lesssim b_n$ and $a_n \gtrsim b_n$. $c, C, c_0, c_1, c_2, \cdots, $ and so on refer to universal constants in the paper, with their specific values possibly varying from place to place.
For a vector $\bm v \in \mathbb{R}^d$ and a subset $S \subseteq [d]$, we use $\bm v_S$ to denote the restriction of vector $\bm v$ to the index set $S$. For a matrix $A\in\mathbb{R}^{n_1\times n_2}$, let $\Lambda_{\max} (A)$ denote the largest singular value of $A$, $\Lambda_{\min}(A)$ denote the smallest singular value of $A$, and $\|A\|_{\infty,\infty}$ denote $\max_{i\leq n_1,j\leq n_2}|A_{i,j}|$. For a random variable $u\in\mathbb{R}$, define its sub-Gaussian norm as $\|u\|_{\psi_2}=\sup_{l\geq 1} l^{-1/2}\mathbb{E}^{1/l}[|u|^l]$. For a random vector $\bm U\in\mathbb{R}^{n}$, define its sub-Gaussian norm as $
\|\bm U\|_{\psi_2}=\sup_{\|\bm v\|_2=1,\bm v\in\mathbb{R}^{n}}\|\langle \bm U,\bm v\rangle\|_{\psi_2}$.
Let $\text{SNR}=\|\bm \Sigma^{1/2}\bm \beta\|_2^2/\sigma^2$ denote the signal-to-noise ratio. Let $\tau_q$ denote the $q$-th quantile of standard normal distribution.
\section{Estimation and prediction based on proxy data}
\label{sec-method}
In this section, we introduce our proposed estimators for prediction and estimation based on proxy data in Section \ref{sec2-1}. We study its theoretical properties and minimax optimality in Section \ref{sec2-2}.
\subsection{Two-sample Lasso method}
\label{sec2-1}
For the estimation and prediction tasks, the methods for proxy data resemble one-sample high-dimensional regression methods. The Lasso estimator \citep{Lasso} provides
a rate optimal estimator of $\bm \beta$ in the conventional setting. Decomposing the empirical loss $\|\bm y-\bm X{\bm b}\|_2^2$ as $\|\bm y\|_2^2-2{\bm b}^{\intercal}\bm X^{\intercal}\bm y+\|\bm X{\bm b}\|_2^2$ and removing the constant term, the Lasso estimator can be written as
\[
\widehat{\bbeta}^{(os)}=\argmin_{\bm b\in\mathbb{R}^p}\left\{\frac{1}{2}{\bm b}^{\intercal}\widehat{\Sig} {\bm b}-{\bm b}^{\intercal}\widehat{\bm S}+\lambda^{(os)}\|{\bm b}\|_1\right\}
\]
with some tuning parameter $\lambda^{(os)}>0$ and the superscript ``os'' is short for ``one-sample''. In fact, we have seen that the Lasso can be equivalently performed based on one-sample summary data $\widehat{\Sig}$ and $\widehat{\bm S}$. Methodology-wise, there is no need to distinguish one-sample summary and individual data for the Lasso. In the sequel, we will refer to $\widehat{\bbeta}^{(os)}$ as one-sample Lasso for simplicity.
With proxy data, it is natural to consider the following estimatior
\begin{align}
\label{eq-cbeta}
\widehat{\bbeta}^{(ts)}=\argmin_{\bm b\in\mathbb{R}^p}\{\frac{1}{2}{\bm b}^{\intercal}\widetilde{\Sig} {\bm b}-{\bm b}^{\intercal}\widehat{\bm S}+\lambda^{(ts)}\|{\bm b}\|_1\},
\end{align}
where we replace the unknown $\widehat{\Sig}$ with its proxy $\widetilde{\Sig}$ and consider a possibly different tuning parameter $\lambda^{(ts)}$.
We will see later that $\lambda^{(ts)}$ should always be chosen larger than $\lambda^{(os)}$ for consistency.
The two-sample Lasso estimator $\widehat{\bbeta}^{(ts)}$ has been considered in \citet{Lassosum} and \citet{Chen20}. However, the choice of $\lambda^{(ts)}$, the convergence rate, and minimax optimality have not been established. Choosing $\lambda^{(ts)}$ is also a practical challenge because cross validation cannot be performed without individual-level data. We provide the theoretical requirement on $\lambda^{(ts)}$ in Section \ref{sec2-2} and discuss some practical choices based on information criteria in Section \ref{sec-simu}.
\subsection{Convergence rates for estimation and prediction}
\label{sec2-2}
We assume the following conditions for theoretical analysis.
\begin{condition}[Gaussian designs]
\label{cond1}
Each row of $\bm X$ and $\tilde{\bm X}$ are i.i.d. Gaussian with mean zero and positive definite covariance $\bm \Sigma$ with bounded eigenvalues.
\end{condition}
\begin{condition}[sub-Gaussian noises]
\label{cond2}
The random noises $\epsilon_i$, $i=1,\dots,n$, are i.i.d. with mean zero and variance $\sigma^2>0$. $\epsilon_i$ and $\bm x_i$ are independent for $i=1,\dots,n$. The sub-Gaussian norms of $\epsilon_i$ are upper bounded by a constant.
\end{condition}
For estimation and prediction, it suffices to relax Conditions \ref{cond1} and \ref{cond2} to assume independent sub-Gaussian designs and independent sub-Gaussian noises. Here we assume slightly stronger regularity conditions, Gaussian designs and \textit{i.i.d.} noises. These assumptions ensure that the asymptotic variance of the debiased estimators only depends on the first two moments of the observations. With individual samples, this assumption is not necessary because one can estimate the variance based on the empirical noises \citep{Dezeure17}. In lack of the individual-level data, we cannot estimate the asymptotic variance empirically and have to rely on the properties of higher moments, which makes Gaussian distribution a natural assumption.
We first derive the rate of convergence for $\widehat{\bbeta}^{(ts)}$ in the two-sample summary setting. Let $\mathbb{E}[y_i^2]=M$ and
\begin{align}
\label{eq-gam}
\gamma_{n,\tilde{n}}=\sigma^2+\|\bm \Sigma^{1/2}\bm \beta\|_2^2(\frac{n}{\tilde{n}}+1)=M+\frac{n}{\tilde{n}}\bm \beta^{\intercal}\bm \Sigma\bm \beta.
\end{align}
Loosely speaking, $ \gamma_{n,\tilde{n}}$ is the variance of the random noises based on proxy data. In Theorem \ref{lem2-est}, we establish the convergence rate of $\widehat{\bbeta}^{(ts)}$ under mild conditions.
\begin{theorem}[Convergence rates for $\widehat{\bbeta}^{(ts)}$]
\label{lem2-est}
Assume Conditions \ref{cond1} and \ref{cond2} and $Ms\log p\ll \min\{n,\tilde{n}\}$.
For $\lambda^{(ts)}= c_1\sqrt{
\gamma_{n,\tilde{n}}\log p/n}$, with large enough $c_1,c_2$, then with probability at least $1-\exp(-c_1\log p)-\exp(-c_2\tilde{n})$,
\begin{align*}
&\|\widetilde{\Sig}^{1/2}(\widehat{\bbeta}^{(ts)}-\bm \beta)\|_2^2\vee \|\widehat{\bbeta}^{(ts)}-\bm \beta\|_2^2\leq C\frac{ \gamma_{n,\tilde{n}}s\log p}{n}\\
& \|\widehat{\bbeta}^{(ts)}-\bm \beta\|_1\leq Cs\sqrt{\frac{ \gamma_{n,\tilde{n}}\log p}{n}}.
\end{align*}
\end{theorem}
Comparing with the one-sample optimal rates in squared $\ell_2$-norm, which is $\sigma^2s\log p/n$, we can see that the ratio of two rates (two-sample over one-sample) is
\begin{align}
\label{ratio1}
1+\text{SNR}(\frac{n}{\tilde{n}}+1),
\end{align}
where $\text{SNR}=\|\bm \Sigma^{1/2}\bm \beta\|_2^2/\sigma^2$.
This implies that the estimation error rate in two-sample case is strictly worse than the one-sample case as long as $\text{SNR}> 0$. Larger $n$ leads to larger relative loss with $\widehat{\bbeta}^{(ts)}$ relative to $\widehat{\bbeta}^{(os)}$. In constrast, larger $\tilde{n}$ implies smaller relative loss.
Finally, a larger SNR implies a larger loss of $\widehat{\bbeta}^{(ts)}$ compared to $\widehat{\bbeta}^{(os)}$.
As a result, the condition for consistency is no weaker in the proxy setting than that with one-sample data.
The two-sample tuning parameter $\lambda^{(ts)}\asymp \sqrt{ \gamma_{n,\tilde{n}}\log p/n}$, whose order is always no smaller than its one-sample counterpart. The choice of $\lambda^{(ts)}$ is crucial in practice and we will discuss this in Section \ref{sec-simu}.
To better understand the unique challenges with proxy data, we consider a special scenario where $\tilde{n}\rightarrow \infty$, or equivalently, $\bm \Sigma$ is known.
\begin{remark}[The scenario of $\tilde{n}\rightarrow \infty$]
\label{re3-1}
{\rm
If $\tilde{n}\rightarrow \infty$, which is equivalent to observing $(\widehat{\bm S},\bm \Sigma)$, then
\[\|\widehat{\bbeta}^{(ts)}-\bm \beta\|_2^2=O_P\left(\frac{Ms\log p}{n}\right).
\]
} \end{remark}
Remark \ref{re3-1} shows that even if $\tilde{n}\rightarrow \infty$, the convergence rate of $\widehat{\bbeta}^{(ts)}$ is still inflated when $\text{SNR}> 0$ in comparison to having one-sample data. This comparison implies that, without the in-sample $\widehat{\bm \Sigma}$, any estimator of $\bm \Sigma$, even the oracle one, can lead to dramatic loss in estimation accuracy. Comparing Remark \ref{re3-1} with Theorem \ref{lem2-est}, we see that the error caused by finite external data is of order $\bm \beta^T\bm \Sigma\bm \beta s\log p/\tilde{n}$.
We now show that the convergence rate of $\widehat{\bbeta}^{(ts)}$ is minimax optimal in $\ell_2$-norm.
Consider the parameter space
\begin{align}
\Xi(s,M_0,\sigma_0^2)&=\left\{\|\bm \beta\|_0\leq s, \bm \beta^{\intercal}\bm \Sigma\bm \beta\leq M_0, ~0<\sigma^2\leq \sigma_0^2,\right.\nonumber\\
&\quad~~\left. 0<1/C_1\leq \Lambda_{\min}(\bm \Sigma)\leq \Lambda_{\max}(\bm \Sigma)\leq C_1<\infty\right\} \label{eq-Xi}
\end{align}
for some constant $C_1>1$ and $\sigma_0^2$ can be any positive constant. We see that $M\leq M_0+\sigma_0^2$ in the space of $ \Xi(s,M_0,\sigma_0^2)$.
Let $\mathcal{Z}=\{\widehat{\bm S},\widetilde{\Sig}\}$ denote the observed data and $\mathcal{F}(\mathcal{Z})$ denote functions based on the summary data $\mathcal{Z}$.
\begin{theorem}[Lower bound for estimating $\bm \beta$]
\label{thm-mini-est}
Consider the parameter space $\Xi(s,M_0,\sigma_0^2)$ in (\ref{eq-Xi}) with $s\geq 2$.
Suppose that $Ms\log p\ll n$, and $(\bm \beta^T\bm \Sigma\bm \beta\vee 1)s\log p\ll \tilde{n}$.
Then there exists some constant $c_2$ that
\[
\min_{\widehat{\bbeta}\in\mathcal{F}(\mathcal{Z})}\sup_{\bm \beta\in\Xi(s,M_0,\sigma_0^2)}\mathbb{P}\left(\|\widehat{\bbeta}-\bm \beta\|_2^2\geq \frac{c_1(M_0+\sigma_0^2)s\log p}{n}+\frac{c_2M_0s\log p}{\tilde{n}}\right)\geq 1/2.
\]
\end{theorem}
In the parameter space $\Xi(s,M_0,\sigma_0^2)$, it holds that $M=\mathbb{E}[y_i^2]\leq M_0+\sigma_0^2$. Hence, the lower bound in Theorem \ref{thm-mini-est} matches the $\ell_2$-upper bound in Theorem \ref{lem2-est} in terms of rates. We mention that the sample size condition in Theorem \ref{thm-mini-est} essentially restricts us to a class of $\widehat{\bm S}$ with distributional regularity, i.e., its distribution conditioning on $\bm y$ has positive definite covariance matrix. As far as we know, this is the first lower bound result based on summary data and the proof is based on some novel analysis of the distribution of the marginal correlation statistics.
\section{Inference for individual coefficient based on proxy data}
\label{sec3}
In this section, we consider statistical inference, such as hypothesis testing and constructing confidence intervals for $\beta_j$ with some fixed $1\leq j\leq p$.
It is known that the $\ell_1$-regularized estimates are biased and cannot be directly used for inference.
For inference based on proxy data, we follow a similar idea as the debiased methods, which have been proposed based on one-sample individual data. Specifically, the debiased Lasso \citep{ZZ14, van14, JM14} can be written as
\begin{align}
\label{os-db}
\hat{\beta}_j^{(os-db)}&=\hat{\beta}^{(os)}_j+\frac{(\bm X\widehat{\bw}_j)^{\intercal}(\bm y-\bm X\widehat{\bbeta}^{(os)})}{n}=\hat{\beta}^{(os)}_j+\widehat{\bw}_j^{\intercal}(\widehat{\bm S}-\widehat{\bm \Sigma}\widehat{\bbeta}^{(os)}),
\end{align}
where $\widehat{\bbeta}^{(os)}$ is the one-sample Lasso estimator and $\widehat{\bw}_j\in\mathbb{R}^p$ is a correction score vector that can be computed based on $\widehat{\bm \Sigma}$. We see that the debiased Lasso for $\beta_j$ can also be realized based on the summary statistics $\widehat{\Sig}$ and $\widehat{\bm S}$. Hence, we refer to the estimate in (\ref{os-db}) as one-sample debiased Lasso (os-db) in the sequel.
This similarly motivates its counterpart with two-sample summary data:
\begin{align}
\label{ts-db}
\hat{\beta}_j^{(ts-db)}&=\hat{\beta}^{(ts)}_j+\tilde{\bw}_j^{\intercal}(\widehat{\bm S}-\widetilde{\Sig} \widehat{\bbeta}^{(ts)}),
\end{align}
where $\widehat{\bbeta}^{(ts)}$ is computed in (\ref{eq-cbeta}) and $\tilde{\bw}_j\in\mathbb{R}^p$ is a correction score vector computed based on $\widetilde{\Sig} $.
Specifically, we consider
\begin{align}
\tilde{\bw}_j&=\argmin_{\bm w\in\mathbb{R}^p}\|\bm w\|_1\label{eq-tw}\\
&\text{subject to}~\|\widetilde{\Sig}\bm w-{\bm e}_j\|_{\infty}\leq \lambda_j,\nonumber
\end{align}
where $\lambda_j=c_1\sqrt{\log p/\tilde{n}}$ for some positive constant $c_1$.
The realization of $\tilde{\bw}_j$ is via a Dantzig selector optimization \citep{CT07}, which induces a sparse solution of the $j$-th column of the inverse covariance matrix $\bm \Omega=\bm \Sigma^{-1}$. Some existing one-sample methods, such as \citet{JM14}, do not look for a sparse estimate $\tilde{\bw}_j$ but choose a different objective function in (\ref{eq-tw}). In the proxy setting, however, the sparsity of $\tilde{\bw}_j$ plays a crucial role in the analysis. Those non-sparse methods for one-sample setting cannot be directly generalize for the current purpose as we will further discuss in Section \ref{sec3-2}.
\subsection{Asymptotic normality for debiased two-sample Lasso}
\label{sec3-2}
We study the asymptotic property of $\hat{\beta}_j^{(ts-db)}$ defined in (\ref{ts-db}) and prove its asymptotic normality under certain conditions.
Let $\bm \Omega_{.,j}$ denote the $j$-th column of $\bm \Omega$ and $s_j=\|\bm \Omega_{.,j}\|_0$.
\begin{theorem}[Asymptotic normality of the debiased estimator]
\label{thm-db}
Assume that Condition \ref{cond1} and Condition \ref{cond2} hold, $n\gg \log p$, and $\tilde{n}\gg (s\vee s_j)\log p$. Then it holds that
\begin{equation}
\label{re1-thm-db}
\hat{\beta}^{(ts-db)}_j-\beta_j=z_j+O_P\left( \gamma_{n,\tilde{n}}^{1/2}\frac{(s+s_j)\log p}{\sqrt{n\tilde{n}}}\right),
\end{equation}
where $ \gamma_{n,\tilde{n}}$ is defined in (\ref{eq-gam}) and
\[
(V_j^{(ts)})^{-1/2}z_j\xrightarrow{D} N(0,1)~~ \text{for}~~ V_j^{(ts)}=\frac{\Omega_{j,j} \gamma_{n,\tilde{n}}}{n}+\frac{\beta_j^2}{n}+\frac{\beta_j^2}{\tilde{n}}.
\]
Further assuming $( s\vee s_j)\log p \ll \sqrt{\tilde{n}}$,
then $(V_j^{(ts)})^{-1/2}(\hat{\beta}^{(ts-db)}_j-\beta_j)\xrightarrow{D} N(0,1)$.
\end{theorem}
Theorem \ref{thm-db} establishes the asymptotic distribution of $\hat{\beta}_j^{(ts-db)}$ in (\ref{re1-thm-db}) and provides the sample size condition for its asymptotic normality. The variance of $\hat{\beta}_j^{(ts-db)}$ is $V_j^{(ts)}$ and the remaining bias of $\hat{\beta}^{(ts-db)}_j$ is shown in the last term on the right hand side of (\ref{re1-thm-db}).
We first bring some details into the magnitude of $V_j^{(ts)}$.
The asymptotically normal component is $z_j=\bm \Omega_{.,j}^{\intercal}(\widehat{\bm S}-\bm \Sigma\bm \beta)+\bm \Omega_{.,j}^{\intercal}(\bm \Sigma-\widetilde{\Sig})\bm \beta$, where the first term comes from the marginal statistics and the second term comes from the proxy matrix. The variance $V_j^{(ts)}$ is obtained based on the moment formula for multivariate Gaussian.
The last two terms of $V_j^{(ts)}$, $\beta_j^2/n$ and $\beta_j^2/\tilde{n}$, are dominated by the first term of $V_j^{(ts)}$ given the positive definiteness of $\bm \Omega$.
Hence, when $\tilde{n}\gg n$, $V_j^{(ts)}\asymp \Omega_{j,j}M/n$; when $n\gg \tilde{n}$, $V_j^{(ts)}\asymp \Omega_{j,j}\|\bm \Sigma^{1/2}\bm \beta\|_2^2/\tilde{n}$. In comparison to its one-sample counterpart, $V_j^{(os)}=\Omega_{j,j}\sigma^2/n$, the relative loss in efficiency is
\[
\frac{V_j^{(ts)}}{V_j^{(os)}}\asymp 1+\text{SNR}(\frac{n}{\tilde{n}}+1),
\]
which is identical to the relative loss in estimation (\ref{ratio1}). When $\tilde{n}\rightarrow \infty$, i.e. $\bm \Sigma$ is known, $V_j^{(ts)}$ is still larger than $V_j^{(os)}$. This shows the significant loss in efficiency for inference problems when the marginal statistics and covariance estimator are not based on the same set of samples.
More importantly, the distinction between $V_j^{(ts)}$ and $V_j^{(os)}$ implies that simply applying the one-sample inference algorithms to the two-sample data could be wrong. We illustrate this point numerically in Section \ref{sec-simu-3}.
The remaining bias of $\hat{\beta}^{(ts-db)}_j$ is controlled by the sample sizes $\sqrt{n\tilde{n}}$ and $\tilde{n}$. This implies that the sample size for external reference panel, $\tilde{n}$ plays a more significant role in controlling the bias while the sample size for GWAS, $n$, plays a milder role. We have seen the same phenomenon in the estimation results in Section \ref{sec2-2}.
In one-sample setting, the remaining bias of debiased $\hat{\beta}_j^{(os-db)}$ in (\ref{os-db}), is of order $s\log p/n$ \citep{van14, CG17}. If $\tilde{n}/n$ is sufficiently large, the remaining bias of $\hat{\beta}^{(ts-db)}_j$ can be \textbf{smaller} than that of its one-sample counterpart. In view of the asymptotic bias and asymptotic variance in (\ref{re1-thm-db}), it suffices to require $(s\vee s_j)\log p\ll \sqrt{\tilde{n}}$ for asymptotic normality. This condition implies that $\tilde{n}$ determines the range of sparsity such that valid inference can be established. In contrast, for one-sample debiased Lasso $\hat{\beta}_j^{(os-db)}$, $n$ determines the range of sparsity for valid inference, which is $s\log p\ll \sqrt{n}$. This can be a blessing of proxy-data scenario. An extreme case is when $\bm \Sigma$ is known, or equivalently $\tilde{n}\rightarrow \infty$ as in the following remark.
\begin{remark}[The scenario of $\tilde{n}\rightarrow \infty$]
\label{re1-inf}
{\rm
When $\bm \Sigma$ is known, $\tilde{\bm w}_j=\bm \Omega_{.,j}$ and $\hat{\beta}_j^{(ts-db)}=\bm \Omega_{j,.}\bm X^{\intercal}\bm y/n$, which is asymptotically normal with mean zero and variance $\Omega_{j,j} \gamma_{n,\infty}/n+\beta_j^2/n$.
} \end{remark}
It may be surprising to see that for fixed $n$ and $p$, the remaining bias of $\hat{\beta}^{(ts-db)}_j$ vanishes when $\tilde{n}\rightarrow \infty$. However, many existing applications often have $\tilde{n}\lesssim n$. This can be due to the less cost of sharing GWAS statistics than sharing the LD matrix. Same pattern holds for distributed inference, in which case $n$ is the total sample size and $\tilde{n}$ is the local sample size. This should raise some caution in applications with two-sample summary data.
We finally discuss the conditions on the sparsity $s_j$. In classical one-sample setting, inference for $\beta_j$ may not require sparse $\bm \Omega_{.,j}$, see, for example, the analysis in \citet{JM14} for linear models. We mention that the condition on $s_j$ cannot be removed using the same idea in our anlaysis. This comes from a unique challenge of proxy data, where $\widehat{\bm S}$ implicitly depends on $\widehat{\Sig}$, which is unobserved but approximated. Nevertheless, the condition on $s_j$ can be avoided by sample splitting. Ideally, one can create two independent estimate of $\bm \Sigma$ and use one for two-sample Lasso and the other one as the debiasing samples.
However, sample splitting is not viable with summary data in most cases. Hence, we focus on the current procedure and the results without sample splitting.
In the next theorem, we establish the minimax lower bound for estimating $\beta_j$.
\begin{theorem}[Minimax lower bound for estimation of $\beta_j$]
\label{thm-mini-inf}
Consider the parameter space $\Xi(s,M_0,\sigma_0^2)$ in (\ref{eq-Xi}).
Suppose that $\max\{1,M_0+\sigma_0^2\}\leq c_1\min\{n, \tilde{n}\}$ for some constant $c_1>0$. Then for any fixed $1\leq j\leq p$, there exists some constant $c_2$ that
\[
\inf_{\hat{\beta}_j\in\mathcal{F}(\mathcal{Z})} \sup_{\bm \beta\in\Xi(s,M_0,\sigma_0^2) }\mathbb{P}\left(|\hat{\beta}_j-\beta_j|\geq c_2\sqrt{\frac{M_0+\sigma_0^2}{n}}+c_2\sqrt{\frac{M_0}{\tilde{n}}}\right)\geq \frac{1}{2}.
\]
\end{theorem}
In Theorem \ref{thm-mini-inf}, we show that the parametric part of the rate for $\hat{\beta}_j^{(ts-db)}$ is minimax optimal. That is, under the sample size condition $(s\vee s_j)\log p\ll \sqrt{\tilde{n}}$, the two-sample debiased estimator $\hat{\beta}_j^{(ts-db)}$ has rate optimal confidence interval length.
Comparing with the minimax rate for one-sample inference, we see that the variance part are inflated with proxy data. For the nonparametric part, the proof based on summary statistics is much more involved.
In the supplements, we provide the minimax lower bound for estimating $\beta_j$ when $\bm \Sigma$ is known and the lower bound matches the upper bound derived in Remark \ref{re1-inf}.
\subsection{Variance estimator and confidence intervals}
In view of $V_j^{(ts)}$, we propose a variance estimator for $\hat{\beta}_j^{(ts-db)}$ as
\begin{align}
\widehat{V}^{(ts)}_j&=\tilde{\bw}_j^{\intercal}\widetilde{\Sig}\tilde{\bw}_j(\frac{\|\bm y\|_2^2}{n^2}+\frac{2(\widehat{\bbeta}^{(ts)})^{\intercal}\widehat{\bm S}-(\widehat{\bbeta}^{(ts)})^{\intercal}\widetilde{\Sig}\widehat{\bbeta}^{(ts)}}{\tilde{n}})\nonumber\\
&\quad+\frac{(\hat{\beta}_j^{(ts-db)})^2}{n}+\frac{(\hat{\beta}_j^{(ts-db)})^2}{\tilde{n}}.\label{eq-cVj}
\end{align}
Notice that $\widehat{V}^{(ts)}_j$ is \textbf{not} the two-sample analogy of variance estimator for the classical debiased Lasso. This is because the probabilistic limit of $\widehat{V}^{(ts)}_j$ is asymptotically larger than the asymptotic variance of the conventional debased Lasso. Hence, if we treat proxy data as one-sample summary data, correct coverages are not guaranteed. We propose the following $(1-\alpha)\times 100\% $-confidence interval for $\beta_j$ as
\begin{equation}
\label{ci}
\hat{\beta}_j^{(ts-db)}\pm \tau_{\alpha/2}\sqrt{\widehat{V}_j^{(ts)}}.
\end{equation}
Once the $z$-statistics $ z_j^{(ts)}=\hat{\beta}_j^{(ts-db)}/\sqrt{\widehat{V}_j^{(ts)}}$ is obtained for $j=1,\dots, p$, we can perform multiple testing with FDR control using the procedure in \citet{JJ19}, which is a refined version based on \citet{Liu13}.
In the next lemma, we prove the consistency of $\widehat{V}_j^{(ts)}$ defined in (\ref{eq-cVj}) and conclude the validness of the confidence interval (\ref{ci}).
\begin{lemma}[A consistent variance estimator]
\label{lem-var}
Under the conditions of Theorem \ref{thm-db},
\[
\frac{|\widehat{V}_j^{(ts)}-V_j^{(ts)}|}{V_j^{(ts)}}=O_P\left(\frac{ \gamma_{n,\tilde{n}}}{\sqrt{n}}+ \gamma_{n,\tilde{n}}\frac{s\log p}{n}+ \gamma_{n,\tilde{n}}\frac{s_j\log p}{\tilde{n}}+\frac{1}{\sqrt{\tilde{n}}}\right).
\]
To summarize, assuming Condition \ref{cond1}, Condition \ref{cond2}, and $(s\vee s_j)\log p\ll \sqrt{\tilde{n}}$, then $(\widehat{V}_j^{(ts)})^{-1/2}(\hat{\beta}^{(ts-db)}_j-\beta_j)\xrightarrow{D} N(0,1)$.
\end{lemma}
So far we have proved the asymptotic validness of the confidence interval in (\ref{ci}) under the conditions of Theorem \ref{thm-db}.
\subsection{Power analysis with proxy data}
\label{sec-st}
In this section, we evaluate the power of hypothesis testing with proxy data. We have seen in Section \ref{sec-method} that it is necessary to focus on the regime that $s\log p\ll \sqrt{\tilde{n}}$ and $s_j\log p\ll \tilde{n}/\sqrt{n}+\tilde{n}^{1/2}$ for valid inference.
For the simplicity of the power analysis, we ignore the asymptotic bias in the debiased estimator.
To avoid confusion, we introduce some new notations.
Let $\sqrt{n}\hat{z}^{(os)}_j$ be the probabilistic limit of conventional debiased estimator $\sqrt{n}\hat{\beta}_j^{(db)}/\sqrt{V_j}$ where $V_j=\Omega_{j,j}\sigma^2$ \citep{van14}. Let $\bm\Omega=\bm \Sigma^{-1}$. The distribution of one-sample $z$-score is
\begin{equation}
\label{z-os}
\sqrt{n}\hat{z}^{(os)}_j\sim N\left(\frac{\sqrt{n}\beta_j}{\sqrt{\Omega_{j,j}\sigma^2}},1\right),~1\leq j\leq p.
\end{equation}
For proxy-data based inference, let $\sqrt{n}\hat{z}_j^{(ts)}$ be the probabilistic limit of two-sample debiased estimator $\sqrt{n}\hat{\beta}^{(ts-db)}_j/\sqrt{V_j^{(ts)}}$. The marginal distribution of each two-sample $z$-score is
\begin{equation}
\label{z-ts}
\sqrt{n} \hat{z}^{(ts)}_j\sim N\left(\frac{\sqrt{n}\beta_j}{\sqrt{\Omega_{j,j} \gamma_{n,\tilde{n}}+\beta_j^2(1+n/\tilde{n})}},1\right),~1\leq j\leq p.
\end{equation}
We see that the distribution of $\hat{z}^{(ts)}_j$ depends not only on $\beta_j$ but all other coefficients $\beta_{-j}$.
We evaluate the power of single hypothesis testing, $\mathcal{H}_{0,j}:~\beta_j=0$ vs $\mathcal{H}_{1,j}:~\beta_j\neq 0$ based on $\hat{z}_j^{(os)}$ and $\hat{z}_j^{(ts)}$, respectively. By definition, the power of these two statistics can be expressed as
\begin{align*}
\text{Power}^{(os)}_j(\alpha;\bm b)=\mathbb{P}(|\hat{z}_j^{(os)}|\geq \tau_{\alpha}|\beta_j=b_j,\bm \beta_{-j}=\bm b_{-j}).\\
\text{Power}^{(ts)}_j(\alpha,\bm b)=\mathbb{P}(|\hat{z}_j^{(ts)}|\geq \tau_{\alpha}|\beta_j=b_j,\bm \beta_{-j}=\bm b_{-j}).
\end{align*}
Based on (\ref{z-os}), we know that $\text{Power}^{(os)}_j(\alpha;\bm b)$ is independent of $\bm b_{-j}$.
\begin{theorem}[Power of two-sided single hypothesis testing]
\label{thm1-st}
For any $\bm b\in\mathbb{R}^p$, it holds that
{\small
\begin{align*}
&\textup{Power}^{(os)}_j(\alpha,\bm b)=\Phi(|\eta_j^{(os)}|-\tau_{\alpha})+\Phi(-|\eta_j^{(os)}|-\tau_{\alpha}),~\text{where}~\eta_j^{(os)}=\frac{\sqrt{n}b_j}{\Omega_{j,j}^{1/2}\sigma}.\\
&\textup{Power}^{(ts)}_j(\alpha,\bm b)=\Phi\left(|\eta_j^{(ts)}|-\tau_{\alpha}\right)+\Phi\left(-|\eta_j^{(ts)}|-\tau_{\alpha}\right),~\text{where}~\\
&\quad\eta_j^{(ts)}=\frac{\sqrt{n}b_j}{\sqrt{(\Omega_{j,j}\bm b^{\intercal}\bm \Sigma\bm b+b_j^2)(1+n/\tilde{n})+\Omega_{j,j}\sigma^2}}.
\end{align*}
}
\end{theorem}
The one-sample and two-sample power functions are increasing functions of $|\eta_j^{(os)}|$ and $|\eta_j^{(ts)}|$, respectively.
The power based on proxy data is smaller than that based on one-sample data for $\beta_j\neq 0$. Furthermore, the power based on $\hat{z}^{(os)}_j$ is independent of effect size distribution but the power based on $\hat{z}^{(ts)}_j$ depends on the effect size distribution. Specifically, if the effect size vector $\bm b$ has sparsity $s$ and approximately equal signal strength, then $\eta_j^{(ts)}$ is approximately $\sqrt{n}b_j/\sqrt{\Omega_{j,j}\{sb_j^2(1+n/\tilde{n})+\sigma^2\}}$. We see that $|\eta_j^{(ts)}|$ is bounded away from infinity for finite $n\wedge \tilde{n}$ even if $b_j\rightarrow\infty$.
To further demonstrate this phenomenon, we consider the equal signal strength model
\begin{equation}
\label{em}
b_j\in\{-b_0,0,b_0\}~\text{and}~\|\bm b\|_0=s.
\end{equation}
\begin{corollary}[Power in the equal signal strength model]
\label{cor1-st}
Consider the equal strength model (\ref{em}) with $\bm \Sigma=I_p$. The results of Theorem \ref{thm1-st} hold with
\begin{align}
\label{re1}
|\eta_j^{(os)}|=\frac{\sqrt{n}|b_0|}{\sigma}~\text{and}~|\eta_j^{(ts)}|\leq\min\left\{\sqrt{\frac{n\wedge \tilde{n}}{s}},\frac{\sqrt{n}|b_0|}{\sigma}\right\}.
\end{align}
\end{corollary}
In the two-sample setting, the signal strength $|\eta_j^{(ts)}|$ and hence the power is upper-truncated. As long as $|b_0|> \sigma \sqrt{(n\wedge \tilde{n})/n/s}$, the power based on proxy-data is strictly lower than the power based on one-sample data. Second, for any finite $n\wedge \tilde{n}$ and $s$, the right-hand side of (\ref{re1}) is strictly below one no matter how large $|b_0|/\sigma$ is. This analysis demonstrates a significant loss of power in proxy-data based inference in comparison to the one-sample based inference.
In Figure \ref{fig1}, we examine the finite sample performance based on one-sample and two-sample data in the equal signal strength model (\ref{em}). We see significant power loss with proxy data-based inference when $s$ is not too small relative to $n\wedge \tilde{n}$.
\begin{figure}[H]
\includegraphics[width=0.495\textwidth,height=5.7cm]{st-power-small.eps}
\includegraphics[width=0.495\textwidth,height=5.7cm]{st-power-large.eps}
\caption{The theoretical power functions for testing $\mathcal{H}_0:\beta_j=0$. The line with ``OS'' is $\textup{Power}^{(os)}(\tau_{0.95},\bm b)$ and the line with ``TS($s_0$)'' is $\textup{Power}^{(ts)}(\tau_{0.95},\bm b)$ with sparsity $s=s_0$ in the equal signal strength model (\ref{em}). The left panel is $(n, \tilde{n},p,\alpha)=(100,100,500,0.05)$ and the right panel is $(n,\tilde{n},p,\alpha)=(10^4,5\times10^3,10^6,0.05)$. The upper dotted line indicates 1 and lower dotted line indicates the nominal level 0.05.}
\label{fig1}
\end{figure}
\section{Inference for linear functionals}
\label{sec3-3}
We now study statistical inference for the PRS $\mu_*=\bm x_*^{\intercal}\bm \beta$ given an individual's feature $\bm x_*$. For prediction, we can use $\widehat{\mu}_*=\bm x_*^{\intercal}\widehat{\bm \beta}$.
We focus on constructing confidence intervals for $\mu_*=\bm x_*^{\intercal}\bm \beta$ in the rest of this section. Inference for linear functionals of $\bm \beta$ have been studied in the classical setting. The minimax rate is established in \citet{CG17} and various methods are established in \citet{CG17}, \citet{cai2019individualized}, and \citet{javanmard2020flexible}. All the afore-mentioned methods consider the debiasing recipe: the correction scores are obtained by constrained minimizations, where the constraints can be directly used to upper bound the the remaining bias of the debiased estimator. Our problem is more challenging as some uncertainty coming from the unobserved covariance matrix $\widehat{\bm \Sigma}$ cannot be directly controlled based on the observed data.
Trading-off multiple sources of bias, we consider a different method. For $\tilde{\bw}_j$ defined in (\ref{eq-tw}), denote
\begin{align}
\label{eq-tOmega}
\widetilde{\bm\Omega}=(\tilde{\bw}_1,\dots,\tilde{\bw}_p)\in\mathbb{R}^{p\times p}.
\end{align}
In fact, $\widetilde{\bm\Omega}$ is an estimate of the inverse covariance matrix. Our estimated $\widetilde{\bm\Omega}$ is equivalent to the CLIME estimator \citep{Cai11}, which can be expressed as
\begin{align*}
\widetilde{\bm\Omega}&=\argmin_{\bm \Omega\in\mathbb{R}^{p\times p}}\|\bm \Omega\|_1\\
&\text{subject to}~\|\widetilde{\Sig}\bm \Omega-I_p\|_{\infty,\infty}\leq \tilde{\lambda},
\end{align*}
where $\tilde{\lambda}=c_1\sqrt{\log p/\tilde{n}}$ for some positive constant $c_1$.
We then obtain an initial bias-correction score $\widetilde{\bm\Omega}\bm x_*$. Next, we refine $\widetilde{\bm\Omega}\bm x_*$ to reduce the bias in the direction of $\bm x_*$:
\begin{align}
\tilde{\bw}_*&=\argmin_{\bm w\in\mathbb{R}^{p}}\|\bm w-\widetilde{\bm\Omega}\bm x_*\|_{1}\label{eq-tOmegas}\\
&\text{subject to}~~\|\widetilde{\Sig}\bm w-\bm x_*\|_{\infty}\leq \|\bm x_*\|_2\tilde{\lambda}.\nonumber
\end{align}
The optimization in (\ref{eq-tOmegas}) can be efficiently solved, because it equivalently computes the Dantzig selector \citep{CT07} by treating $\bm w-\widetilde{\bm\Omega}\bm x_*$ as the target parameter.
For $\hat{\bm \beta}^{(ts)}$ defined in (\ref{eq-cbeta}) and $\tilde{\bw}_*$ defined in (\ref{eq-tOmegas}), define the debiased estimator for $\mu_*=\bm x_*^{\intercal}\bm \beta$ as
\begin{align}
\label{eq-db2}
\hat{\mu}_*^{(ts-db)}&=\bm x_*^{\intercal}\hat{\bm \beta}^{(ts)}+\tilde{\bw}_*^{\intercal}(\widehat{\bm S}-\widetilde{\Sig} \widehat{\bbeta}^{(ts)}).
\end{align}
Some more comments on $\tilde{\bw}_*$ defined in (\ref{eq-tOmegas}) are warranted. Our proposed $\tilde{\bw}_*$ distinguishes from the constrained minimizations based on one-sample individual data, say, expressions (7) and (8) of \citet{cai2019individualized} or expression (12) of \citet{javanmard2020flexible}, directly control the bias in the direction of $\bm x_*$ based on the observed $\widehat{\Sig}$. Their correction scores have no sparse guarantees and their objective functions are quadratic. In the proxy setting, the analysis for debiasing has more remainder terms to control, which involve the discrepancy between the observed $\widetilde{\Sig}$ and the unobserved $\widehat{\Sig}$. To control the bias term involving $\widehat{\Sig}$, we rely on the sparsity of the precision matrix $\bm \Omega$. The estimator $\tilde{\bw}_*$ in (\ref{eq-tOmegas}) can simultaneously leverage the sparsity structure of $\bm \Omega$ and control the bias in the direction of $\bm x_*$.
The estimators established in Section \ref{sec3-3} can be understood as a generalization of the estimators in Section \ref{sec3}. Especially, the canonical basis ${\bm e}_j$ in (\ref{eq-tw}) is replaced by a generic linear coefficient $\bm x_*$ in (\ref{eq-tOmegas}).
\subsection{Asymptotic normality for two-sample debiased $\mu_*$}
\label{sec3-4}
We study the theoretical properties of $\hat{\mu}_*^{(ts-db)}$ defined in (\ref{eq-db2}). The theoretical analysis for a generic linear functional is more challenging, because $\bm x_*$ is non-sparse in general while canonical basis $\bm e_j$ has sparsity one. Let $s_{\Omega}=\max_{j\leq p}\|\bm \Omega_{.,j}\|_0$.
\begin{theorem}[Asymptotic normality of $\hat{\mu}_*^{(ts-db)}$]
\label{thm-db2}
Assume Condition \ref{cond1} and Condition \ref{cond2} hold true, $n\gg \log p$ and $\tilde{n}\gg (s\vee s_{\Omega})\log p$. It holds that
\[
\hat{\mu}_*^{(ts-db)}-\mu_*=z_*+O_P\left( \gamma_{n,\tilde{n}}^{1/2}\|\bm x_*\|_2\frac{(s+s_{\Omega}^{3/2})\log p}{\sqrt{n\tilde{n}}}\right),
\]
where $ \gamma_{n,\tilde{n}}$ is defined in (\ref{eq-gam}) and
\[
(V_*^{(ts)})^{-1/2}z_*\xrightarrow{D} N(0,1)~~ \text{for}~~ V_*^{(ts)}=\frac{\bm x_*^{\intercal}\bm \Omega\bm x_* \gamma_{n,\tilde{n}}}{n}+\frac{\mu_*^2}{n}+\frac{\mu_*^2}{\tilde{n}}.
\]
Hence, given that \begin{align}
\label{ss2}
s\log p \ll \sqrt{\tilde{n}}~~\text{and}~~s_{\Omega}^{3/2}\log p\ll \tilde{n}/\sqrt{n},
\end{align}
$ (V_*^{(ts)})^{-1/2}(\hat{\mu}_*^{(ts-db)}-\mu_*)\xrightarrow{D} N(0,1)$.
\end{theorem}
Theorem \ref{thm-db2} establishes the limiting distribution of $ \hat{\mu}_*^{(ts-db)}$ and the asymptotic normality for $\hat{\mu}_*^{(ts-db)}$ given (\ref{ss2}).
The sparsity condition on $s$ is the same as in Section \ref{sec3-2} but the condition on $s_{\Omega}$ is stricter. This comes from the challenge of dealing with a non-sparse loading $\bm x_*$.
We now connect Theorem \ref{thm-db2} with the method in (\ref{eq-tOmegas}). We see that the remaining bias of $\hat{\mu}_*^{(ts-db)}$ depends on the sparsity of $\bm \Omega$, $s_{\Omega}$. We leverage the sparsity of $\bm \Omega$ by first initialize $\widetilde{\bm\Omega}\bm x_*$ and compute $\tilde{\bw}_*$ as its projection to the $\ell_{\infty}$-constrained space. Again, the number of proxy data $\tilde{n}$ determines the range of sparsity condition for constructing confidence intervals. The asymptotic variance $V_*^{(ts)}$ is determined by $n$ and $\tilde{n}$ simultaneously.
We introduce the variance estimator of $\hat{\mu}_*^{(ts-db)}$, which is
\begin{align}
\widehat{V}^{(ts)}_*&=\tilde{\bw}_*^{\intercal}\widetilde{\Sig}\tilde{\bw}_*(\frac{\|\bm y\|_2^2}{n^2}+\frac{2(\widehat{\bbeta}^{(ts)})^{\intercal}\widehat{\bm S}-(\widehat{\bbeta}^{(ts)})^{\intercal}\widetilde{\Sig}\widehat{\bbeta}^{(ts)}}{\tilde{n}})\nonumber\\
&\quad+\frac{(\hat{\mu}_*^{(ts-db)})^2}{n}+\frac{(\hat{\mu}_*^{(ts-db)})^2}{\tilde{n}}.\label{eq-Vnew}
\end{align}
We can similarly show that $\widehat{ V}_*^{(ts)}$ defined in (\ref{eq-Vnew}) is a consistent estimator of $V_*^{(ts)}$. Hence, we propose the following $(1-\alpha)\times 100\% $-confidence interval for $\mu_*$ as
\begin{equation}
\label{ci2}
\hat{\mu}_*^{(ts-db)}\pm \tau_{\alpha/2}\sqrt{ \widehat{V}^{(ts)}_*}.
\end{equation}
\section{Numerical results}
\label{sec-simu}
In this section, we evaluate the empirical performance of the estimation and inference procedures developed in previous sections.
A practical issue is the choice of tuning parameter $\lambda^{(ts)}$. Without the individual data, cross-validation cannot be used. Alternative strategies include using some information criteria including Akaike information criterion (AIC) and Bayesian information criterion (BIC), or generalized information criterion. Some theoretical guarantees based on theses criteria have been studied, see, for example, \citet{zhang2010regularization} and \citet{fan2013tuning}. Here we use BIC to select $\lambda^{(ts)}$ in (\ref{eq-cbeta}). Specifically, we consider
\[
\textup{BIC}(\lambda)= \log\left((\hat{\bm \beta}^{(ts)}_{\lambda})^{\intercal}\widetilde{\Sig}\hat{\bm \beta}^{(ts)}_{\lambda}- 2(\hat{\bm \beta}^{(ts)}_{\lambda})^{\intercal}\widehat{\bm S}+M\right)+\frac{\log (n\wedge \tilde{n})}{n\wedge \tilde{n}} \|\hat{\bm \beta}^{(ts)}_{\lambda}\|_0,
\]
where $\hat{\bm \beta}^{(ts)}_{\lambda}$ is the two-sample Lasso estimate with tuning parameter $\lambda$.
The variance of $\hat{\beta}_j^{(ts-db)}$ depends on $M$, the second moment of $y_i$. In fact, $M$ can be approximated from the variance estimators of $\widehat{\bm S}$. Specifically, it is easy to show that
\[
\textup{Var}(\widehat{\bm S}_j)=\textup{Var}\left(\frac{\bm X_j^{\intercal}\bm y}{n\bm \Sigma_{j,j}}\right)(1+o(1))=\frac{\bm \Sigma_{j,j}M+(\bm \Sigma_{j,S}\bm \beta_S)^2}{n\bm \Sigma_{j,j}^2}(1+o(1)).
\]
When the correlation between $\bm x_{.,j}$ $j\notin S$ and $\bm X_{.,S}$ is sparse, say, $|\{j\leq p: \bm \Sigma_{j,S}\neq 0\}|\ll p$, then a consistent estimate of $M$ is
\[
\widehat{M}=\frac{n}{p}\sum_{j=1}^p \widehat{var}(\widehat{\bm S}_j)\widetilde{\Sig}_{j,j}.
\]
For one-sample method, the tuning parameter in Lasso is chosen based on 10-fold cross validation and the tuning parameter in debiased Lasso is set as $\sqrt{2\log p/n}$.
We consider $p=500$ and $s\in\{4,8,12\}$.
Let $\beta_{4(k-1)+1: 4k}=(0.5,-0.5,0.2,-0.2)^{\intercal}$ for $k=1,\dots, s/4$ and $\beta_k=0$ otherwise.
We consider $(n,\tilde{n})\in\{(100,400),$ $(200,400),(200,200),(400,200),(400,100)\}$, which corresponds to $n/\tilde{n}\in\{0.25,0.5,1,2,4\}$. In the main paper, we consider the identity covariance matrix and an equi-correlated $\bm \Sigma$ with $\bm \Sigma_{j,k}=1$ if $j=k$ and $\bm \Sigma_{j,k}=0.3$ if $j\neq k$. In the Supplemental Materials, we also provide results based on a block diagonal $\bm \Sigma$ where each block is Toeplitz with the first row being $(0.6^0,0.6,\dots, 0.6^4)$.
\subsection{Estimation and prediction results}
\label{sec-simu-1}
In Figure \ref{fig-SSE}, we report the estimation results with one-sample and two-sample Lasso. The prediction results are similar and are given in the supplements. As one-sample method only uses $n$ individual samples, we see that the estimation errors (SSE) decrease as $n$ increases for any given $s$.
For the methods based on proxy data, the estimation errors do not monotonically decrease as $n/\tilde{n}$ increase. This is because the estimation error is proportional to $ \gamma_{n,\tilde{n}}/n$, which depends on $n\wedge \tilde{n}$. Second, the estimation errors are larger than the corresponding errors in one-sample case. As the sparsity $s$ increases, both methods have SSE increasing. However, the SSE of two-sample Lasso increases more significantly. This is because, according to Theorem \ref{lem2-est}, as $s$ increases, SNR increases and a larger $\lambda^{(ts)}$ is needed which leads to larger SSE. In the conventional one-sample case, the tuning parameter $\lambda^{(os)}$ can be chosen independent of SNR theoretically.
For equi-correlated $\bm \Sigma$, we observe similar patterns on estimation errors as for $\bm \Sigma=I_p$.
\begin{figure}[H]
\includegraphics[width=0.99\textwidth, height=4.7cm]{Ident-SSE}
\includegraphics[width=0.99\textwidth, height=4.7cm]{Equi-SSE}
\caption{\label{fig-SSE}Sum of squared errors (SSE) for estimating $\bm \beta$ based on one-sample Lasso (OS) and two-sample Lasso (TS) with identify covariance matrix (first row) and equi-correlated covariance matrix (second row). Each point is the mean based on 200 independent experiments. }
\end{figure}
\subsection{Inference for the regression coefficients}
\label{sec-simu-2}
We evaluate the performance of different methods for statistical inference for the regression coefficients. Besides the one-sample and two-sample debiased Lasso, we also evaluate the effects of treating the two-sample summary data as one-sample data on statistical inference. We have discussed in Section \ref{sec3} that it can lead to incorrect variance estimators for the debiased Lasso. Hence, we consider a method with misspecification, which applies one-sample formula to two-sample data, shorthanded as TS2.
In Figure \ref{fig-ident-coef0}, we see that the one-sample and two-sample debiased Lasso for a zero coefficient has coverage probabilities close to the nominal level in various configurations. The misspecified method, however, has coverage probabilities significantly lower than the nominal level. These results highlight the importance of dealing with proxy data with the methods introduced in Section \ref{sec-method}. Ignoring the fact that the covariance matrix is estimated from the proxy data can lead to incorrect statistical inference. On the other hand, the standard deviations based on proxy data are significantly larger than those based on one-sample data especially when $n$ is large and $\tilde{n}$ is small. The gap increases as SNR increases. This agrees with our analysis in Theorem \ref{thm-db}. We also point out that in contrast to the one-sample case, the standard deviations in two-sample case are not monotonic functions of $n/\tilde{n}$.
In Figure \ref{fig-equi-coef1}, we report the one-sample and two-sample debiased Lasso for $\beta_{3}=0.2$ with non-sparse $\bm \Omega$.
The coverage probabilities close to the nominal level in most configurations but are slightly lower when $s$ is large. This is mainly due to a non-sparse $\bm \Omega$ can cause larger remaining bias in the debiased Lasso method.
The results for $\beta_1=0.5$ are provided in the Supplemental Materials. We see that for strong signals, the coverage probabilities are slightly lower than the nominal level in the equi-correlated setting and block-diagonal setting. This is because the debiased estimators for strong signals are subject to larger remaining bias and similar patterns have been observed and studied in one-sample setting. In all the settings, we see that the coverage probabilities for $\beta_{20}=0$ are close to the nominal level. Hence, testing $H_{0,j}:\beta_j=0$ can always have accurate Type-I error in all the scenarios in consideration.
\begin{figure}[H]
\includegraphics[width=0.99\textwidth, height=6cm]{Ident-coef0}
\caption{\label{fig-ident-coef0}Average coverage probabilities (first row) and average standard deviations (second row) with identify covariance matrix for $\beta_{20}=0$. Three method in comparison are one-sample debiased Lasso (OS), two-sample debiased Lasso (TS), and the application of one-sample debiased Lasso to two-sample data (TS2). The solid line is the nominal confidence level 0.95. Each point is the mean based on 200 independent experiments.}
\end{figure}
\begin{figure}[H]
\includegraphics[width=0.99\textwidth, height=6cm]{Equi-coef1}
\caption{\label{fig-equi-coef1}Average coverage probabilities (first row) and average standard deviations (second row) with equi-correlated matrix for $\beta_3=0.2$. Three method in comparison are one-sample debiased Lasso (OS), two-sample debiased Lasso (TS), and the application of one-sample debiased Lasso to two-sample data (TS2). The solid line is the nominal confidence level 0.95. Each point is the mean based on 200 independent experiments. }
\end{figure}
\subsection{Inference for the linear functionals}
\label{sec-simu-3}
In this subsection, we present the inference results for $\bm x_*^{\intercal}\bm \beta$ where $\bm x_*$ is randomly generated from $N(0,\bm \Sigma)$ and $\bm \Sigma$ is taken to be the identity or the equi-correlated matrix, respectively. With $\bm \Sigma=I_p$ (Figure \ref{fig-mu-ident}), the coverage probabilities of one-sample and two-sample methods have coverage probabilities close to the nominal level. However, mis-fitting two-sample data with one-sample method still lead to low coverage For non-sparse $\bm \Omega$ (Figure \ref{fig-mu-equi}), the two-sample coverage probabilities are lower than the nominal level. For block diagonal $\bm \Sigma$, which corresponds to a sparse $\bm \Omega$, the coverage probabilities are close to the nominal level. These observations reveal that the two-sample methods are more severely affected by the sparsity of precision matrix.
Moreover, the coverage gets lower as $n/\tilde{n}$ increases. This again witnesses the critical role of $\tilde{n}$ for two-sample inference. See the sample size condition (\ref{ss2}) and the discussion follows.
\begin{figure}[H]
\includegraphics[width=0.99\textwidth, height=6cm]{Ident-mu}
\caption{\label{fig-mu-ident}Average coverage probabilities (first row) and average standard deviations (second row) with identify covariance matrix for $\mu_*$. Three method in comparison are one-sample debiased Lasso (OS), two-sample debiased Lasso (TS), and the application of one-sample debiased Lasso to two-sample data (TS2). The solid line is the nominal confidence level 0.95. Each point is the mean based on 200 independent experiments. }
\end{figure}
\begin{figure}[H]
\includegraphics[width=0.99\textwidth, height=6cm]{Equi-mu}
\caption{\label{fig-mu-equi}Average coverage probabilities (first row) and average standard deviations (second row) with equi-correlated matrix for $\mu_*$. Three method in comparison are one-sample debiased Lasso (OS), two-sample debiased Lasso (TS), and the application of one-sample debiased Lasso to two-sample data (TS2). The solid line is the nominal confidence level 0.95. Each point is the mean based on 200 independent experiments. }
\end{figure}
\section{Data analysis}
\label{sec-data}
We apply the proposed methods to a GWAS study in outbred Carworth Farms White (CFW) mice population \citep{parker2016genome}. \cite{parker2016genome} showed no widespread population structure or cryptic relatedness in the CFW mice and therefore, we view these mice as independent of each other. The primary pre-processing of phenotypes and genotypes, including outliers removal and basic transformation, was conducted using the original paper's code. After the pre-processing, the data set consists of 1,038 mice with 79,824 genetic variants (SNPs) and 71 different phenotypes. We study the genetic associations for the weights of four hindlimb muscles. Specifically, the responses include the weight of TA (transverse abdominal), EDL(extensor digitorum longus), gastroc (gastrocnemius), and soleus, respectively.
\subsection{Prediction of hindlimb muscle weights using genotype data}
In prediction tasks, we use all the SNPs to predict each response and evaluate the out-of-sample prediction accuracy. Take the TA response as an example. In each experiment, we randomly split the samples into two folds and use one fold to compute the GWAS statistic, $\widehat{\bm S}$, and the other fold to compute the sample covariance matrix, $\widetilde{\Sig}$. We consider different sample size ratios of the GWAS and the empirical covariance matrix. Specifically, we consider the sample size for GWAS, $n\in\{250,500,750\}$, and $\tilde{n}=1038-n$, which gives $n/\tilde{n}$ is approximated one of $\{1/3,1,3\}$. For each sample size configuration, we repeat independent splitting and predictions 30 times.
The prediction results are plotted in Figure \ref{CFW-pred}. We observe that the SNPs are predictive for the EDL and Soleus weight, but are not predictive for the TA and Gastroc weight for any sample size ratios. For EDL and Soleus, we see that as $n/\tilde{n}$ in creases, the test errors decrease significantly in one-sample case.
In two-sample case, the test errors have the smallest median when $n/\tilde{n}=1$. This can be understood through a simple analysis. For a fixed total sample size $n+\tilde{n}=N$ and some $0<\rho<1$, the term
\[
\gamma_{n,\tilde{n}}/n= \gamma_{\rho N,(1-\rho)N}/(\rho N)=\frac{M-\rho\sigma^2}{N\rho(1-\rho)}.
\]
Approximating the numerator by $cM$ for some $0<c<1$, it gives that $\rho=1/2$ minimizes $ \gamma_{\rho N,(1-\rho)N}$. Recall that $ \gamma_{n,\tilde{n}}/n$ determines the convergence rate of the two-sample Lasso as shown in Theorem \ref{lem2-est}. This explains why $n/\tilde{n}=1$ has the smallest test errors in two-sample case. Comparing the one-sample and two-sample results, we see that one-sample Lasso has more accurate predictions on average for different sample size ratios.
\begin{figure}[H]
\includegraphics[height=5.2cm,width=0.99\textwidth]{CFW-pred}
\caption{Test errors based on one-sample and two-sample Lasso prediction for four muscle weights. The x-axis denotes three settings corresponding to $n/\tilde{n}\in\{1/3,1,3\}$ and y-axis reports the relative test error $\|\tilde{\bm y}-\tilde{\bm x}^{\intercal}\bm{b}\|_2^2/\|\tilde{\bm y}\|_2^2$. Each boxplot is based on 30 random splits.}
\label{CFW-pred}
\end{figure}
\subsection{Inference for polygenic risk scores}
\label{sec6-3}
In order to build a polygenic risk prediction, we first perform a pre-processing step to remove highly correlated covariates and to reduce the computational cost. We divide all the SNPs along chromosomes 1 to 19 into five blocks with each block containing about 15,000 SNPs. We perform principle component analysis to each block so that the principal components (PCs) account for 90\% of variation of the SNPs of that block.
This gives 3,302 principal components and we use them as the design matrix. We mention that the PCs of SNPs are linear transformations of SNPs and hence the corresponding regression coefficients are also linear transformations of the original coefficients. However, it is easy to see that the definition of $\mu_*$ is invariant to the linear transformations on the designs.
In each experiment, we randomly select 10 pairs of $(\bm x_*,\tilde{y}_*)$ from $\{(\tilde{\bx}_i,\tilde{y}_i)\}_{i=1}^{\tilde{n}}$ and report the average coverage of proposed confidence intervals on $\tilde{y}_*$. The results are given in Figure \ref{CFW-PRS}. We mention that as $\tilde{y}_*\neq \mu_*$ and $\tilde{y}_*$ has larger variance from noise, the coverage probabilities can be lower than the nominal level.
We see that the one-sample method has median coverage of above 90\% in all the settings. The two-sample method has coverage probabilities close to one in most settings.
In view of the lengths of the confidence intervals, we see that the one-sample method results in confidence intervals that are significantly shorter than those of two-sample method.
\begin{figure}[H]
\includegraphics[height=6.5cm,width=0.99\textwidth]{CFW-PRS}
\caption{Inference for PRS based on one-sample and two-sample debiased Lasso for four muscle weights. The x-axis denotes three settings corresponding to $n/\tilde{n}\in\{1/3,1,3\}$ and y-axis reports the average probabilities of the 95\% confidence intervals covering $\tilde{y}_*$ (top) and the average standard errors (bottom) in each experiment. Each boxplot is based on 30 random splits.}
\label{CFW-PRS}
\end{figure}
Finally, we present in the Supplemental Materials the inference results for the individual coefficients. We observe that the confidence intervals based on one-sample data are much shorter than those based on proxy data, further indicating the power limit for proxy-data-based inference.
\section{Discussion}
\label{sec-diss}
Statistical learning with summary data has attracted significant interests in genetic, epidemiology, and other health-related studies.
In this work, we have provided statistical inference methods and theoretical guarantees with proxy data in high-dimensional linear models. Some new phenomenon are observed in the asymptotic normality conditions and power functions.
We emphasize that the challenges stem from both the proxy and summary properties of the available data. One challenge with summary data-based regression is on tuning parameter selection and we consider BIC in Section \ref{sec-simu} and discuss some other options. Another challenge emerges from existence of heteroscedastic noises and model misspecification. In these two cases, the asymptotic variance of the debiased Lasso need to be estimated empirically \citep{buhlmann2015high,Dezeure17}, which cannot be applied with summary data. These are important and interesting future research directions.
We conclude by pointing out some other related proxy-data problems. First, it is interesting to study classification based on the proxy data. The corresponding models, such as the logistic regression model, have nonlinear link functions and hence the external covariance matrix cannot be directly used as a proxy for the Hessian. Further approximations are needed. Second, in genetic studies, LD-score regression \citep{bulik2015ld,speed2019sumher} based on GWAS summary statistics has been widely applied to estimate the heritability and co-heritability, where linear mixed-effects models are assumed. Finally, causal effect estimation using two-sample Mendelian randomization \citep{hartwig2016two,bowden2017framework} is widely studied in epidemiology. It leverages the GWAS summary data for the exposure and for the outcome, which can be collected based on different samples, to conduct causal inference. These problems may exhibit similar theoretical properties as investigated in this work but are not included in our linear model setting. It is of significant interest to study the potential bias and power loss in these problems when only summary statistics are available.
\section*{Acknowledgments}
S.L. was supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China.
T.T.C and H.L. were supported by NIH grants GM129781 and GM123056.
\bibliographystyle{chicago}
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"redpajama_set_name": "RedPajamaArXiv"
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В настоящее время программа GPA не существует отдельно от программы Международного бакалавриата. Системы IB Diploma и IB MYP ввели единую шкалу оценок от 1 до 7, где 7 — высшая оценка, 1 — низшая. Оценки всегда являются целыми числами.
Страны СНГ, Российская империя, СССР и РФ
В истории российского просвещения изначально, как и в Европе, существовала трёхразрядная система оценок. В списке студентов Киевской духовной академии (1737 г.) высший разряд обозначает хорошие выдающиеся успехи: «учения изрядного, надежного, доброго, честного, хорошего, похвального». Средний разряд обозначает успехи «учения посредственного, мерного, нехудого». Низший разряд характеризует успехи ниже среднего: «учения слабого, подлого, прехудого, безнадёжного, ленивого».
Постепенно словесная оценка становилась однообразней и короче, она чаще заменялась цифровой, причём направление шкалы установилось противоположным германскому.
Традиция обозначать цифрами прилежание и успехи учеников утвердилась в России ещё в начале XIX века. Тогда в гимназиях употреблялись цифры от 0 до 5. Нуль показывал, что ученик совсем не исполнил своих обязанностей; если он получал два «нуля» подряд, то он подвергался телесному наказанию (до 1864 г.) «Единицу» и «двойку» ставили тогда, когда ученик неудовлетворительно приготовил урок; «тройку» ставили за посредственное прилежание; «четыре» — когда ученик хорошо исполнил свои обязанности; «пять» он получал только за отличное знание урока. Учитель был обязан ставить баллы в классе, характеризуя только знание заданного на дом урока, и не имел права учитывать внимание или рассеянность учеников во время занятия, а также временное или постоянное прилежание ученика, его возраст и способности.
В разное время в России применялись 3-, 5-, 8-, 10-, 12-балльные системы оценки знаний. Из них прижилась 5-балльная, которая и была в 1837 году официально установлена Министерством народного просвещения: «1» — слабые успехи; «2» — посредственные; «3» — достаточные; «4» — хорошие; «5» — отличные. В течение XX века оценка «1» постепенно вышла из употребления, в результате 5-балльная система трансформировалась в современную 4-балльную. В последние годы по России в некоторых учебных заведениях возвращается 5-балльная система («1» — балл за невыполненную работу). Эта традиционная для советского образования система сейчас повсеместно применяется в России и многих странах постсоветского пространства, хотя в последние годы заметен отход от неё:
Белоруссия перешла на 10-балльную шкалу;
Украина — на 12-балльную и буквенную систему;
Прибалтика предпочла англосаксонскую систему (в Эстонии до сих пор используется пятибалльная шкала, «1» — оценка за невыполненную работу) и т. д.;
Молдавия перешла на 10-балльную шкалу;
Грузия перешла на 10-балльную шкалу;
Армения перешла на 10-балльную шкалу.
Казахстан
В школах используются 5-балльная (оценивание за четверть, оценка «1» не используется), 10-балльная (формативное оценивание за работу на уроке) и зачетная системы оценивания (четвертное или полугодовое оценивание факультативов и кружков). Срезы проводятся по итогам каждого раздела и каждой четверти, которые называются СОР-ы (Суммативное оценивание за раздел) и СОЧ-и (Суммативное оценивание за четверть). Количество СОР-ов в одной четверти или в одном полугодии (в случае, если количество часов предмета в неделю не превышает одного) составляет от одного до четырёх. СОЧ проводится один раз за четверть или полугодие, обычно в конце данного периода. Четвертная оценка выводится по специальной формуле, которая учитывает в себе формативные оценки, то есть оценки, полученные учеником в течение четверти за ответы на уроках, и оценки за СОР-ы и СОЧ-и.
В колледжах Казахстана используется пятибалльная шкала оценок. В вузах — 100-балльная, наряду с буквенной, где оценки ниже 60 процентов (ниже C-) являются непроходными.
Киргизия
Киргизия использует 5-балльную шкалу оценивания:
Республика Молдова
В Молдове в начальных классах (1-4) применяется критериальная система оценивания. За каждый продукт ребёнок самостоятельно оценивает себя по каждому критерию тремя цветами: зелёный (выполнил самостоятельно) — соответствует букве «с», жёлтый (был руководим учителем) — соответствует букве «р», красный (с постоянной поддержкой учителя) — соответствует букве «п». Суммативное оценивание выражается заглавными буквами ОХ — очень хорошо, Х — хорошо, У — удовлетворительно. В последующих классах используют 10-балльную шкалу, где 5 — минимальная удовлетворительная оценка:
10 (Превосходно)
9 (Очень хорошо)
8 (Хорошо)
6-7 (Средне)
5 (Удовлетворительно)
1-4 (Неудовлетворительно)
Россия
Система оценки знаний в школе
С 11 января 1944 года в российских школах введена пятибалльная система оценки успеваемости учащихся согласно Постановлению Совета народных комиссаров РСФСР № 18 от 10 января 1944 года и Приказу Народного комиссара просвещения РСФСР № 24 от 10 января 1944 года.
В соответствии с инструкцией Управления начальных и средних школ Наркомпроса РСФСР, утверждённой Народным комиссаром просвещения РСФСР 29 февраля 1944 года, установлены следующие критерии оценивания учащихся:
Согласно Инструкции Управления начальных и средних школ Наркомпроса РСФСР, утверждённой Народным комиссаром просвещения РСФСР 29 февраля 1944 года, при определении четвертных и итоговых (в конце учебного года) оценок не допускается выведение их как средних арифметических. Эти итоговые оценки должны соответствовать уровню знаний учащегося к моменту его аттестации.
Эксперимент оценивания знаний учеников буквенной шкалой «ABCDF» в РФ проводился в посёлке Урдома Архангельской области (без разрешений от Министерства образования, без согласия родителей учеников) и не подтвердил свою эффективность.
Оценивание знаний учащихся начальной и средней школы исходя из процентного отношения в Российской Федерации
В течение учебного года преподаватель начальной и средней школы должен оценивать знания учащихся по таблице, приведенной ниже, и в данном случае оценки с минусом обязательны. Балл «1-» не существует в России. Если учащийся получает балл «1», то это говорит о его полном незнании пройденного материала.
При проверке работ, написанных на государственном экзамене, проверяющий преподаватель не может ставить учащемуся оценку с минусом. В этом случае таблица оценивания знаний меняется, и преподаватель должен оценивать знания учащихся по таблице, приведенной ниже.
Выставление годовых оценок в средней и начальной школе
Четыре четверти
При выставлении годовых оценок учащемуся преподаватель, исходя из четырёх четвертных, должен поставить балл, равный среднему арифметическому из итоговых оценок. Если среднее арифметическое четырёх чисел не является целым и две категории оценок стоят в равном количестве, то годовая оценка будет являться спорной и выставляется по этим данным:
1) Спорная между 5 и 4
2) Спорная между 4 и 3
2) Спорная между 3 и 2
Два семестра/ полугодия
При выставлении годовых оценок учащемуся преподаватель, исходя из двух семестровых/ полугодичных, должен поставить балл, равный среднему арифметическому из итоговых оценок двух семестров/ полугодий. Если среднее арифметическое двух чисел не является целым, то годовая оценка является спорной и выставляется по этим данным:
1) Спорная между 5 и 4
2) Спорная между 4 и 3
2) Спорная между 3 и 2
Преподаватель может выставлять годовые оценки по выше представленным данным только в том случае, если учебное пособие, по которому учащийся обучался в течение года, соответствует Федеральному Государственному образовательному стандарту (по дате на 2014 год), так как все учебники, соответствующие этому стандарту, были составлены по данному принципу:
4 четверти
2 семестра/полугодия
Система оценки знаний в средних и высших учебных заведениях
В вузах и ссузах России оценки знаний установлены Приказом Государственного комитета СССР по народному образованию от 22 июня 1990 года № 432 «Об утверждении Положения о формах контроля учебной работы учащихся дневных и вечерних отделений средних специальных учебных заведений». Согласно данному нормативному документу знания, умения и навыки учащихся по всем формам контроля учебной работы, включая учебную и технологическую практики, оцениваются в баллах: 5 (Отлично); 4 (Хорошо); 3 (Удовлетворительно); 2 (Неудовлетворительно). Лабораторные работы, практические занятия и преддипломная практика оцениваются: «Зачтено», «Не зачтено». Учебные заведения культуры и искусства могут использовать другие системы оценок успеваемости учащихся, согласованные с вышестоящим органом.
Украина
Украина осенью 2000 года представила свою новую шкалу оценивания, которая заменила советскую.
Новая система оценивания базируется на основе существовавшей ранее 5-балльной шкалы оценок, которая соотносится с 12-балльной системой оценивания. Оценка 12 выставляется только за выдающиеся успехи либо за какую-либо творческую работу.
Четвёртый уровень — высокий (10-12 баллов). Знания ученика глубокие, твёрдые, системные; ученик умеет использовать их для выполнения творческих заданий, его учебная деятельность отличается умением самостоятельно оценивать разнообразные ситуации, явления и факты, проявлять и отстаивать личную позицию.
Третий уровень — достаточный (7-9 баллов). Ученик знает существенные признаки понятий, явлений, связи между ними, умеет объяснить основные закономерности, а также самостоятельно использует знания в стандартных ситуациях, владеет умственными операциями (анализом, абстрагированием, обобщением). Ответ правильный, логически обоснованный, но ученику недостает собственных суждений.
Средний уровень(4-6 балла)-Ученик воспроизводит основной учебный материал, способен выполнять задания по образцу, владеет элементарными умениями учебной деятельности
Первый уровень — начальный (1-3 балла). Ответ ученика фрагментарный, характеризуется начальными представлениями о предмете изучения.
С 2022 года, в рамках реформы НУШ, оценивание учеников 1-4 классов будет происходить буквами.
Беларусь
В Беларуси используются оценки от 1 до 10.
10 (Превосходно)
9 (Отлично)
8 (Почти отлично)
7 (Очень хорошо)
6 (Весьма хорошо)
5 (Хорошо)
4 (Весьма удовлетворительно)
3 (Удовлетворительно)
2 (Почти удовлетворительно)
1 (Неудовлетворительно)
Европа
Система оценки знаний баллами зародилась в иезуитских школах в XVI—XVII веках и имела гуманную цель заменить принятые в те времена телесные наказания на поощрения. Первая трёхбалльная шкала оценок возникла в Германии, она получилась в результате разделения всех учеников на три нумерованных разряда: лучших, средних и худших, причём переход из одного разряда в другой, более высокий, знаменовал собой приобретение целого ряда преимуществ и привилегий. Первоначально единица имела значение высшей оценки. Со временем средний разряд, к которому принадлежало наибольшее число учеников, дополнительно разделили на подразряды — так сформировалась многоуровневая ранговая шкала, с помощью которой стали оценивать познания учащихся.
Австрия
В Австрии используются оценки от 5 до 1.
Албания
В Албании используются оценки от 1 до 10. Некоторые школы разрешают использовать более высокие оценки, а также другую систему оценивания знаний.
Болгария
В Болгарии в школах используется следующая шкала оценок:
Для экзаменов и тестов для более точного результата выставляется оценка с учётом двух десятичных знаков после запятой:
Такие оценки, как, например, Хорошо (3,50) или Превосходно (5,75) являются общими. Любая оценка, равная либо большая 50, считается удовлетворительной. Минимальной является оценка 2,00; оценки ниже 3,00 являются неудовлетворительными, и максимальной является оценка 6,00. Оценки наподобие Очень хорошо (5-) и Средне (3+) также возможны, но они не учитываются в подсчётах.
Грубо говоря, болгарская шкала может приравниваться к американской следующим образом: 6=A, 5=B, 4=C, 3=D и 2=F. Также её можно сравнить с австралийской шкалой оценивания: 6=HD, 5=D, 4=Cr, 3=P и 2=F.
Наиболее распространённая формула подсчёта оценок в болгарских школах в настоящее время следующая: итоговая оценка = (6 * число правильных ответов) / общее число вопросов. Таким образом, если учащийся ответил правильно на 7 вопросов из 10, его оценка будет следующей: (6*7)/10 = 4,20, которая оценивается как Хорошо (4), что соответствует среднему уровню знаний.
Босния и Герцеговина
В Боснии и Герцеговине в начальных и средних школах используются оценки от 5 до 1, в университетах используется шкала оценивания от 10 до 5.
В начальных и средних школах используется следующая шкала:
Шкала оценивания в университетах:
Венгрия
С 1950 года в Венгрии используется 5-балльная шкала оценивания. В ней присутствует всего одна неудовлетворительная оценка: 1 — elégtelen (Неудовлетворительно). В целом, нижний предел шкалы варьируется от 50 % до 60 % или на один балл выше. Также в этой шкале присутствуют следующие оценки: 2 — elégséges (Удовлетворительно или Достаточно), 3 — közepes (Средне), 4 — jó (Хорошо) и 5 — jeles (Превосходно).
Пятибалльная система оценивания в большинстве случаев используется в конце семестра, а также на других образовательных уровнях (например, в начальной школе, старшей школе, университете). Во время учебного года учитель имеет право использовать шкалу оценок, применяемую в начальной школе. Также после оценки может использоваться знак (,) («alá»), а также апостроф (') («fölé»). Существуют и промежуточные оценки (например, 3/4) («háromnegyed»), что является эквивалентом оценки 3,5; 4/5 — оценка между 4 и 5 и т. д. Иногда, для того чтобы показать, что за семестр у обучающегося произошёл большой прогресс, может использоваться и оценка «5*» («csillagos ötös»).
Германия
В Германии в среднем образовании используется 6-балльная система оценок с обратной зависимостью, которая имеет следующие количественные и качественные обозначения:
1 — ausgezeichnet / sehr gut (Отлично)
2 — gut (Хорошо)
3 — befriedigend (Удовлетворительно)
4 — ausreichend (Достаточно)
5 — mangelhaft (Неудовлетворительно)
6 — ungenügend (Недостаточно).
В некоторых учебных заведениях для выведения средней оценки используется пересчёт приведённых выше оценок в баллы по следующему соотношению: оценка 1 = 15 баллов, 2 = 12 баллов, 3 = 9 баллов, 4 = 6 баллов, 5 = 3 балла и оценка 6 = 0 баллов.
В высшем образовании используется 5-балльная система оценок, пять количественных и качественных параметров которой полностью идентичны приведённым выше пяти первым параметрам 6-балльной системы оценок среднего образования.
Дания
В Дании в 2007 году была принята 7-балльная система оценивания (syv-trins-skalaen), которая стала заменой старой 13-балльной системы оценивания (13-skala). Новая шкала была создана и совмещена в соответствии со стандартами ECTS-шкалы. Syv-trins-skalaen состоит из семи различных оценок в диапазоне от 12 до −3, с максимальной оценкой в 12 баллов. Эта новая шкала является остатком «чистой» шкалы, а это означает, что оценка не всегда соответствует заслугам.
Исландия
В Исландии используются оценки от 0 до 10, где 5 является наименьшей удовлетворительной оценкой, однако в некоторых случаях наименьшей удовлетворительной оценкой является 3,5.
Литва и Латвия
В Литве и Латвии используются баллы от 1 до 10. Наименьшей удовлетворительной оценкой является 4. В большинстве случаев ученику нельзя исправлять оценки от 5 баллов до 10, однако бывают школы, где ученики имеют возможность исправить эти оценки. Баллы от 1 до 4 являются неудовлетворительными и должны быть исправлены. Ученик не может перейти в следующий класс, если имеет в табеле оценки ниже 4 баллов.
Северная Македония
Шкала для начальных и средних школ:
Университетская шкала:
Норвегия
В начальной школе (Barneskole, в возрасте от 6 до 13 лет) официально оценки не ставятся. Конечно, преподаватели пишут собственные комментарии или анализы тестов в конце каждой четверти. В младших и старших классах средней школы используется шкала от 1 до 6, где 6 — высшая оценка, а 2 — самая низкая удовлетворительная оценка. Для обычных тестов и четвертных результатов оценки часто употребляются со знаками «+» и «−» (кроме 6+ и 1−). Также в общей практике используются такие оценки, как 5/6 или 4/3, что указывает на результат между этими двумя оценками. Конечно, те оценки, которые учащиеся получают в свой диплом (Vitnemål), содержат целое число: 1, 2, 3, 4, 5 или 6. Нецелочисленный средний балл учащегося также указывается в дипломе Vitnemål.
В высшем образовании в соответствии с системой ECTS оценки за университетские и аспирантские экзамены выставляются по шкале от A (высший балл) до F (низший балл), где E является минимальным проходным баллом. Система ECTS была введена в университетах и колледжах Норвегии в начале 2000-х годов, в школах оценки переводятся в систему ECTS с 2003 года. До 2003 года наиболее распространённой шкалой оценок, используемой на университетском уровне, была шкала 1,0 (высший балл) до 6,0 (низший балл) с минимальным удовлетворительным результатом в 4,0.
Румыния
В начальных школах Румынии используется следующая шкала:
Foarte Bine (FB) — Очень хорошо
Bine (B) — Хорошо
Satisfăcător (S) — Удовлетворительно
Nesatisfăcător (I) — Неудовлетворительно
В средних и старших школах и академических институтах используется 10-балльная шкала, где 5 — наименьшая удовлетворительная оценка:
10 (Превосходно)
9 (Очень хорошо)
8 (Хорошо)
6-7 (Нормально)
5 (Удовлетворительно)
1-4 (Неудовлетворительно)
В этой системе нет оценки 0, а 1 ставится только за списывание. Если учащийся выполнил 86 % заданий, то он будет иметь результат в 8,60 балла, который будет округлён до 9.
Сербия
В Сербии имеется такая же шкала оценивания, что и в бывшей Югославии. В начальных и средних школах используется 5-балльная шкала:
5 (одлично, odlično) — Превосходно
4 (врло добро, vrlo dobro) — Очень хорошо
3 (добро, dobro) — Хорошо
2 (довољно, dovoljno) — Удовлетворительно, минимальная удовлетворительная оценка.
1 (недовољно, nedovoljno) — Неудовлетворительно, минимально возможная оценка.
Чехия
В Чехии 5-балльная шкала используется в начальных и средних школах:
Турция
В Турции применяется шкала оценивания от 1 до 5.
Финляндия
Школьная шкала оценок в Финляндии формально имеет диапазон от 0 до 10, но оценки ниже 4 не ставятся. Таким образом, в настоящее время шкала оценок делится на неудовлетворительные (4) и оценки от 5 до 10, являющиеся удовлетворительными. Данная шкала оценивания похожа на румынскую шкалу:
10 (Превосходно) — данную оценку получают 5 % лучших учащихся
9 (Очень хорошо)
8 (Хорошо)
7 (Удовлетворительно)
6 (Малоудовлетворительно)
5 (Посредственно (проходной минимум))
4 (Неудовлетворительно)
На отдельных экзаменах, но не в качестве итоговой оценки, шкала оценок может делиться с точностью до ½ балла, что представляет собой промежуточные оценки, которые обозначаются знаками + и -, которые означают, что учащийся получит на 1/4 более высокую либо более низкую оценку. Например: 9 < 9+ < 9½ < 10- < 10. Оценка 10+ может быть поставлена при условии, что учащийся приложит дополнительные усилия.
Гимназии используют такую же шкалу оценок, что и обычные школы, но выпускные экзамены оцениваются по шкале с традиционными латинскими наименованиями, опирающейся на нормальное распределение результатов выпускников.
Оценка magna cum laude approbatur была введена в 1970 году, а eximia cum laude approbatur в 1996 году. Оценки laudatur, полученные до 1996 года, сегодня считаются эквивалентными eximia cum laude approbatur.
Университеты и институты используют шкалу оценок от 0 (Неудовлетворительно) до 5 (Блестяще) или Неудовлетворительно/ Удовлетворительно (). Короткие практические курсы оцениваются, как правило, по второй шкале.
Франция
Во Франции 20-балльная система, причём оценки в 20 и 19 баллов используются крайне редко.
Хорватия
В школах Хорватии оценки выставляются по следующей шкале:
В конце каждой четверти высчитывается средний балл всех оценок (prosječna ocjena), где результат определяется согласно следующей шкале:
В разговорном хорватском языке оценки могут соотноситься с их числовыми значениями: jedinica, dvojka, trojka, četvorka и petica. В хорватской области Кварнер jedinica также известна как komad либо kolac, а dvojka также известна как duja.
Швейцария
Швейцария использует шкалу оценок от 1 до 6. 6 является наивысшей оценкой, а 4 — минимальной удовлетворительной оценкой.
Европейская система оценивания
В Лихтенштейне используется такая же шкала оценок, что и в Швейцарии и Молдове. Там используют румынскую шкалу оценивания вследствие того, что в Европе существуют некоторые стандарты систем оценивания. Большинство из них включает в себя комбинации разных шкал оценивания.
Азия
Афганистан
Оценка A+ по всем предметам приравнивается к «Золотому A+»
Вьетнам
Школы и университеты во Вьетнаме используют 10-балльную шкалу оценивания, где 10 — высшая оценка, а 0 — низшая. Как правило, минимальной удовлетворительной оценкой является 4.
В разных школах шкалы оценивания могут различаться между собой. Это зависит от сложности каждой шкалы.
Обычно данная шкала подразделяется на 7 уровней успеваемости.
Распределение уровней отличается в соответствии со стандартами в западных странах, и сильно зависит от университета. Во вьетнамских университетах получить оценки 9 или 10 практически невозможно. Студенты с трудом получают итоговые оценки выше, чем 8,0. В целом, все государственные университеты дают уровень Cs большинству студентов, которые в состоянии сдать экзамены на удовлетворительный результат.
Израиль
В Израиле используется 100-балльная шкала оценивания, где применяются следующие оценки:
Индия
Оценивание в университетах
Индийские университеты следуют процентной шкале, а Индийский институт технологий следует 10-балльной шкале со средним баллом. Данная процентная система работает следующим образом:
Максимальная оценка: 100
Минимальная оценка: 0
Минимальный проходной балл: 40 или 30 (в зависимости от университета)
* В некоторых институтах Индии более низкий процент может считаться удовлетворительным.
Новая 8-балльная шкала с учётом среднего балла была представлена в Мумбайском университете в 2012—2013 учебном году:
10-балльная шкала оценивания, представленная Индийским технологическим институтом, выглядит следующим образом:
Оценивание в старшей школе
Для выставления оценок в старшей школе используется усреднённый процент. Показатель выше 90 процентов считается превосходным; между 70-89 процентами — первый уровень; между 50-69 % — второй уровень, 40-49 % являются минимальным проходным баллом; однако данная терминология и классификация зависит от Совета образования.
Индонезия
Примечания
Школы
Образование
Педагогика | {
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} | 1,631 |
Anette Langner (* 15. August 1961 in Stuttgart) ist eine deutsche Politikerin (SPD).
Leben und Beruf
Nach dem Abitur 1980 begann Anette Langner 1981 ein Studium der Rhetorik, Literaturwissenschaft und Geschichte an der Eberhard-Karls-Universität Tübingen, welches sie 1988 als Magister Artium (M.A.) beendete. Anschließend absolvierte sie bis 1990 eine Ausbildung zur Marketingassisstentin im Verlagswesen und war danach bis 1996 als Marketingassistentin und Medienberaterin bei Zeitschriftenverlagen und Agenturen tätig. 1996 wechselte sie als Referentin für Presse- und Öffentlichkeitsarbeit zur Kieler Beschäftigungs- und Ausbildungsgesellschaft KIBA GmbH und übernahm hier 1999 die Leitung des Bereichs Personalentwicklung.
Anette Langner trat 2002 in die SPD ein und war von 2005 bis 2013 Vorsitzende des SPD-Kreisverbandes Plön.
Abgeordnete
Sie gehörte zwischen 2003 und 2010 der Gemeindevertretung von Schönberg an.
Von 2005 bis 2012 war Anette Langner Mitglied des Landtages von Schleswig-Holstein. Hier war sie stellvertretende Vorsitzende des Wirtschaftsausschusses und außerdem Beisitzerin im Vorstand der SPD-Landtagsfraktion. Bei der Landtagswahl 2005 erreichte Anette Langner 44,4 % der Erststimmen im Wahlkreis Plön-Nord, setzte sich damit mit knappem Vorsprung gegen den CDU-Kandidaten Werner Kalinka (44,3 %) durch und zog direkt in den Landtag ein.
Bei der vorgezogenen Landtagswahl am 27. September 2009 unterlag sie mit 33,3 % der Erststimmen dem CDU-Kandidaten Werner Kalinka (35,7 %), sie kam über die Landesliste in den Landtag. 2012 wurde sie im Wahlkreis Plön-Nord/Malente mit 39,9 % der Stimmen direkt in den Landtag gewählt. Mit der Ernennung zur Staatssekretärin legte sie ihr Landtagsmandat nieder.
Öffentliche Ämter
Langner war vom 13. Juni 2012 bis zum 28. Juni 2017 Staatssekretärin im Ministerium für Soziales, Gesundheit, Wissenschaft und Gleichstellung des Landes Schleswig-Holstein für die Ministerin Kristin Alheit. Nach der Landtagswahl 2017 und dem Regierungswechsel wurde sie in den Ruhestand versetzt.
Weblinks
Website von Anette Langner
Einzelnachweise
Deutscher
SPD-Mitglied
Landtagsabgeordneter (Schleswig-Holstein)
Staatssekretär (Schleswig-Holstein)
Geboren 1961
Frau | {
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Q: How to remove lines from TSV file where columns are empty or all whitespace? I've a tab-delimited file, e.g. myfile.tsv:
abc\tfoo
xyz\tbar
but sometimes, it has some blank columns, e.g.
abc\tfoo
xyz\tbar
what\t
\tthe
bleep\tsleep
i.e.
$ printf "abc\tfoo\n" > myfile.tsv
printf "xyz\tbar\n" >> myfile.tsv
printf "what\t\n" >> myfile.tsv
printf "\tthe\n" >> myfile.tsv
printf "bleep\tsleep\n" >> myfile.tsv
$ cat myfile.tsv
abc foo
xyz bar
what
the
bleep sleep
I could write a python script to remove the lines where the columns are empty, e.g.
with open('myfile.tsv') as fin:
for line in fin:
x, y = line.strip().split('\t')
x = x.strip()
y = y.strip()
if x and y:
print(line)
But how do I do the same with some unix shell commands, e.g. grep, sed, awk or something.
I've tried also something like this in grep:
grep -e ".\t." myfile.tsv
That seems to work but if the columns have spaces, it won't.
$ printf "abc\tfoo\n" > myfile.tsv
printf "xyz\tbar\n" >> myfile.tsv
printf "what\t \n" >> myfile.tsv
printf " \tthe\n" >> myfile.tsv
printf "bleep\tsleep\n" >> myfile.tsv
$ grep -e ".\t." myfile.tsv
abc foo
xyz bar
what
the
bleep sleep
A: Using Miller (mlr):
$ cat -t myfile.tsv
abc^Ifoo
xyz^Ibar
^I
what^I
^Ithe
bleep^Isleep
$ mlr --tsv filter 'bool empty=false ; for (k,v in $*) { empty = is_empty(v); empty { break } } !empty' myfile.tsv
abc foo
xyz bar
bleep sleep
The equivalent thing in awk:
$ awk -F '\t' '{ empty = 1; for (i = 1; i <= NF; ++i) if (empty = (length($i) == 0)) break }; !empty' myfile.tsv
abc foo
xyz bar
bleep sleep
A: Using sed
$ sed -E '/^\t|\t$/d' myfile.tsv
abc foo
xyz bar
bleep sleep
A: To remove lines where the ALL fields on that line either contain only spaces, tabs, or are empty, you can match and exclude lines containing only whitespace:
grep -v '^[[:space:]]*$'
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,426 |
Q: Classic ASP: parent.parent.getelementbyID from iframe in Chrome I have inherited an old classic ASP website, where part of the interface is done in nested iframes. In the nested iframe, the value of a text-field in the parent of the parent is set, but this only works in Internet Explorer
A simple snippet:
function Test() {
parent.parent.document.getElementById("myField").value = "Test";
}
...
<input id="myField" type="button" onclick="Test();"/>
How to I get this to work in Chrome? Or Edge for that matter..
A: Changing
parent.parent.document.getElementById("myField").value = "Test";
To
parent.parent.document.forms["formName"]["myField"].value = "Test";
seemed to do the job. There are multiple forms in the parent.parent page, and eventhough the ID and names are unique, specifying the form seems necessary.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,696 |
Pier Francesco Mazzucchelli, allgemein bekannt als il Morazzone (* 29. Juli 1573 in Morazzone (bei Varese); † um 1625) war ein italienischer Maler und Freskant im Übergang vom Spätmanierismus zum Frühbarock, der unter anderem für seine Malereien in mehreren Sacri Monti der Lombardei bekannt ist.
Leben
Er war ein Sohn von Cesare di Mazuchi del Tachino und der Ermelina da Fagnano. Sein Künstlername Morazzone leitet sich von seinem Geburtsort ab.
Während seiner Kindheit übersiedelte die Familie nach Rom, wo Pier Francesco seine malerische Ausbildung bei Ventura Salimbeni erhielt. Von seinen römischen Jugendwerken – laut Giovanni Baglione arbeitete er u. a. im Cortile von San Giovanni in Laterano und in der Sakristei des Petersdoms – sind allein die Fresken mit der Anbetung der Könige und der Visitation in einer Kapelle der Kirche San Silvestro erhalten, die er 1596 für Antonio Maria Manzoli, den Bischof von Gravina, malte. Darin zeigt er Einflüsse nicht nur von Salimbeni, sondern auch von Cavalier d'Arpino und Federico Barocci.
1598 verließ er Rom, nachdem er – wahrscheinlich bei einem Streit um eine Frau – einen Alessandro del Rio verwundet hatte und deshalb in einen Prozess verwickelt worden war.
Ab September 1598 war er in Varese, wo er am 13. November Anna Castiglioni heiratete, die aus einer angesehenen Familie stammte und mit der er bis zu ihrem Tode im Jahr 1621 acht Kinder hatte.
Sein wohl erstes Werk in Varese waren Fresken mit einer Marienkrönung und musizierenden Engeln in der Rosenkranzkapelle der Kirche San Vittore (1598-99).
Morazzone stieg bald zu einem der bedeutendsten Maler der Lombardei auf, neben Giulio Cesare Procaccini und Cerano (Giovan Battista Crespi), mit denen er ab 1602 an einem Bilderzyklus über das Leben des Hl. Carlo Borromeo für den Mailänder Dom arbeitete. Morazzone werden dabei die zwei Gemälde Carlo Borromeo verzichtet auf den Kirchenschatz und Begegnung des Hl. Carlo Borromeo mit Emanuele Filiberto von Savoyen zugeschrieben, die er eventuell zusammen mit Paolo Camillo Landriani, gen. il Duchino, malte.
1603 erhielt er den Auftrag für Fresken und sechs Ölbilder über das Leben der Jungfrau Maria in der Collegiata von Arona, der Taufkirche des Hl. Carlo Borromeo.
Bereits im August 1602 hatte er einen Vertrag für Fresken in der Kapelle mit dem Gang auf den Kalvarienberg im Sacro Monte di Varallo unterzeichnet, an denen er bis 1607 arbeitete. Auf Wunsch des Bischofs Carlo Bascapè sollte er diese Bilder ausdrücklich im Stile der 1528 von Gaudenzio Ferrari geschaffenen Fresken in der Kalvarienbergkapelle malen. Seine Malereien mussten außerdem mit den Skulpturen in der Kapelle zu einem harmonischen Gesamtkunstwerk abgestimmt werden. Diese Aufgabe gab Morazzones Kunst, die bis dahin deutlich vom römischen Spätmanierismus beeinflusst war, einen wichtigen Impuls in eine natürlichere Richtung auf der Basis der lombardischen Tradition. Auch entwickelte er eine starke Ausdruckskraft. Im Sacro Monte di Varallo malte er später auch die Fresken in den Kapellen des Ecce Homo (1610) und der Verurteilung Jesu (1610–1616). Seine Arbeit in Varallo endete nach einem Streit, in dessen Folge man ihm den jungen Tanzio da Varallo vorzog.
Die Jahre etwa zwischen 1608 und 1615 waren die fruchtbarsten seiner ganzen Karriere. Neben den Fresken im Sacro Monte di Varallo malte er 1608–1609 auch die Kapelle der Geißelung im Sacro Monte di Varese aus und schuf 1616–1617 die Fresken in der elften Kapelle (Institution der Portiuncola) im Sacro Monte di Orta, bei denen Einflüsse der lombardischen und venezianischen Tradition sichtbar sind, besonders von Pordenone und Paolo Veronese.
Morazzone arbeitete auch mehrfach für Carlo Emanuele I. von Savoyen, nachdem er im Jahr 1608 bei den Festdekorationen zur Hochzeit von Margherita und Isabella von Savoyen mit den Herzögen von Mantua und Modena mitgewirkt hatte.
Daneben malte er eine ganze Reihe von Werken für Kirchen in Como und Umgebung, die er teilweise der Protektion des dortigen Bischofs Quintilio Lucini Passalacqua verdankte, darunter das Banner des Hl. Abbondio im Dom zu Como (1608).
All diese (und andere) Aufträge konnte er nur mithilfe seiner gut funktionierenden Werkstatt bewältigen. Mittlerweile hatte Morazzone den Höhepunkt der Anerkennung erreicht. Der Dichter Giovan Battista Marino verglich ihn in seinem Tempio panegirico di Maria de' Medici (Lyon, 1615, S. 105) mit dem "unsterblichen Apelles" ("immortale, Apelle Insubro"), und erwähnte zwei Bilder von Morazzone in seiner Galeria (Venedig, 1620). Auch Girolamo Borsieri erwähnte den Maler in seinem Supplemento alla nobiltà di Milano von 1619.
Zu einem nicht bekannten Zeitpunkt, eventuell um 1617, malte Morazzone gemeinsam mit Cerano und Giulio Cesare Procaccini das berühmte sogenannte "Bild der drei Hände" ("quadro delle tre mani"), eine Darstellung des Martyriums der Hl. Rufina und Seconda, das sich heute in der Pinacoteca di Brera in Mailand befindet. Von Morazzone sind die Figuren im Zentrum mit dem Henker, einem Pagen und einem Engel mit Palme; Procaccini malte die Hl. Rufina mit dem Engel im Vordergrund rechts; von Cerano sind der Cavalier im Hintergrund, und die enthauptete Seconda mit einem Engel und einem Hund unten links.
Zu Morazzones Hauptwerken zählen auch die 1620 fertiggestellten Fresken in der Cappella della Buona Morte in der Kirche San Gaudenzio von Novara, die er komplett mit Malereien über das Thema Memento mori und mit einem spektakulären Jüngsten Gericht ausmalte.
1622 für Carlo Emanuele I von Savoyen geschaffene Malereien im Castello di Rivoli gingen tragischerweise 1691 bei einem Brand verloren. Um dieselbe Zeit malte er auch die sogenannte Honigmadonna (Madonna del miele) in der Galleria Sabauda (Turin), von der es mehrere Repliken gibt.
1623 hielt er sich am Hofe des Herzogs Ferdinando Gonzaga in Mantua auf, aber aus einem Brief des Malers vom 29. März geht hervor, dass er schon seit September 1622 massive gesundheitliche Probleme hatte und deshalb ein bei ihm bestelltes Bild der Hochzeit zu Kana nicht mehr ausführen konnte. Trotzdem folgte er noch im Jahr 1623 einem Ruf nach Piacenza, wo er die Kuppel des Domes ausmalen sollte. Er musste jedoch nach den beiden ersten Bildern der Propheten David und Jesaja aufhören, und der Auftrag für die Kuppeldekoration wurde 1626 endgültig an Guercino (Giovanni Francesco Barbieri) übergeben.
Der genaue Zeitpunkt und Ort des Todes von Pier Francesco Morazzone sind nicht bekannt, laut Baglione soll er in seinem Geburtsort gestorben sein.
Bildergalerie
Werke (Auswahl)
Anbetung der Könige und Visitation, Fresken in der Cappella della Concezione der Kirche San Silvestro (?), Rom, 1596
Marienkrönung und musizierende Engel, Fresken in der Rosenkranzkapelle der Kirche San Vittore, Varese, 1598–99 und 1617
Der Hl. Carlo Borromeo verzichtet auf den Kirchenschatz und Begegnung des Hl. Carlo Borromeo mit Emanuele Filiberto von Savoyen, Mailänder Dom, 1602
Fresken im Sacro Monte di Varallo:
in der Kalvarienbergkapelle, 1602–1607
in der Kapelle des Ecce Homo, 1610
in der Kapelle der Verurteilung Jesu, 1610–16
Fresken und Ölgemälde in der Collegiata von Arona, um 1603
Pfingsten (Deckengemälde), Pinacoteca del Castello Sforzesco, Mailand, 1605
das Marchesato di Susa, Galleria Sabauda, Turin
das Banner des Hl. Abbondio, Dom von Como, 1608
Fresken im Sacro Monte di Varese:
in der Kapelle der Geißelung, 1608–09
Maria Magdalena, San Vittore, Varese, 1611
Anbetung der Könige, Sant'Antonio Abate, Mailand, ca. 1610
Caritas, Chiesa della Carità, Como, 1608–10 (für Giovan Pietro Odescalchi)
Stigmatisation des Hl. Franziskus und Kain und Abel (urspr. in der Villa des Abtes Marco Gallio in Borgo Vico), Sakristei der Mansionari von Como, 1609–10
vierteiliger Marienzyklus in der Cappella della Cintura in Sant'Agostino, Como, 1612
5 biblische Szenen als Allegorien der 5 Sinne (Öl auf Kupfer), Dekor im Büro des Bischofs Passalacqua, Como, 1613
Dekoration der Sankt-Georgs-Kapelle im Santuario di Rho, 1614
Fresken und Ölgemälde in den Kapellen der Hl. Rochus und Carlo in der Collegiata di San Bartolomeo, Borgomanero, 1612 bis 1620
Fresken in der Kapelle XI (Institution der Porziuncola) im Sacro Monte di Orta, 1616–1617
Bethlehemitischer Kindermord, Museo diocesano, Mailand, nach 1616
Rosenkranzmadonna mit den Hl. Dominikus und Katharina von Siena, Certosa di Pavia, 1617
Fresken in der Kirche Sant'Ambrogio Olona (bei Varese), um 1617 (schlecht erhalten)
"Bild der drei Hände" ("quadro delle tre mani"): Martyrium der Hl. Rufina und Seconda, Pinacoteca di Brera, Mailand, um 1617 (?) (zusammen mit Cerano und Giulio Cesare Procaccini)
Jüngstes Gericht und andere Fresken in der Cappella della Buona Morte in San Gaudenzio, Novara, 1620
Honigmadonna (Madonna del Miele), Galleria Sabauda, Turin, ca. 1622
die Propheten David und Jesaia, Fresken im Dom von Piacenza, 1623–25 (?)
Literatur
Morazzone (eigt. Mazzucchelli), Pier Francesco. In: Lexikon der Kunst. Bdand 8, Karl Müller Verlag, Erlangen 1994, S. 233.
Weblinks
Einzelnachweise
Maler (Italien)
Maler des Manierismus
Maler des Barock
Künstler (Mailand)
Freskant
Geboren 1573
Gestorben im 17. Jahrhundert
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,603 |
\section{Introduction}
Increasingly within the sciences, networks and network methodologies are being used to answer research questions. Such networks might be observed, such as connections
in communication network or information flows within, or they could be unobserved:
inferred networks that can explain a process or effect.
Given the increase in the size of data sets,
it may also be useful to infer a network from data to efficiently summarise the data generating process.
We consider time series observations recorded at different nodes of a network, or graph. Our
\pkg{GNAR} package \citep{leeming18:GNAR} and its novel generalised network autoregressive (GNAR) statistical models
focus on partnering a network with a multivariate
time series and modelling them jointly. One can find an association network, see,
e.g., Chapter~7 of \cite{Kolaczyk2009},
or Granger causality network, e.g., \cite{Dahlhaus2003},
between different variables by analysing a multivariate time series and its properties.
However,
here we assume the existence of an underlying network and use it during the analysis of the time series, although sometimes its complete structure is unknown.
Networks can provide
strong information about the dependencies between variables.
Within our generalised network autoregressive (GNAR) model, each node depends on its previous values as in the univariate autoregressive
framework, but also may depend on the previous values at its neighbours, neighbours of neighbours, and so on. Our GNAR modelling framework is flexible, allowing for different types of network,
networks that change their structure over time (time-varying networks),
and also can be powerfully applied in the important practical
situation where the time series feature missing
observations.
Driven in part by the increased popularity and recent research activity in the field of statistical network analysis, there has been a concurrent growth in software for analysing such data. An exhaustive list of these packages is beyond the scope of this article, but we review some relevant ones here.
Existing software in this area predominantly focusses on the various models for network-structured data. In the static network setting, these include packages dedicated to latent space network models, such as \pkg{collpcm} \citep{wyse17:collpcm}, \pkg{HLSM} \citep{adhikari18:HLSM}, \pkg{latentnet} \citep{krvitsky18:latentnet} amongst others; exponential random graph models and their variants, for example \pkg{ergm} \citep{handcock18:ergm}, \pkg{GERGM} \citep{denny18:GERGM} or \pkg{hergm} \citep{schweinberger18:hergm}; and block models in e.g., \pkg{blockmodels} \citep{leger15:blockmodels}. For dynamic networks, packages for time-varying equivalents of these network models are also available, see e.g., the \pkg{tergm} package \citep{krvitsky18:tergm} or \pkg{dynsbm} \citep{matias18:dynsbm}. There are also a multitude of more general packages for network analysis, e.g., for network summary computation or implementations of methodology in specific applications of interest.
Despite this, software dedicated to the analysis of time series and other \emph{processes} on networks is sparse. A number of packages implement epidemic (e.g., SIR) models of disease spread, notably \pkg{epinet} \citep{groendyke18:epinet}, \pkg{EpiLM}/\pkg{EpiLMCT} \citep{warriyar18:EpiLM, almutiry18:EpiLMCT} and \pkg{hybridModels} \citep{marquez18:hybridModels}; these use transmission rates to model processes as opposed to temporal and network dependence through time series models as in \pkg{GNAR}. Similarly, the \pkg{NetOrigin} software \citep{manitz18:NetOrigin} is dedicated to source estimation for propagation processes on networks, rather than fitting time series models. Packages such as \pkg{networkTomography} \citep{blocker14:networkTomography} deal with time-varying models for (discrete) count processes or flows on links of a \emph{fixed} routing network; the \pkg{tnam} package \citep{leifeld17:tnam} fits models using \emph{spatial} (and not network-node) dependence. Both of these are in contrast to the \pkg{GNAR} package,
which implements time series models which account for known \emph{time-varying network structures}.
Other packages can implicitly develop network-like structured time series models through penalised or constrained variable selection, such as \pkg{autovarCore}
\citep{emerencia18:autovarCore}, \pkg{nets} \citep{brownlees17:nets}, \pkg{sparsevar} \citep{vazoller16:sparsevar}, as well as the \pkg{vars} package \citep{Pfaff2008}.
Packages that take a graphical modelling approach to the dependence structure within time series include \pkg{gimme} \citep{lane19:gimme}, \pkg{graphicalVAR}
\citep{epskamp18:graphicalVAR}, \pkg{mgm} \citep{haslbeck19:mgm}, \pkg{mlVAR} \citep{epskamp19:mlVAR}, and \pkg{sparseTSCGM} \citep{abegaz16:sparseTSCGM}. These approaches also differ fundamentally from the GNAR models since the network is constructed during analysis, as opposed to \pkg{GNAR},
which specifically incorporates information on the network structure into the model \emph{a priori}. The \pkg{vars} package features in Section~\ref{sec:results}, where we highlight the differences between the GNAR models and this existing class of techniques.
Section~\ref{Model} introduces our model, and demonstrates how \pkg{GNAR}
can be used to fit network models to simulated
network time series in Section~\ref{exsim}.
Order selection and prediction are discussed in Section~\ref{Implementation}, which includes
an example of how to use BIC to select model order for a wind speed network
time series in Section~\ref{wind}.
An extended example, concerning constructing a network to aid GDP forecasting,
is presented in Section~\ref{gdp}.
Section~\ref{Discussion} discusses different network modelling options that could be chosen, and presents a summary of the article.
All results were calculated using version~3.5.1 of the statistical software \proglang{R} (\cite{Rcore2017}).
\section{Network time series processes} \label{Model}
We assume that our multivariate time series follows an autoregressive-like model at each node, depending both on the previous values of the process at that node, and on neighbouring nodes at previous time steps. These neighbouring nodes are included as part of the network structure, as defined below.
\subsection{Network terminology and notation}
Throughout we assume the presence of one or more networks, or graphs, associated with the observed time series. Each univariate time series that makes up the multivariate time series occurs, or is observed at,
a node, or location on the graph(s). These nodes are connected by a set of edges, which may be directed, and/or weighted.
We denote a graph by $\mathcal{G} = (\mathcal{K},\mathcal{E})$, where $\mathcal{K}=\{1,...,N\}$ is the set of nodes, and $\mathcal{E}$ is the set of edges. A directed edge from node $i \in \mathcal{K}$ to $j \in \mathcal{K}$ is denoted $i \rightsquigarrow j$, and an un-directed edge between the nodes is denoted $i \leftrightsquigarrow j$. The edge set of a directed graph is $\mathcal{E} = \{ (i,j): i \rightsquigarrow j; i,j \in \mathcal{K} \}$, and similarly for the set of un-directed edges.
\subsubsection{Stage-$r$ neighbourhoods}
We introduce the notion of neighbours and stage-neighbours in the graph structure as follows; for a subset $A \subset \mathcal{K}$ the neighbour set of $A$ is given by $\mathcal{N}(A) = \{j \in \mathcal{K}/A: i \rightsquigarrow j ; i \in A \}$. These are the first neighbours, or stage-1 neighbours of $A$. The $r$th stage neighbours of a node $i \in \mathcal{K}$ are given by $\mathcal{N}^{(r)}(i) = \mathcal{N} \{ \mathcal{N}^{(r-1)}(i) \} / [ \{\cup_{q=1}^{r-1} \mathcal{N}^{(q)}(i)\} \cup \{i\}] $, for $r=2,3,...$ and $\mathcal{N}^{(1)}(i) = \mathcal{N}(\{i\})$.
Figure~\ref{jss3594-pltnet} shows an example graph, where node E has stage-1 neighbour A, stage-2 neighbour D, and stage-3 neighbours B and C. Neighbour sets for this example include $\mathcal{N}^{(1)}(D)= \{A, B, C\}$, and $\mathcal{N}^{(3)}(E)= \{B, C \}$. In the time-varying network setting, a subscript $t$ is added to the neighbour set notation.
\subsubsection{Connection weights} \label{connectionweights}
Each network can have connection weights $\omega \in [0,1]$ associated with every pair of nodes.
This connection weight can depend on the size of the neighbour set and also encodes any edge-weight information.
Formally, the values of the connection weights from a node $i \in \mathcal{K}$ to its stage-$r$ neighbour $j \in \mathcal{N}^{(r)}(i)$ will be the reciprocal of the number of stage-$r$ neighbours; $\omega_{i,j} = |\mathcal{N}^{(r)}(i) |^{-1} $, where $|\cdot |$ denotes the cardinality of a set. In Figure~\ref{jss3594-pltnet} the connection weights would be, for example, $\omega_{E,A} = 1$, $\omega_{A, E}=\omega_{A,D}=0.5$.
Connection weights are not necessarily symmetric, even for an un-directed graph.
We note that this choice of these inverse distance weights is one of many possibilities,
and some other
means of creating connection weights could be used.
When the edges are weighted, or have a distance associated with them, we use the concept of distance to find the shortest path between two vertices. Let the distance from node $i$ to $\ell$ be denoted $d_{i,\ell} \in \mathbb{R}_{+}$, and if there is an un-normalised weight between these nodes, denote this $\mu_{i,\ell}\in \mathbb{R}_{+}$. To find the length of connection between a node $i$ and its stage-$r$ neighbour, $k$, we sum the distances on the paths with $r$ edges from $i$ to $k$ and take the minimum (note that there are no paths with fewer edges than $r$ as $k$ is a stage-$r$ neighbour). If the network includes weights rather than distances, we find the shortest $r$ length path where $d_{i,\ell}=\mu_{i,\ell}^{-1}$. Then the connection weights between node $i$ and its stage-$r$ neighbour $k$ are either $\omega_{i,k} = d_{i,k}^{-1} \{ \sum_{\ell \in \mathcal{N}^{(r)}(i)} d_{i,\ell}^{-1} \}^{-1}$ for distances, or $\omega_{i,k} = \mu_{i,k} \{ \sum_{\ell \in \mathcal{N}^{(r)}(i)} \mu_{i,\ell} \}^{-1}$ for a network with weights. This definition means that all nodes will have connection weights that sum to one for any non-empty neighbour set, whether they are in a sparse or dense part of the graph.
\subsubsection{Edge or node covariates}
A further important innovation permits
a covariate that can be used to encode edges effects (or nodes) into certain types.
Our covariate will take $C \in \mathbb{N}$ discrete values and be indexed by $c$. A more general covariate could be
considered, but we wish to keep our notation simple in the definition that follows. For example,
in an epidemiological network we might have two edge types:
one that carries information about windborne spread of
infection and the other carries information about identified direct infections.
The covariates do not change our neighbour sets or connection weight definitions,
so we have the property $\sum\limits_{q \in \mathcal{N}^{(r)}(i)} \sum\limits_{c=1}^C \omega_{i,q,c} = 1$ for all $i \in \mathcal{K}$ and $r \in \mathbb{N}$ such that $\mathcal{N}^{(r)}(i)$ is non-empty.
\subsection{The generalised network autoregressive model}\label{NARmod}
Consider an $N\times 1$ vector of nodal time series, $\mathbf{X}_t=(X_{1,t},\hdots,X_{N,t})'$, where $N$ is considered fixed. Our aim is to model the dependence structure within and between the nodal series using the network structure provided by (potentially time-varying) connection weights, $\omega$.
For each node $i \in \{1,\hdots,N\}$ and time $t\in \{1,\hdots,T\}$, our generalised autoregressive model of order $(p,[\mathbf{s}])\in \mathbb{N}\times\mathbb{N}_0^p$
for $\mathbf{X}_t$ is
\begin{equation} \label{NAR1}
X_{i,t} = \sum_{j=1}^p \left( \alpha_{i,j} X_{i,t-j} +\sum_{c=1}^C \sum_{r=1}^{s_j} \beta_{j,r, c}\sum_{q \in \mathcal{N}^{(r)}_t(i)} \omega_{i,q,c}^{(t)} X_{q,t-j} \right) + u_{i,t},
\end{equation}
where $p\in\mathbb{N}$ is the maximum time lag, $[\mathbf{s}]=(s_1,\hdots,s_p)$ and $s_j \in \mathbb{N}_0$ is the maximum stage of neighbour dependence for time lag $j$, with $\mathbb{N}_0 = \mathbb{N}\cup \{0\}$,
$\mathcal{N}^{(r)}_t(i)$ is the $r$th stage neighbour set of node $i$ at time $t$, $\omega_{i,q,c}^{(t)} \in [0,1]$ is the connection weight between node $i$ and node $q$ at time $t$ if the path corresponds
to covariate $c$. Here, we consider a sum from one to zero to be zero, i.e., $\sum_{r=1}^0 ( \cdot ) \coloneqq 0$.
The $\alpha_{i, j} \in \mathbb{R}$ are `standard' autoregressive parameters at lag $j$ for node $i$.
The $\beta_{j, r, c} \in \mathbb{R}$ correspond to the effect of the $r$th stage neighbours, at lag $j$, according to covariate
$c= 1, \ldots, C$. Later, we derive conditions on the model parameters to achieve process stationarity over the network.
Here the noise, $\{u_{i,t}\}$, is assumed to be independent and identically distributed at each node $i$, with mean zero and variance $\sigma_i^2$.
Our model meaningfully enhances that of the arXiv publication
\cite{Knight2016} by now additionally including
different autoregressive parameters,
connection weights at each node
and, particularly, parameters $\beta$ that depend on covariates. Note that the IID assumption on the noise $\{u_{i,t}\}$ could of course be relaxed to include correlated innovations.
We note that crucially, the time-dependent network topology is integral to the model parametrisation through the use of time-varying weights and neighbours. These features yield a model that is sensitive to the network structures and captures contemporaneous as well as autoregressive relationships, as defined by equation~\eqref{NAR1}. The network should therefore be viewed not as an estimable quantity, but as a time-dependent known structure.
In the GNAR model, the network may change over time, but the covariates stay fixed. This means that the underlying network can be altered over time,
for example, to allow for nodes to drop in and out of the series but model fitting can still be carried out. Practically, this is extremely useful, as shown by the example in Section~\ref{gdp}.
Our model allows for the $\alpha$ parameters may be different at each node,
however the interpretation of the network regression parameters, $\beta_{j, r, c}$, is the same throughout the network.
A more restrictive version of the above model is the global-$\alpha$ GNAR$(p, [\textbf{s}])$ model, which has the same autoregressive covariate at each node,
where the $\alpha_{i,j}$ are replaced by $\alpha_j$.
This defines a process with the same behaviour at every node, with differences being present only due to the graph structure.
\subsection{GNAR network example}
Networks in the \pkg{GNAR} package are stored in a list with two components
\code{edges} and \code{dist}. The \code{edges} component is itself a list with $N$ slots
each containing a vector whose entries are indices to their neighbouring nodes. For example,
if $3 \leftrightsquigarrow 4$ denotes an undirected edge between nodes $3$ and $4$ then
the vector \code{edges[[3]]} will contain a \code{4} and \code{edges[[4]]} will contain a \code{3}.
If the network is undirected this will mean that each edge is `double counted' in summary information. A directed edge $3 \rightsquigarrow 4$ would be listed in
\code{edges[[3]]} as a \code{4}, but not \code{edges[[4]]} if there is no edge in the opposite direction.
The \code{dist} component is of the same format as \code{edges}, and contains the distances corresponding to the edge links, if they exist.
For example, in an un-weighted setting, the connection weights are such that all neighbours of a node have equal effect on the node. This is achieved by setting all entries of the \code{dist} component to one, and the software calculates the connection weight from these.
A \pkg{GNAR} network is stored in a \code{GNARnet} object, and an object
can be checked using the \code{is.GNARnet} function.
The S3 methods \code{plot}, \code{print}, and \code{summary} are available for \code{GNARnet} objects.
Figure~\ref{jss3594-pltnet} shows an example that is stored as a \code{GNARnet}
object called \code{fiveNet} and can be reproduced using
\begin{Schunk}
\begin{Sinput}
R> library("GNAR")
R> library("igraph")
R> plot(fiveNet, vertex.label = c("A", "B", "C", "D", "E"))
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-pltnet}{\textwidth}{An example un-directed, un-weighted graph with five nodes labelled A to E.}
The basic structure of the \code{GNARnet} object is, as usual, displayed with
\begin{Schunk}
\begin{Sinput}
R> summary(fiveNet)
\end{Sinput}
\begin{Soutput}
GNARnet with 5 nodes and 10 edges
of equal length 1
\end{Soutput}
\end{Schunk}
\subsubsection{Converting a network to GNARnet form} \label{GNARnet}
Our \code{GNARnet} format integrates with other methods of specifying a network via a set of functions that generate a \code{GNARnet} from others, such as an
\code{igraph} object.
An \code{igraph} object can be converted to and from the \code{GNARnet} structure using the functions \code{igraphtoGNAR} and \code{GNARtoigraph}, respectively.
For example, starting with the \code{fiveNet} \code{GNARnet} object,
\begin{Schunk}
\begin{Sinput}
R> fiveNet2 <- GNARtoigraph(net = fiveNet)
R> summary(fiveNet2)
\end{Sinput}
\begin{Soutput}
IGRAPH 41ddef8 U-W- 5 5 --
+ attr: weight (e/n)
\end{Soutput}
\begin{Sinput}
R> fiveNet3 <- igraphtoGNAR(fiveNet2)
R> all.equal(fiveNet, fiveNet3)
\end{Sinput}
\begin{Soutput}
[1] TRUE
\end{Soutput}
\end{Schunk}
whereas the reverse conversion would be performed as
\begin{Schunk}
\begin{Sinput}
R> g <- make_ring(10)
R> print(igraphtoGNAR(g))
\end{Sinput}
\begin{Soutput}
GNARnet with 10 nodes
edges:1--2 1--10 2--1 2--3 3--2 3--4 4--3 4--5 5--4 5--6
6--5 6--7 7--6 7--8 8--7 8--9 9--8 9--10 10--1 10--9
edges of each of length 1
\end{Soutput}
\end{Schunk}
We can also use the \code{GNARtoigraph} function to extract graphs
involving higher-order neighbour structures,
for example, creating a network of third-order neighbours.
In addition to interfacing with \code{igraph}, we can convert between \code{GNARnet} objects and adjacency matrices using functions \code{as.matrix} and
\code{matrixtoGNAR}. We can produce an adjacency matrix for the \code{fiveNet} object with
\begin{Schunk}
\begin{Sinput}
R> as.matrix(fiveNet)
\end{Sinput}
\begin{Soutput}
[,1] [,2] [,3] [,4] [,5]
[1,] 0 0 0 1 1
[2,] 0 0 1 1 0
[3,] 0 1 0 1 0
[4,] 1 1 1 0 0
[5,] 1 0 0 0 0
\end{Soutput}
\end{Schunk}
and an example converting a weighted adjacency matrix to a \code{GNARnet} object is
\begin{Schunk}
\begin{Sinput}
R> adj <- matrix(runif(9), ncol = 3, nrow = 3)
R> adj[adj < 0.3] <- 0
R> print(matrixtoGNAR(adj))
\end{Sinput}
\begin{Soutput}
GNARnet with 3 nodes
edges:1--1 1--3 2--2 3--1 3--2
edges of unequal lengths
\end{Soutput}
\end{Schunk}
\subsection{Example: GNAR model fitting}\label{exsim}
The \code{fiveNet} network has a simulated multivariate time series associated
with it of class \code{ts} called \code{fiveVTS}.
The pair together are a network time series. The object can be loaded in the usual way using the \code{data} function.
\pkg{GNAR} contains functions for fitting and predicting from GNAR models: \code{GNARfit} and the \code{predict} method, respectively. These make use of the familiar \proglang{R} command \code{lm}, since the GNAR model can be essentially re-formulated as a linear model, as we shall see in Section~\ref{Implementation} and Appendix~\ref{consistency}. As such, least squares variance / standard error computations are also readily obtained, although other, e.g., HAC-type variance estimators could also be considered for GNAR models.
Suppose we wish to fit the global-$\alpha$ network
time series model GNAR$(2, [1, 1])$, a model with four parameters in total.
We can fit this model with the following code.
\begin{Schunk}
\begin{Sinput}
R> data("fiveNode")
R> answer <- GNARfit(vts = fiveVTS, net = fiveNet, alphaOrder = 2,
+ betaOrder = c(1, 1))
R> answer
\end{Sinput}
\begin{Soutput}
Model:
GNAR(2,[1,1])
Call:
lm(formula = yvec ~ dmat + 0)
Coefficients:
dmatalpha1 dmatbeta1.1 dmatalpha2 dmatbeta2.1
0.20624 0.50277 0.02124 -0.09523
\end{Soutput}
\end{Schunk}
In this fit, the global autoregressive parameters are $\hat{\alpha}_1 \approx 0.206$ and
$\hat{\alpha}_2 \approx 0.021$ and the $\beta$ network parameters are
$\hat{\beta}_{1,1, 1} \approx 0.503$ and $\hat{\beta}_{2,1, 1} \approx -0.095$. Also,
the network edges only have one type of covariate so $C = c=1$.
We can just look at one node. For example, the model at node~A is
\begin{displaymath}
X_{A,t} = 0.206X_{A,t-1} + 0.503 (X_{E,t-1} + X_{D, t-1})/2
+ 0.021X_{A, t-2} - 0.095 (X_{E,t-2} + X_{D, t-2})/2 + u_{E,t}.
\end{displaymath}
The model coefficients can be extracted from a \code{GNARfit} object
using the \code{coef} function as is customary.
The \code{GNARfit} object returned by \code{GNARfit} function also has methods to extract fitted values and the residuals.
For example, Figure~\ref{jss3594-gnarfvp} shows the first node time series and the
residuals from fitting the model. Figure~\ref{jss3594-gnarfvp} was produced by
\begin{Schunk}
\begin{Sinput}
R> plot(fiveVTS[, 1], ylab = "Node A Time Series")
R> lines(fitted(answer)[, 1], col = 2)
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-gnarfvp}{\textwidth}{Time series of first node (black) with fitted values from `answer' model overlaid in red.}
Alternatively, we can examine the associated residuals:
\begin{Schunk}
\begin{Sinput}
R> myresiduals <- residuals(answer)[, 1]
R> layout(matrix(c(1, 2), 2, 1))
R> plot(ts(residuals(answer)[, 1]), ylab = "`answer' model residuals")
R> hist(residuals(answer)[, 1], main = "",
+ xlab = "`answer' model residuals")
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-gnarres}{\textwidth}{Residual plots from `answer' model fit. Top:
Time series; Bottom: Histogram.}
By altering the input parameters in the \code{GNARfit} function, we can fit a range of different GNAR models and the reader can consult Appendix~\ref{extrasims} for further examples.
\subsection{Example: GNAR data simulation on a given network}
\label{exsimexample}
The following example demonstrates network time series simulation using the network in Figure~\ref{jss3594-pltnet}.
Model \code{(a)} is a GNAR$(1,[1])$ model with individual $\alpha$ parameters, $(\alpha_{A,1}, \alpha_{B,1}, \alpha_{C,1}, \alpha_{D,1}, \alpha_{E,1})=(0.4, 0, -0.6, 0, 0)$, and the same $\beta$ parameter throughout, $\beta_1 = 0.3$. Model \code{(b)} is a
\mbox{global-$\alpha$} GNAR$(2,[2,0])$ model with parameters $\alpha_1=0.2$, $\beta_{1,1}=0.2$, $\beta_{1,2}=0.3$ and $\alpha_2=0.3$. Both simulations are created using standard
normal noise whose standard deviation is controlled using the \code{sigma} argument.
\begin{Schunk}
\begin{Sinput}
R> set.seed(10)
R> fiveVTS2 <- GNARsim(n = 200, net = fiveNet,
+ alphaParams = list(c(0.4, 0, -0.6, 0, 0)), betaParams = list(c(0.3)))
\end{Sinput}
\end{Schunk}
By fitting an individual-alpha GNAR$(1,[1])$ model to the simulated data with the \code{fiveNet} network, we
can see that these estimated parameters are similar to the specified ones of 0.4, 0, -0.6, 0, 0 and 0.3. This agreement does not come as a surprise given that we show theoretical consistency for parameter estimators (see Appendix~\ref{consistency}).
\begin{Schunk}
\begin{Sinput}
R> print(GNARfit(vts = fiveVTS2, net = fiveNet, alphaOrder = 1,
+ betaOrder = 1, globalalpha = FALSE))
\end{Sinput}
\begin{Soutput}
Model:
GNAR(1,[1])
Call:
lm(formula = yvec ~ dmat + 0)
Coefficients:
dmatalpha1node1 dmatalpha1node2 dmatalpha1node3 dmatalpha1node4
0.45902 0.13133 -0.49166 0.03828
dmatalpha1node5 dmatbeta1.1
0.02249 0.24848
\end{Soutput}
\end{Schunk}
Repeating the experiment for the GNAR(2, [2, 0]) Model \code{(b)}, the estimated parameters are again similar
to the generating parameters:
\begin{Schunk}
\begin{Sinput}
R> set.seed(10)
R> fiveVTS3 <- GNARsim(n = 200, net = fiveNet,
+ alphaParams = list(rep(0.2, 5), rep(0.3, 5)),
+ betaParams = list(c(0.2, 0.3), c(0)))
R> print(GNARfit(vts = fiveVTS3, net = fiveNet, alphaOrder = 2,
+ betaOrder = c(2,0)))
\end{Sinput}
\begin{Soutput}
Model:
GNAR(2,[2,0])
Call:
lm(formula = yvec ~ dmat + 0)
Coefficients:
dmatalpha1 dmatbeta1.1 dmatbeta1.2 dmatalpha2
0.2537 0.1049 0.3146 0.2907
\end{Soutput}
\end{Schunk}
Alternatively, we can use the \code{simulate} S3 method for \code{GNARfit} objects to simulate time series associated to a GNAR model, for example
\begin{Schunk}
\begin{Sinput}
R> fiveVTS4 <- simulate(GNARfit(vts = fiveVTS2, net = fiveNet,
+ alphaOrder = 1, betaOrder = 1, globalalpha = FALSE), n = 200)
\end{Sinput}
\end{Schunk}
\subsection{Missing data and changing connection weights with GNAR models} \label{changing}
Standard multivariate time series models, including vector autoregressions (VAR), can have significant problems
in coping with certain types of missingness and imputation is often used,
see \cite{Guerrero10}, \cite{Honaker10}, \cite{Bashir16}. While in VAR modelling successful solutions have been found to cope with specific missingness scenarios, such as implemented in the \pkg{gimme} \proglang{R} package \citep{lane19:gimme}, however, if a variable has e.g., block
missing data, the coefficients corresponding that variable can be difficult to calculate, and
impossible if their partner variable is missing at cognate times. In addition, due to computational burden \pkg{gimme} is limited to modelling a single time lag.
A key advantage
of our parsimonious GNAR model is that it models via neighbourhoods across the entire data set.
If a node is missing for a given time, then it does not contribute to the estimation of neighbourhood
parameters that the network structure suggests it should, and there are plenty of other
nodes that do contribute, generally resulting in a high number of observations to estimate
each coefficient. In GNAR models, missing data of this kind is not a problem.
The flexibility of GNAR modelling means that we can also model missing data as a changing network, or alternatively, as changing connection weights. In the situation where the overall network is considered fixed, but when observations are missing at particular nodes, the connections and weightings need altering accordingly. Again, using the graph in Figure~\ref{jss3594-pltnet}, consider the situation where node~A does not have any data recorded. Yet, we want to preserve the stage-2 connection between D and E, and the stage-3 connection between E and both B and C.
To do this, we do not redraw the graph and remove node A and its connections, instead we reweight the connections that depend on node~A. As node~A does not feature in the stage-2 or stage-3 neighbours of E, the connection weights $\omega_{E,D}, \omega_{E,B}, \omega_{E, C}$ do not change, but the connection weight $\omega_{E,A}$ drops to zero in the absence of observation from node~A. Similarly, the stage-1 neighbours of D are changed without A, so $\omega_{D,A}$ drops to zero and the other two connection weights from node~D increase accordingly; $\omega_{D,B}=\omega_{D,C} = 0.5$.
Missing data of this kind is handled automatically by the \code{GNAR} functions using
customary \code{NA} missing data values present in the \code{vts} (vector time series)
component of the overall network time series. For example, inducing some (artificial) missingness in the
\code{fiveVTS} series, we can still obtain estimates of model parameters:
\begin{Schunk}
\begin{Sinput}
R> fiveVTS0 <- fiveVTS
R> fiveVTS0[50:150, 3] <- NA
R> nafit <- GNARfit(vts = fiveVTS0, net = fiveNet, alphaOrder = 2,
+ betaOrder = c(1, 1))
R> layout(matrix(c(1, 2), 2, 1))
R> plot(ts(fitted(nafit)[, 3]), ylab = "Node C fitted values")
R> plot(ts(fitted(nafit)[, 4]), ylab = "Node D fitted values")
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-NAs}{\textwidth}{Fitted values of \mbox{global-$\alpha$} GNAR$(1,[1])$ fit to the `fiveVTS' data, with observations 50--150 removed from node C.
Fitted values: Top: Node~C; Bottom: Node~D.}
As shown in Figure~\ref{jss3594-NAs}, after removing observations from the time series at
node~C, its neighbour, node~D, still has a complete set of fitted values.
\subsection{Stationarity conditions for a GNAR process with fixed network}
\begin{theorem} \label{thstatcon}
Given an unchanging network, $\mathcal{G}$, a sufficient condition for the GNAR model (\ref{NAR1}) to be stationary is
\begin{equation}\label{statcon}
\sum_{j=1}^p \left( |\alpha_{i,j}| + \sum\limits_{c=1}^C \sum\limits_{r=1}^{s_j}|\beta_{j,r,c}| \right)<1 \quad \forall i \in 1,...,N.
\end{equation}
\end{theorem}
\noindent The proof of Theorem~\ref{thstatcon} can be found in Appendix~\ref{appstatcon}.
For the global-$\alpha$ model this condition reduces to
\begin{equation}
{\sum_{j=1}^p \left( |\alpha_{j}| + \sum\limits_{c=1}^C \sum\limits_{r=1}^{s_j}|\beta_{j,r,c}| \right)<1 }.
\end{equation}
We can explore these conditions using the~\code{GNARsim} function.
The following example uses parameters whose absolute value sums to greater than one and
then we calculate the mean over successive time periods. The mean increases rapidly
indicating nonstationarity.
\begin{Schunk}
\begin{Sinput}
R> set.seed(10)
R> fiveVTS4 <- GNARsim(n = 200, net = fiveNet,
+ alphaParams = list(rep(0.2, 5)), betaParams = list(c(0.85)))
R> c(mean(fiveVTS4[1:50, ]), mean(fiveVTS4[51:100, ]),
+ mean(fiveVTS4[101:150, ]), mean(fiveVTS4[151:200, ]))
\end{Sinput}
\begin{Soutput}
[1] -120.511 -1370.216 -15725.884 -180319.140
\end{Soutput}
\end{Schunk}
\subsection{Benefits of our model and comparisons to others}
Conditioned on a given network fixed in time and with a known (time-dependent) weight- and neighbourhood structure, the
GNAR model can be mathematically formulated as a specific restricted VAR model, where the restrictions are imposed by the network and thus impact model parametrisation,
as mathematically encoded by equation~\eqref{NAR1}. This is explored in more depth in Appendix~\ref{consistency} and contrasts with a VAR model where any restrictions
can only be imposed on the parameters themselves.
An unrestricted VAR model with dimension $n$ has $\mathcal{O}(n^2)$ parameters, whereas a GNAR model with known network (usually) has $\mathcal{O}(n)$ parameters, and a global-$\alpha$ GNAR model can have $\mathcal{O}(1)$ parameters.
The large, and rapidly increasing, number of parameters in VAR often make it a challenging
model to fit and non-problem-specific mathematical constraints are often used to mitigate those
challenges. Further, the large number of VAR parameters usually mean that it fits multivariate
time series well, but then performs poorly in out-of-sample prediction.
An example of this is shown in Section~\ref{gdp}.
Our model has similarities with the network autoregression
introduced by \cite{Zhu2017}, motivated by social networks
In our notation, the \cite{Zhu2017} model can be written as a special case as
\begin{equation}
X_{i,t} = \beta_0 + Z_i^\top\gamma + \sum_{j=1}^p \left( \alpha_j X_{i,t-j} + \beta_j \sum_{q \in \mathcal{N}^{(1)}(i)} \omega_i X_{q,t-j} \right) + u_{i,t},
\end{equation}
where $\beta_0$ is a global intercept term, $Z_i$ is a vector of node-specific covariates with corresponding parameters $\gamma$, $\omega_i$ is the reciprocal of the out-degree of node $i$, and the innovations are independent and identically distributed, with zero mean, such that $\operatorname{var}(u_{i,t})=\sigma^2$. Hence, the \cite{Zhu2017} model without intercept and node-specific covariates is a special case of our GNAR model, with $\max\limits_{j \in \{1,...,p\}} s_j = 1$, i.e., dependencies limited to stage-1 immediate neighbours, and un-weighted edges.
Our model is designed to deal with a time-varying network, and our $\beta_{j, r, c}$ parameters can include general edge-based covariate information.
A further important advantage is that our GNAR model in Section~\ref{NARmod} can express dependence on stage-$r$ neighbour sets for any $r$
An earlier model with similarities to the generic network autoregression
is the Dynamic Bayesian Network (DBN) model considered in \cite{Spencer2015}. Their model can be written as
\begin{equation}
X_{i,t} = \beta_{0,i} + \sum\limits_{q \in \mathcal{N}^{(1)}(i)} \beta_{i,q} X_{q,t-1} + u_{i,t},
\end{equation}
where $\beta_{0,i}$ is a node-specific intercept term, the other $\beta$ parameters describe the network autoregression, and $u_{i,t} \sim N(0, \sigma_i^2)$.
The DBN model is also a constrained VAR model, but with no univariate autoregression terms,
and the network autoregression only includes the stage-1 neighbours.
Unlike our model and the \cite{Zhu2017} model, there are no restrictions on the parameters other than parameters only being present when there is an edge between two nodes. The \cite{Spencer2015} framework does not allow for a range of networks, as their underlying network is assumed to be a Directed Acyclic Graph. With these assumptions, the network and parameters are inferred by considering potential predictors for each node in turn. A key difference between our model and the \cite{Spencer2015} model is that we assume that the behaviour of connected nodes is the same throughout the network, whereas the DBN model allows for different $\beta$ parameters for different connections, including allowing a change of sign.
The benefits of the GNAR model compared to these, and other models, include the ability to deal with a time-changing network, missing observations,
and using network information to reduce the number of parameters.
As detailed in Section~\ref{changing}, we can incorporate missing data information with the GNAR model by allowing the connection weights to change. Allowing for a changing network structure enables us to model new nodes being added to the system, or connections between nodes changing over time. Adding autoregressive parameters to neighbours with stage greater than one results in our model being able to capture more network relationships than just those of immediate neighbours.
\section{Estimation} \label{Implementation}
In modelling terms, our GNAR model is a linear model and we employ standard techniques such as least squares estimation to fit them and to provide
statistically consistent estimators, as verified in Appendix~\ref{consistency}. An important practical consideration for fitting GNAR models is the choice of
model order. Specifically, how do we select $p$ and $\mathbf{s}$?
\subsection{Order selection} \label{BIC}
We use the Bayesian information criterion (BIC) proposed by~\cite{Schwarz1978}
to select the GNAR model order.
Under the assumption of a constant network, and that the innovations are independent
and identically distributed white noise with bounded fourth moments, this criterion is consistent,
as shown in~\cite{Lutkepohl2005}. The BIC allows us to select both the lag and neighbourhood orders simultaneously by selecting the model with smallest BIC from a set of candidates.
For a general candidate GNAR$(p, [\mathbf{s}])$ model with $N$ nodes,
the BIC is given by
\begin{equation} \label{BICeq}
\operatorname{BIC}(p,\mathbf{s}) = \ln | \hat{\varSigma}_{p,\mathbf{s}} | + T^{-1} M\ln(T) ,
\end{equation}
where ${\hat{\varSigma}_{p,\mathbf{s}} = T^{-1} \hat{\mathit{U}}'\hat{\mathit{U}}}$, $\hat{\mathit{U}}$ is the residual matrix from the
NAR$(p, [\mathbf{s}])$ fit, and $M$ is the number of parameters. In the general case $M=Np +C \sum_{j=1}^p s_j$, and in the global-$\alpha$ model $M=p+C\sum_{j=1}^p s_j$. The covariance matrix estimate, $\hat{\varSigma}_{p,\mathbf{s}}$, is also the maximum likelihood estimator of the innovation covariance matrix under the assumption of Gaussian innovations.
\pkg{GNAR} enables us to easily compute the BIC for any model by using
the \code{BIC} method for \code{GNARfit} objects. For example, on the default model fitted by
\code{GNARfit}, and an alternative model that additionally includes second-order
neighbours at the first lag into the model, we can compare their BICs by
\begin{Schunk}
\begin{Sinput}
R> BIC(GNARfit())
\end{Sinput}
\begin{Soutput}
[1] -0.003953124
\end{Soutput}
\begin{Sinput}
R> BIC(GNARfit(betaOrder = c(2, 1)))
\end{Sinput}
\begin{Soutput}
[1] 0.02251406
\end{Soutput}
\end{Schunk}
Whilst we focus on the BIC for model selection for the remainder of this article, the \pkg{GNAR} package also
include functionality for the Akaike information criterion (AIC) proposed by \cite{akaike73:information} as
\begin{equation} \label{AICeq}
\operatorname{AIC}(p,\mathbf{s}) = \ln | \hat{\varSigma}_{p,\mathbf{s}} | + 2 T^{-1} M,
\end{equation}
where ${\hat{\varSigma}_{p,\mathbf{s}}}$ is as defined in equation~\eqref{BICeq} and $M$ is again the number of
model parameters. Similar to above, the AIC can be obtained by using the code
\begin{Schunk}
\begin{Sinput}
R> AIC(GNARfit())
\end{Sinput}
\begin{Soutput}
[1] -0.06991947
\end{Soutput}
\begin{Sinput}
R> AIC(GNARfit(betaOrder = c(2, 1)))
\end{Sinput}
\begin{Soutput}
[1] -0.05994387
\end{Soutput}
\end{Schunk}
Similar to the BIC, the model with the lowest AIC is preferred. Note that the likelihood of the data associated to the model fit can also be obtained using e.g., \code{logLik(GNARfit())}.
Various models can be tried to obtain a good fit whilst, naturally, attending to
the usual aspects of good model fitting, such as residual checks. A thorough simulation study that displays the numerical performance of our proposed method appears in Section 4.5 of \cite{LeemingPhD}.
\subsection{Model selection on a wind network time series}
\label{wind}
\pkg{GNAR} incorporates the data suite \code{vswind} that contains a number of
\proglang{R} objects pertaining to 721 wind speeds taken at each of 102 weather
stations in England and Wales. The suite contains the vector time
series \code{vswindts}, the associated network \code{vswindnet},
a character vector of the weather station location names in \code{vswindnames}
and coordinates of the stations in the two column matrix \code{vswindcoords}.
The data originate from the UK Met Office site
\url{http://wow.metoffice.gov.uk} and full details can be found in
the \code{vswind}
help file in the \pkg{GNAR} package.
Figure~\ref{jss3594-windnetplot} shows a picture of the meteorological station network with distances
created by
\begin{Schunk}
\begin{Sinput}
R> oldpar <- par(cex = 0.75)
R> windnetplot()
R> par(oldpar)
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-windnetplot}{\textwidth}{Plot of the wind speed network. Blue numbers
are relative distances between sites; labels are the site name.}
We investigate fitting a network time series model. We first fit a simple GNAR$(1, [0])$ model using
a single $\alpha$, followed by an equivalent model with potentially individually distinct $\alpha$s
\begin{Schunk}
\begin{Sinput}
R> BIC(GNARfit(vts = vswindts, net = vswindnet, alphaOrder = 1,
+ betaOrder = 0))
\end{Sinput}
\begin{Soutput}
[1] -233.3848
\end{Soutput}
\begin{Sinput}
R> BIC(GNARfit(vts = vswindts, net = vswindnet, alphaOrder = 1,
+ betaOrder = 0, globalalpha = FALSE))
\end{Sinput}
\begin{Soutput}
[1] -233.1697
\end{Soutput}
\end{Schunk}
Interestingly, the model with the single $\alpha$ gives the better fit, as judged by BIC.
The single $\alpha$ model with \code{alphaOrder = 2} and \code{betaOrder = c(0, 0)} gives a lower BIC of
$-243$, so we investigate this next. Note that this model also gives the lowest AIC score. In particular, we explore a set of GNAR$(2, [b1, b2])$ models
with $b1$, $b2$ ranging from zero to 14 using the following code:
\begin{Schunk}
\begin{Sinput}
R> BIC.Alpha2.Beta <- matrix(0, ncol = 15, nrow = 15)
R> for(b1 in 0:14)
+ for(b2 in 0:14)
+ BIC.Alpha2.Beta[b1 + 1, b2 + 1] <- BIC(GNARfit(vts = vswindts,
+ net = vswindnet, alphaOrder = 2, betaOrder = c(b1, b2)))
R> contour(0:14, 0:14, log(251 + BIC.Alpha2.Beta),
+ xlab = "Lag 1 Neighbour Order", ylab = "Lag 2 Neighbour Order")
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-windcontour}{\textwidth}{Contour plot of BIC values for the two-lag autoregressive model
incorporating \mbox{$b1$-stage} and \mbox{$b2$-stage} neighbours at time lags one and two.
Values shown are $\log(251 + \operatorname{BIC})$ to display clearer contours.}
The results of the BIC evaluation for incorporating different and deeper neighbour sets, at lags one
and two, are
shown in the contour plot in Figure~\ref{jss3594-windcontour}. The minimum value of the BIC occurs in the bottom-right
part of the plot, where it seems incorporating five or sixth-stage neighbours for the first time lag
is sufficient to achieve the minimum BIC, and incorporating further lag one stages does not reduce the BIC. Moreover,
increasing the lag two neighbour sets beyond first stage neighbours would appear to increase the BIC for those
lag one neighbour stages greater than five (the horizontal contour at $0$ in the bottom right hand corner of
the plot). A fit of a possible model is
\begin{Schunk}
\begin{Sinput}
R> goodmod <- GNARfit(vts = vswindts, net = vswindnet, alphaOrder = 2,
+ betaOrder = c(5, 1))
R> goodmod
\end{Sinput}
\begin{Soutput}
Model:
GNAR(2,[5,1])
Call:
lm(formula = yvec ~ dmat + 0)
Coefficients:
dmatalpha1 dmatbeta1.1 dmatbeta1.2 dmatbeta1.3 dmatbeta1.4
0.56911 0.10932 0.03680 0.02332 0.02937
dmatbeta1.5 dmatalpha2 dmatbeta2.1
0.04709 0.23424 -0.04872
\end{Soutput}
\end{Schunk}
We investigated models with \code{alphaOrder} equal to two, three, four and five, but with no
neighbours. As judged by BIC, \code{alphaOrder = 3}
gives the best model. We could extend the example above to investigate differing stages of neighbours at
time lags one, two and three. However, a more comprehensive BIC investigation would examine
all combinations of neighbour sets over a large number of time lags. This would be feasible, but computationally
intensive for a single CPU machine, but could be coarse-grain parallelized.
Further analysis would proceed with model diagnostic checking and further modelling
as necessary.
\subsection{Constructing a network to aid prediction}\label{construction}
Whilst some multivariate time series have actual, and sometimes obvious, networks associated with them, our methodology can be useful for series without a clear or
supplied network. We propose a network construction method that uses prediction error, but note here that our scope is not to estimate an underlying network, but
merely to find a structure that is useful in the task of prediction. Here, we use a prediction error measure, understood as the sum of squared differences between the
observations and the estimates: $\sum_{i=1}^N (X_{i,t} - \hat{X}_{i,t})^2$.
The \code{predict} S3 method for GNAR models takes an input \code{GNARfit} model object and from this predicts the nodal time series at the next timepoint, similar to the S3 method for the \code{Arima} class. This allows for a `ex-sample' prediction evaluation. The \code{predict} function outputs the prediction as a vector. For example, to predict the series at the last timepoint
\begin{Schunk}
\begin{Sinput}
R> prediction <- predict(GNARfit(vts = fiveVTS[1:199,], net = fiveNet,
+ alphaOrder = 2, betaOrder = c(1, 1)))
R> prediction
\end{Sinput}
\begin{Soutput}
Time Series:
Start = 1
End = 1
Frequency = 1
Series 1 Series 2 Series 3 Series 4 Series 5
1 -0.6427718 0.2060671 0.2525534 0.1228404 -0.8231921
\end{Soutput}
\end{Schunk}
For a small-dimensional multivariate series, any and all
potential un-weighted networks can be constructed and the corresponding prediction errors compared using the \code{predict} method. Next, we consider the larger data setting where it is computationally infeasible to investigate all possible networks.
Erd\H{o}s-R\'{e}nyi random graphs can be generated with $N$ nodes, and a fixed probability of including each edge between these nodes, see Chapter~11 of \cite{Grimmett2010} for further details. The probability parameter controls the overall sparsity of the graph. Many random graphs of this type can be created, and then our GNAR model can be used for within-sample prediction. The prediction error can then be used to identify networks that aid prediction. We give an example of
this process in the next section.
\section{OECD GDP: Network structure aids prediction}
\label{gdp}
We obtained the annual gross domestic product (GDP) growth rate time series for
35 countries from the OECD website\footnote{OECD (2018), Quarterly GDP (indicator). doi: 10.1787/b86d1fc8-en (Accessed on 29 January 2018)}. The series
covers the years 1961--2013, but not all countries are included from the start.
The values are annual growth rates expressed as a percentage change compared to the previous year. We differenced the time series for each country to remove
the gross trend.
We use the first $T=52$ time points and designate each of the 35 countries as nodes to investigate the potential of modelling this time series using a network. In this data set 20.8\% (379 out of 1820) of the observations were missing due to some nodes not being included from the start.
We model this by changing the network connection weights as described in
Section~\ref{changing}. In this example, we do not use covariate information, so $C=1$.
The pattern of missing data along with the time series values is shown
graphically in Figure~\ref{jss3594-gdpheat}, produced by the following code.
\begin{Schunk}
\begin{Sinput}
R> library("fields")
R> layout(matrix(c(1, 2), nrow = 1, ncol = 2), widths = c(4.5, 1))
R> image(t(apply(gdpVTS, 1, rev)), xaxt = "n", yaxt = "n",
+ col = gray.colors(14), xlab = "Year", ylab = "Country")
R> axis(side = 1, at = seq(from = 0, to = 1, length = 52), labels = FALSE,
+ col.ticks = "grey")
R> axis(side = 1, at = seq(from = 0, to = 1, length = 52)[5*(1:11)],
+ labels = (1:52)[5*(1:11)])
R> axis(side = 2, at = seq(from = 1, to = 0, length = 35),
+ labels = colnames(gdpVTS), las = 1, cex = 0.8)
R> layout(matrix(1))
R> image.plot(zlim = range(gdpVTS, na.rm = TRUE), legend.only = TRUE,
+ col = gray.colors(14))
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-gdpheat}{\textwidth}{Heat plot (greyscale) of the differenced time series, where the initial white space indicates missing time series observations.}
\subsection{Finding a network to aid prediction}
\label{sec:findnetaidp}
This section considers the case where we observe data up to $t=51$,
and then wish to predict the values for each node at $t=52$. We begin by exploring `within-sample' prediction at $t=51$, and identify a good network for prediction. We use randomly generated Erd\H{o}s-R\'{e}nyi graphs using the \pkg{GNAR} function \code{seedToNet}.
To demonstrate this, the \pkg{GNAR} package contains the \code{gdp} data and a
set of seed values, \code{seed.nos} so that the random graphs can be reproduced for use with the time series object \code{gdpVTS}
here.
\begin{Schunk}
\begin{Sinput}
R> net1 <- seedToNet(seed.no = seed.nos[1], nnodes = 35, graph.prob = 0.15)
R> net2 <- seedToNet(seed.no = seed.nos[2], nnodes = 35, graph.prob = 0.15)
R> layout(matrix(c(2, 1), 1, 2))
R> par(mar=c(0,1,0,1))
R> plot(net1, vertex.label = colnames(gdpVTS), vertex.size = 0)
R> plot(net2, vertex.label = colnames(gdpVTS), vertex.size = 0)
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-seedToNet1}{\textwidth}{Erd\H{o}s-R\'{e}nyi random graphs constructed from the first two elements of the \code{seed.nos} variable with 35 nodes and connection probability 0.15.}
Figure~\ref{jss3594-seedToNet1} shows two of these random graphs.
As well as investigating which network works best for prediction, we also need to identify the number of parameters in the GNAR model. Initial analysis of the autocorrelation function at each node indicated that a second-order autoregressive component should be sufficient, so GNAR models with orders up to $p=2$ were tested, and we included at most two neighbour sets at each time lag. The GNAR models are: GNAR$(1,[0])$, GNAR$(1,[1])$, GNAR$(2,[0,0])$, GNAR$(2,[1,0])$, GNAR$(2,[1,1])$,
GNAR$(2,[2,0])$, GNAR$(2,[2,1])$, and GNAR$(2,[2,2])$,
each fitted as individual-$\alpha$ and global-$\alpha$ GNAR models,
giving sixteen models in total.
For the GDP example, we simulate 10,000 random un-directed networks, each with connection probability 0.15, and predict using the GNAR model with the orders above. Hence, this example requires significant computation time (about 90 minutes on
a desktop PC),
so only a segment of the analysis is included in the code below. For computational reasons, we first divide through by the standard deviation at each node so that we can model the residuals as having equal variances at each node. The function \code{seedSim} outputs the sum of squared differences between the prediction and original values, and we use this as our measure of prediction accuracy.
\begin{Schunk}
\begin{Sinput}
R> gdpVTSn <- apply(gdpVTS, 2, function(x){x / sd(x[1:50], na.rm = TRUE)})
R> alphas <- c(rep(1, 2), rep(2, 6))
R> betas <- list(c(0), c(1), c(0, 0), c(1, 0), c(1, 1), c(2, 0), c(2, 1),
+ c(2, 2))
R> seedSim <- function(seedNo, modelNo, globalalpha){
+ net1 <- seedToNet(seed.no = seedNo, nnodes = 35, graph.prob = 0.15)
+ gdpPred <- predict(GNARfit(vts = gdpVTSn[1:50, ], net = net1,
+ alphaOrder = alphas[modelNo], betaOrder = betas[[modelNo]],
+ globalalpha = globalalpha))
+ return(sum((gdpPred - gdpVTSn[51, ])^2))
+ }
R> seedSim(seedNo = seed.nos[1], modelNo = 1, globalalpha = TRUE)
\end{Sinput}
\begin{Soutput}
[1] 23.36913
\end{Soutput}
\begin{Sinput}
R> seedSim(seed.nos[1], modelNo = 3, globalalpha = TRUE)
\end{Sinput}
\begin{Soutput}
[1] 11.50739
\end{Soutput}
\begin{Sinput}
R> seedSim(seed.nos[1], modelNo = 3, globalalpha = FALSE)
\end{Sinput}
\begin{Soutput}
[1] 18.96766
\end{Soutput}
\end{Schunk}
Prediction error boxplots over simulations from
all sixteen models and 10,000 random networks are shown in Figure~\ref{fig:0209boxres}
(accompanying code not shown due to significant computation time).
The global-$\alpha$ model resulted in lower prediction error in general, so we use this version of the GNAR model. For GNAR$(1,[0])$ and GNAR$(2,[0,0])$, the first and third model
in Figure~\ref{fig:0209boxres} the ``boxplots'' are short horizontal lines as the
results for each graph are identical, as no neighbour parameters are fitted.
\begin{figure}
\centering
\includegraphics[angle=270, scale=1.2]{jss3594-boxres2.pdf}
\caption{Prediction error boxplots at $t=51$ over 10,000 randomly generated networks using \code{seed.nos} and different GNAR models, where `g-$\alpha$' indicates a global-$\alpha$ GNAR model.}
\label{fig:0209boxres}
\end{figure}
As the other global-$\alpha$ models are nested within it, we select the randomly generated graph that minimises the prediction error for global-$\alpha$ GNAR$(2,[2,2])$; this turns out
to be the network generated from \code{seed.nos[921]}.
\begin{Schunk}
\begin{Sinput}
R> net921 <- seedToNet(seed.no = seed.nos[921], nnodes = 35,
+ graph.prob = 0.15)
R> layout(matrix(c(1), 1, 1))
R> plot(net921, vertex.label = colnames(gdpVTS), vertex.size = 0)
\end{Sinput}
\end{Schunk}
\myincfig{jss3594-seed921net}{\textwidth}{Randomly generated un-weighted and un-directed graph over the OECD countries that minimises the prediction error at $t=51$ using GNAR$(2,[2,2])$.}
The network generated from \code{seed.nos[921]} is plotted in Figure~\ref{jss3594-seed921net},
where all countries have at least two neighbours, with 97 edges in total.
This ``921''
network was constructed with GDP prediction in mind, so we would not necessarily
expect any interpretable structure in our found network (and presumably, there
were other networks with not too dissimilar predictive power). However,
the USA, Mexico and Canada are extremely well-connected with eight, eight
and six edges, respectively. Sweden and Chile are also well-connected,
with eight and seven edges, respectively. This might seem surprising, but,
e.g., the McKinsey Global Institute MGI Connectedness Index, see~\cite{McKinsey2016},
ranks Sweden and Chile 18th and 45th respectively out of 139 countries, and
each country is most connected within their regional bloc (Nordic and South America,
respectively).
Each of these edges, or subgraphs of the ``921''
network could be tested to find a sparser network with a similar predictive performance, but we continue with the full chosen network here.
Using this network, we can select the best GNAR order using the BIC.
\begin{Schunk}
\begin{Sinput}
R> res <- rep(NA, 8)
R> for(i in 1:8){
+ res[i] <- BIC(GNARfit(gdpVTSn[1:50, ],
+ net = seedToNet(seed.nos[921], nnodes = 35, graph.prob = 0.15),
+ alphaOrder = alphas[i], betaOrder = betas[[i]]))
+ }
R> order(res)
\end{Sinput}
\begin{Soutput}
[1] 6 3 4 7 8 5 1 2
\end{Soutput}
\begin{Sinput}
R> sort(res)
\end{Sinput}
\begin{Soutput}
[1] -64.44811 -64.32155 -64.18751 -64.12683 -64.09656 -63.86919
[7] -60.67858 -60.54207
\end{Soutput}
\end{Schunk}
The model that minimised BIC in this case was the sixth model, GNAR$(2,[2,0])$,
a model with two autoregressive parameters and network regression parameters on the first two neighbour sets at time lag one.
\subsection{Results and comparisons}
\label{sec:results}
We use the previous section's model to predict the values at $t=52$
and compare its prediction errors to those found using standard
AR and VAR models.
The GNAR predictions are found by fitting a GNAR$(2,[2,0])$ model with the chosen network (corresponding to \code{seed.nos[921]}) to data up to $t=51$, and then predicting values at $t=52$. We first normalise the series, and then compute the total squared error from the model fit.
\begin{Schunk}
\begin{Sinput}
R> gdpVTSn2 <- apply(gdpVTS, 2, function(x){x / sd(x[1:51], na.rm = TRUE)})
R> gdpFit <- GNARfit(gdpVTSn2[1:51,], net = net921, alphaOrder = 2,
+ betaOrder = c(2, 0))
R> summary(gdpFit)
\end{Sinput}
\begin{Soutput}
Call:
lm(formula = yvec2 ~ dmat2 + 0)
Residuals:
Min 1Q Median 3Q Max
-3.4806 -0.5491 -0.0121 0.5013 3.1208
Coefficients:
Estimate Std. Error t value Pr(>|t|)
dmat2alpha1 -0.41693 0.03154 -13.221 < 2e-16 ***
dmat2beta1.1 -0.12662 0.05464 -2.317 0.0206 *
dmat2beta1.2 0.28044 0.06233 4.500 7.4e-06 ***
dmat2alpha2 -0.33282 0.02548 -13.064 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8926 on 1332 degrees of freedom
(23 observations deleted due to missingness)
Multiple R-squared: 0.1859, Adjusted R-squared: 0.1834
F-statistic: 76.02 on 4 and 1332 DF, p-value: < 2.2e-16
GNAR BIC: -62.86003
\end{Soutput}
\begin{Sinput}
R> sum((predict(gdpFit) - gdpVTSn2[52, ])^2)
\end{Sinput}
\begin{Soutput}
[1] 5.737203
\end{Soutput}
\end{Schunk}
The fitted parameters of this GNAR model were $\hat{\alpha}_1 \simeq -0.42, \hat{\beta}_{1,1} \simeq -0.13, \hat{\beta}_{1,2} \simeq 0.28,$ and $ \hat{\alpha}_2 \simeq -0.33$.
We compared our methods with results from fitting an AR model individually
to each node using the \code{forecast.ar()} and \code{auto.arima()} functions from version~8.0 of the CRAN \pkg{forecast} package \citep{hyndman17:forecast},
for further details see \cite{Hyndman2008}.
Due to our autocorrelation analysis from Section~\ref{sec:findnetaidp}
we set the maximum AR order for each of the 35 individual models
to be $p=2$. Conditional on this, the actual order selected was chosen using
the BIC.
\begin{Schunk}
\begin{Sinput}
R> library("forecast")
R> arforecast <- apply(gdpVTSn2[1:51, ], 2, function(x){
+ forecast(auto.arima(x[!is.na(x)], d = 0, D = 0, max.p = 2, max.q = 0,
+ max.P = 0, max.Q = 0, stationary = TRUE, seasonal = FALSE, ic = "bic",
+ allowmean = FALSE, allowdrift = FALSE, trace = FALSE), h = 1)$mean
+ })
R> sum((arforecast - gdpVTSn2[52, ])^2)
\end{Sinput}
\begin{Soutput}
[1] 8.065491
\end{Soutput}
\end{Schunk}
Our VAR comparison was calculated using version~1.5--2 of the
CRAN package \pkg{vars}, \cite{Pfaff2008}. The missing values at the beginning of the series cannot be handled with current software, so are set to zero. The number of parameters in a zero-mean VAR($p$) model is
of order $pN^2$.
In this particular example, the dimension of the observation data matrix is
$T\times N$, with $T < 2N$, so only a first-order VAR can be fitted.
We fit the model using the \code{VAR} function and then use
the \code{restrict} function to reduce dimensionality further, by setting to zero
any coefficient whose associated absolute \mbox{$t$-statistic} value is less than two.
\begin{Schunk}
\begin{Sinput}
R> library("vars")
R> gdpVTSn2.0 <- gdpVTSn2
R> gdpVTSn2.0[is.na(gdpVTSn2.0)] <- 0
R> varforecast <- predict(restrict(VAR(gdpVTSn2.0[1:51, ], p = 1,
+ type = "none")), n.ahead = 1)
\end{Sinput}
\end{Schunk}
This results in forecast vectors for each node, so we extract the point forecast (the first element of the
forecast vectors) and compute the prediction error as follows
\begin{Schunk}
\begin{Sinput}
R> getfcst <- function(x){return(x[1])}
R> varforecastpt <- unlist(lapply(varforecast$fcst, getfcst))
R> sum((varforecastpt - gdpVTSn2.0[52, ])^2)
\end{Sinput}
\begin{Soutput}
[1] 26.19805
\end{Soutput}
\end{Schunk}
\begin{table}[]
\centering
\begin{tabular}{lrr}\hline
Model & \# Parameters & Prediction error \\ \hline
GNAR$(2,[2,0])$ & 4 & 5.7 \\
Individual AR$(2)$ & 38 & 8.1\\
VAR$(1)$ & 199 & 26.2 \\ \hline
\end{tabular}
\caption{Estimated prediction error of differenced real GDP change at $t=52$ for all 35 countries.
\label{tab:gdp}}
\end{table}
Our GNAR model gives a lower prediction error than both the AR and VAR results, reducing the error by 29\% compared to AR and by 78\% compared to VAR. Table~\ref{tab:gdp} summarises these results and also shows the number of
parameters fitted. It is clear that GNAR is particularly parsimonious.
We repeat the procedure above to perform analysis based upon two-step ahead forecasting. In this case, a different network minimises the prediction error for model GNAR(2,[2,2]). However, the BIC step identified that the GNAR(2,[0,0]) model had the best fit, which is a model that does not include network regression parameters.
\begin{Schunk}
\begin{Sinput}
R> gdpVTSn3 <- apply(gdpVTS, 2, function(x){x / sd(x[1:50], na.rm = TRUE)})
R> gdpPred <- predict(GNARfit(gdpVTSn2[1:50,], net = net921, alphaOrder = 2,
+ betaOrder = c(0, 0)), n.ahead=2)
R> sum((gdpPred[1,] - gdpVTSn3[51, ])^2)
\end{Sinput}
\begin{Soutput}
[1] 11.7874
\end{Soutput}
\begin{Sinput}
R> sum((gdpPred[2,] - gdpVTSn3[52, ])^2)
\end{Sinput}
\begin{Soutput}
[1] 8.067577
\end{Soutput}
\begin{Sinput}
R> arforecast <- apply(gdpVTSn3[1:50, ], 2, function(x){
+ forecast(auto.arima(x[!is.na(x)], d = 0, D = 0, max.p = 2, max.q = 0,
+ max.P = 0, max.Q = 0, stationary = TRUE, seasonal = FALSE, ic = "bic",
+ allowmean = FALSE, allowdrift = FALSE, trace = FALSE), h = 2)$mean
+ })
R> sum((arforecast[1,] - gdpVTSn3[51, ])^2)
\end{Sinput}
\begin{Soutput}
[1] 18.56074
\end{Soutput}
\begin{Sinput}
R> sum((arforecast[2,] - gdpVTSn3[52, ])^2)
\end{Sinput}
\begin{Soutput}
[1] 11.31722
\end{Soutput}
\begin{Sinput}
R> gdpVTSn3.0 <- gdpVTSn3
R> gdpVTSn3.0[is.na(gdpVTSn3.0)] <- 0
R> varforecast <- predict(restrict(VAR(gdpVTSn3.0[1:50, ], p = 1,
+ type = "none")), n.ahead = 2)
R> getfcst <- function(x){return(x[,1])}
R> varforecastpt <- matrix(unlist(lapply(varforecast$fcst, getfcst)),
+ nrow=2, ncol=35)
R> sum((varforecastpt[1,] - gdpVTSn3[51,])^2)
\end{Sinput}
\begin{Soutput}
[1] 114.9876
\end{Soutput}
\begin{Sinput}
R> sum((varforecastpt[2,] - gdpVTSn3[52,])^2)
\end{Sinput}
\begin{Soutput}
[1] 120.4467
\end{Soutput}
\end{Schunk}
Table~\ref{tab:gdp2steps} shows that the GNAR model is again the best performing, although in the two-step ahead prediction the fitted model is a special case of GNAR model with no neighbourhood parameters.
\begin{table}[]
\centering
\begin{tabular}{lrr}\hline
Model & Prediction error at $t=51$ & Prediction error at $t=52$ \\ \hline
GNAR$(2,[0,0])$ & 11.8 & 8.1 \\
Individual AR$(2)$ & 18.6 & 11.3\\
VAR$(1)$ & 115.0 & 120.4 \\ \hline
\end{tabular}
\caption{Estimated prediction error of differenced real GDP change at $t=51,52$, for all 35 countries.
\label{tab:gdp2steps}}
\end{table}
Results in Tables~\ref{tab:gdp} and \ref{tab:gdp2steps} indicate that the VAR model works particularly poorly here, despite using thresholding to reduce the number of parameters. This example highlights that, for a multivariate series with many observations per time point, the VAR framework is restricted by the number of parameters that have to be fitted per time lag, thus reducing the
AR-order, $p$, it can capture. In addition, we were unable to find software to fit
VAR models with for missing data at the start of a series.
We end this section by noting that using Erd\H{o}s-R\'{e}nyi graphs are not the only type of network that could be used to aid prediction. As suggested by a referee,
models such the Chung-Lu model \citep{aiello01:a, chung02:connected} could also be used to simulate random networks for this task; these graphs would allow for more
flexible network generation, for example using node-specific connection probabilities proportional to a country's size.
\section{Discussion and summary} \label{Discussion}
The \pkg{GNAR} package can be used to model network
time series using a network autoregressive structure.
Estimation under the proposed model is informed by the, potentially time-varying, structure of the network, assumed known.
Network time series models are in an early stage of development, but there is
enormous potential, especially as network data are increasingly being
collected and analysed in many fields. As far as possible, we attempt
to integrate our methods with existing valuable R functionality, such as its
linear modelling capability and the \code{fit} / \code{summary} / \code{predict} methods that are familiar
with \proglang{R} users.
Within our model a network is formed using edges of all covariates simultaneously, and the connection weights of this single network can be calculated e.g.,
as described in Section~\ref{connectionweights}. Another approach is to consider a separate network for each covariate, and then calculate connection weights for each of
these networks. This would result in different (known) weightings, $\omega$, and consequently different fitted coefficients, $\beta$. The single network
approach is more appropriate for sparse networks and when different types of edge are closely related. In comparison, when covariates relate completely separate link
information between the nodes, use of different networks would be appropriate.
When covariates are present, the neighbour set structure is more complex, as different edge types can be included in a path between nodes. For example, in a network with
\emph{event} and \emph{proximal} edges, network paths between stage-2 neighbours could include edges \emph{event-event}, \emph{event-proximal / proximal-event}, or
\emph{proximal-proximal}. These different types of path could be represented separately in the model using additional $\beta$ parameters. We note that the number of such
parameters would increase greatly for large covariate cardinality $C$ or high neighbour set stage $s_j$, so, in these cases, the large number of additional parameters may
not enhance the model.
Our model permits regression on any non-empty stage neighbour set, so models with high $s_j$ can be fitted.
For large $s_j$, the
neighbour sets may not be scientifically interpretable so small $s_j$ is recommended,
to favour parsimony and interpretability.
Trend is another factor that can seriously affect modelling and estimation, just as in
the regular time series situation. However, trend can be successfully modelled and estimated
by using second-generation wavelet (lifting) techniques before stochastic modelling,
as in \cite{Nunes2015}.
With the option of having different covariates and high order neighbourhood structures included, our GNAR model as presented in Section~\ref{Model} is incredibly flexible. In this article a sufficient condition for stationarity and consistency of the fitted parameters have been shown for the fixed network scenario. In addition, practical suggestions for order selection, and connection weights in the case of missing data have been discussed.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,708 |
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{"url":"https:\/\/www.esaral.com\/relation-between-magnetic-permeability-and-susceptibility\/","text":"Relation Between Magnetic Permeability and Susceptibility \u2013 Class 12 Physics\n\nPermeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. The magnetic susceptibility is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field. Relative permeability, sometimes denoted by the symbol \u03bcr, is the ratio of the permeability of a specific medium to the permeability of free space. Here we will study about the Relation Between Relative Magnetic Permeability and Susceptibility:\n\n### Relation Between Relative Magnetic Permeability and Susceptibility\n\nWhen a magnetic material is kept in a magnetising field (H).\n\nThen total number of magnetic lines of force inside the material = magnetic lines of force due to magnetising field + magnetic lines of force due to magnetisation of specimen\n\ni.e. Magnetic induction (B) = $B_{0}$ (no. of lines of force due to H) + $\\mu_{0} I$ (no. of lines due to magnetisation of specimen0029\n\nor $B = B _{0}+\\mu_{0} I =\\mu_{0} H +\\mu_{0} I =\\mu_{0}( H + I )$\n\n$B =\\mu_{0}( H + I )=\\mu_{0} H \\left(1+\\chi_{ m }\\right)$ as $\\chi_{ m }=\\frac{ I }{ H }$\n\n$B =\\mu H =\\mu_{0} H \\left(1+\\chi_{ m }\\right)$ or $\\mu=\\mu_{0}\\left(1+\\chi_{m}\\right)$\n\nor $\\mu_{ r }=1+\\chi_{ m }$.\n\nEx. A tungsten rod of length 10 cm and area of cross-section $0.25 cm ^{2}$ is placed in a magnetising field of 314 oersted, with its length parallel to the field. The magnetic susceptibility of tungsten is $6.8 \\times 10^{-5}$. Calculate the (i) intensity of magnetisation (ii) magnetic moment and (iii) absolute permeability.\n\nSol. Given H = 314 oersted = $\\frac{314 \\times 10^{3}}{4 \\pi}$ amp. $\/ m .=\\frac{10^{5}}{4}$ amp. $\/ m$\n\n1. Intensity of magnetisation $$I = {{{\\chi _m}} \\over H} = {{6.8 \\times {{10}^{ \u2013 5}} \\times {{10}^5}} \\over 4} = 1.7\\,amp.\/1m$$\n2. Magnetic moment M = $\\mathbf{I}$V = $1.7 \\times 0.1 \\times 0.25 \\times 10^{-4}=4.25 \\times 10^{-6}$ amp.\/m.\n3. Absolute permeability $\\mu=\\mu_{ r } \\mu_{0}=\\mu_{0}\\left(1+\\chi_{ m }\\right)=4 \\pi \\times 10^{-7}\\left[1+6.8 \\times 10^{-5}\\right]=12.56 \\times 10^{-7}$Wb\/A-m\n\nEx. A solenoid of 500 turns\/m is carrying a current of 3A. Its core is made of iron which has a relative permeability of 5000. Determine the magnitude of magnetic intensity, magnetisation and magnetic field inside the core.\n\nSol. Magnetic intensity H = ni = 500 \u00d7 3 = 1500 A\/m\n\n$\\mu_{ r }=1+\\chi_{ m }$ so $\\chi_{ m }=\\mu_{ r }-1=4999 \\approx 5000$\n\nIntensity of magnetisation $I =\\chi H =5000 \\times 1500=7.5 \\times 10^{6} A \/ m$\n\nMagnetic field $B =\\mu_{ r } \\mu_{0} H =5000 \\times 4 \\pi \\times 10^{-7} \\times 1500=9.4$ tesla.\n\nEx. An electron in an atom revolves around the nucleus in an orbit of radius 0.53 A. Calculate the equivalent magnetic moment if the frequency of revolution is $6.8 \\times 10^{9} MHz$\n\nSol. Magnetic moment $M=I A=\\frac{e}{T} \\times \\pi r^{2}=e v \\pi r^{2}$\n\nSo M = $1.6 \\times 10^{-19} \\times 6.8 \\times 10^{9} \\times 3.14 \\times\\left(0.53 \\times 10^{-10}\\right)^{2}=9.6 \\times 10^{-24} Am ^{2}$\n\nEx. An iron rod 0.2 m long $10^{-2}$ m in diameter and of permeability 1000 is placed inside a long solenoid wound with 300 turns per meter. If a current of 0.5 A is passed through the rod find magnetic moment of rod.\n\nSol. $B =\\mu_{0}( H + I )$ so $I =\\frac{ B }{\\mu_{0}}- H =\\frac{\\mu H }{\\mu_{0}}- H =\\left(\\mu_{ r }-1\\right) H$\n\nFor a solenoid $B =\\mu_{0} ni$\n\nSo $H =\\frac{ B }{\\mu_{0}}= ni$ so $I=\\left(\\mu_{r}-1\\right) n i$\n\n$I =(1000-1) 300 \\times 0.5=999 \\times 150 Am ^{2}$\n\nMagnetic moment of rod $M = I \\times V = I \\pi r ^{2} \\ell=999 \\times 150 \\times 3.14 \\times\\left(0.5 \\times 10^{-2}\\right) \\times 0.2$\n\n$=0.2325 JT ^{-1}$","date":"2020-06-04 20:37:53","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\": 0.8601180911064148, \"perplexity\": 611.2328087010724}, \"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-2020-24\/segments\/1590347458095.68\/warc\/CC-MAIN-20200604192256-20200604222256-00348.warc.gz\"}"} | null | null |
package org.jetbrains.plugins.textmate.language.syntax.highlighting;
import com.intellij.openapi.editor.colors.EditorColorsScheme;
import com.intellij.openapi.editor.ex.util.DataStorage;
import com.intellij.openapi.editor.ex.util.LexerEditorHighlighter;
import com.intellij.openapi.editor.highlighter.EditorHighlighter;
import com.intellij.openapi.fileTypes.EditorHighlighterProvider;
import com.intellij.openapi.fileTypes.FileType;
import com.intellij.openapi.fileTypes.SyntaxHighlighter;
import com.intellij.openapi.fileTypes.SyntaxHighlighterFactory;
import com.intellij.openapi.project.Project;
import com.intellij.openapi.vfs.VirtualFile;
import org.jetbrains.annotations.NotNull;
import org.jetbrains.annotations.Nullable;
import org.jetbrains.plugins.textmate.language.syntax.lexer.TextMateLexerDataStorage;
public class TextMateEditorHighlighterProvider implements EditorHighlighterProvider {
@Override
public EditorHighlighter getEditorHighlighter(@Nullable Project project,
@NotNull FileType fileType,
@Nullable VirtualFile virtualFile,
@NotNull EditorColorsScheme colors) {
return new TextMateLexerEditorHighlighter(SyntaxHighlighterFactory.getSyntaxHighlighter(fileType, project, virtualFile), colors);
}
private static final class TextMateLexerEditorHighlighter extends LexerEditorHighlighter {
private TextMateLexerEditorHighlighter(@Nullable SyntaxHighlighter highlighter, @NotNull EditorColorsScheme colors) {
super(highlighter != null ? highlighter : new TextMateHighlighter(null), colors);
}
@NotNull
@Override
protected DataStorage createStorage() {
return new TextMateLexerDataStorage();
}
}
}
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By StageLight | February 2, 2015
ADDITIONAL BROADWAY CASTING ANNOUNCED FOR KANDER AND EBB'S FINAL MUSICAL
CHITA RIVERA AND ROGER REES
New York (January 30, 2015) – Additional Broadway casting was announced today for the highly anticipated musical, The Visit, soon to open on Broadway. The Visit will begin performances at the Lyceum Theatre (149 West 45th Street) on Thursday, March 26 with an Opening Night set for Thursday, April 23. General on sale will begin on January 31, 2015.
Joining the previously announced Tony Award® Winners Chita Rivera and Roger Rees, will be Jason Danieley, Matthew Deming, Diana Dimarzio, David Garrison, Rick Holmes, Tom Nelis, Chris Newcomer, Aaron Ramey, Timothy Shew, and Michelle Veintimilla.
Based on the satirical play by Friedrich Dürrenmatt as adapted by Maurice Valency, The Visit features music and lyrics by the Tony Award® winning team John Kander & Fred Ebb, book by four-time Tony Award® winning playwright Terrence McNally. It will open on Broadway this spring, starring two-time Tony Award® winning Broadway legend Chita Rivera and Tony Award® winner Roger Rees. Helmed by Tony Award® winning director John Doyle and choreographed by Tony Award® nominee Graciela Daniele. Tickets will be available on Telecharge.com beginning January 31.
In her juiciest role yet, Chita Rivera is Claire Zachannassian, the world's wealthiest woman, who returns home to Anton (Roger Rees) who captured her heart then shattered her dreams. What she does next shocks the town and makes for the most thrilling and intriguing musical in years. From the team that brought you "Cabaret", "Chicago", "Kiss of the Spider Woman", this tale of romance, seduction and betrayal proves that revenge is the best revenge.
The Visit is produced by Tom Kirdahy, Tom Smedes, Hugh Hayes, Peter Stern, Judith Ann Abrams, Hunter Arnold, Carl Daikeler, Ken Davenport, Rich Affannato, Gabrielle Palitz, in association with Williamstown Theatre Festival.
Probably written by one our amazing StageLight staff writers!
← Previous Story TOP 10 THEATRE STORIES YOU NEED TO KNOW
Next Story → THE PUBLIC THEATER ANNOUNCES INITIAL CASTING FOR BUZZER | {
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{"url":"https:\/\/chemistry.stackexchange.com\/questions\/133032\/ozone-in-ozone-layer\/133034","text":"# Ozone in ozone layer [closed]\n\nIf ozone is a gas, and gases have the highest entropy, then how does the ozone gas stay within a few layers of the atmosphere, even though they span for kilometers?\n\n\u2022 Entropy and ozone at the top of the atmosphere are two unconnected things. Harmful UV rays from the Sun are unable to penetrate deep into the atmosphere because they encounter oxygen molecules. They absorb the UV and form ozone on the top. If ozone reached us, nobody could survive. It is a harmful thing. Read more about the en.wikipedia.org\/wiki\/Ozone%E2%80%93oxygen_cycle \u2013\u00a0M. Farooq May 3 at 14:39\n\u2022 \u2013\u00a0Faded Giant May 3 at 15:33\n\u2022 @M.Farooq - For clarification - there's plenty of ground level ozone around. It's not good for us, but it's not so bad at low concentrations that we don't survive. I suspect most people are familiar with the smell of ozone, even if they don't know what they're smelling. \u2013\u00a0Andrew May 3 at 15:47\n\u2022 Right, Andrew, ozone at part per million levels it is not that great! It immediately hurts the throat if you are sensitive. If you recall the smell of older photocopy machines, the smell is not that pleasant but fresh rain is of wonderful but that must be parts per billion conc. \u2013\u00a0M. Farooq May 3 at 15:58\n\u2022 Plus some rare highly reactive agens like peroxyacetylnitrate can be created, what reportedly acted in past during some Los Angeles smog situations as plant defoliant. Additionally, ozone is the starter of immission damage to conifer trees by SO2. \u2013\u00a0Poutnik May 3 at 16:07\n\nIt does not have much to do with entropy, rather with the way and place of ozone creation.\n\nStratospheric ozone is produced typically at altitude 20-30 km by UVC radiation with $$\\lambda \\lt \\pu{280 nm}$$:\n\n$$\\ce{O2 + \\nu -> 2 O}$$ $$\\ce{O + O2 -> O3}$$\n\nSee ozone cycle as courtesy of @M. Farooq.\n\nOzone in lower troposphere troposphere is created by UVB (280-320 nm) (+UVA >320 nm ??) mostly due catalytic effects of nitrogen oxides that come typically of the oxidative smog of the Los Angeles type:\n\n$$\\ce{NO2 + \\nu -> NO2^{*}}$$ $$\\ce{NO2^{*} + O2 -> NO + O3}$$ $$\\ce{2 NO + O2 -> 2 NO2}$$\n\nTechnically, the concentration and positioning of ozone in the atmosphere likely relates directly to both ozone formation, stabilization and a destruction cycle.\n\nWhile formation creation cycles has already been outlined, I would like to add more detail on the chemistry based on this source:\n\nCreation:\n\n$$\\ce{O2 + \u210e\u03bd \u2192 2 O\u2022 }$$\n\n$$\\ce{O\u2022 + O2 \u2192 O3 }$$\n\nwhich is followed by the 'ozone\u2013oxygen cycle' where the ozone molecules formed by the reaction above absorbs radiation in wavelengths between UV-C and UV-B.\n\n$$\\ce{O3 + \u210e\u03bd(240\u2013310 nm) \u2192 O2 + O }$$\n\nforming atomic oxygen which can further react with dioxygen:\n\n$$\\ce{O + O2 \u2192 O3 + kinetic energy }$$\n\nThe cited net effect of the ozone\u2013oxygen cycle is the conversion of penetrating UV-B light into heat, with no net loss of O3. This cycle reputedly keeps the ozone layer stable and results in the protection of the lower atmosphere from harmful UV radiation.\n\nRemoval of Ozone:\n\n$$\\ce{O3 + O\u00b7 \u2192 2 O2}$$\n\n$$\\ce{2 O\u00b7 \u2192 O2 }$$\n\nAlso, per a source, on the net effect of the two reactions, to quote:\n\n$$\\ce{2 O3 \u2192 3 O2 }$$\n\nThis reaction proceeds more rapidly with increasing temperature.\n\nSo, higher colder altitudes are more conducive to the preservation of created ozone.\n\nOn its destruction cycles, to quote a source:\n\nOzone is a highly reactive molecule that easily reduces to the more stable oxygen form with the assistance of a catalyst. Cl and Br atoms destroy ozone molecules through a variety of catalytic cycles. In the simplest example of such a cycle,[11] a chlorine atom reacts with an ozone molecule (O3), taking an oxygen atom to form chlorine monoxide (ClO) and leaving an oxygen molecule (O2). The ClO can react with a second molecule of ozone, releasing the chlorine atom and yielding two molecules of oxygen. The chemical shorthand for these gas-phase reactions is:\n\n$$\\ce{Cl\u00b7 + O3 \u2192 ClO + O2}$$\n\nA chlorine atom removes an oxygen atom from an ozone molecule to make a ClO molecule\n\n$$\\ce{ClO + O3 \u2192 Cl\u00b7 + 2 O2}$$\n\nThis ClO can also remove an oxygen atom from another ozone molecule; the chlorine is free to repeat this two-step cycle.\n\nNow, pollutants in the atmosphere include chloro-(and bromo-) organics, which can, under UV photolysis, liberate a chlorine (or bromine) radical.\n\nSo, as one gets closer to the surface of the earth, radiation-induced ozone formation is abated, warmer temperatures are unfavorable and any diffused ozone entering the lower layers, could be further subject to attack by increasing concentration of photosensitized pollutants engaging in catalytic destruction cycles.\n\nAnd, that is why the ozone layer resides at a high altitude, maintains itself and is not found closer to the surface of the earth.","date":"2020-08-13 09:44:25","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\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 15, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.39417320489883423, \"perplexity\": 3183.296391242827}, \"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-2020-34\/segments\/1596439738964.20\/warc\/CC-MAIN-20200813073451-20200813103451-00002.warc.gz\"}"} | null | null |
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Longchamp Abbey (), known also as the Convent of the Humility of the Blessed Virgin, was a convent of Poor Clares founded in 1255 in Auteuil, Paris, by Saint Isabelle of France. The site is now occupied by Longchamp Racecourse.
Royal Foundation
Isabelle was the daughter of Louis VIII of France and Blanche of Castile, and the younger sister of King Louis IX of France (Saint Louis). Though betrothed to Hugh, eldest son and heir of Hugh X of Lusignan, Isabelle refused to celebrate the formal wedding due to her fixed determination to remain a virgin, although she never became a nun.
In furtherance of Isabelle's wish to found a nunnery of Poor Clares, her brother King Louis IX of France began in 1255 to acquire the necessary land in the Forest of Rouvray, not far from the Seine, west of Paris. On 10 June 1256, the first stone of the monastic church was laid. The building appears to have been completed about the beginning of 1259. The less rigorous Rule of Mansuetus allowed the community to hold property. The abbey was named the "Convent of the Humility of the Blessed Virgin". Subject to the Order of Friars Minor, some of the first nuns came from the Poor Clares in Reims. Isabelle never joined the community herself, but did live in the abbey, in a room separate from the nuns' cells. The King visited often and remembered the Abbey in his will. Isabelle died at Longchamp on 23 February 1270, and was buried in the abbey church.
Abbesses of Longchamp
Agnès I d'Anneri 1259-1262
Mathilde de Guyencourt 1262-1263
Agnès II d'Harcourt 1264-1275
Julienne de Toyes 1275-1279
Agnès II d'Harcourt 1279-1287
Jeanne I de Nevers 1288-1294
Jeanne II de Grèce 1294-1303
Jeanne III de Vitry 1303-1312
Jeanne IV d'Harcourt 1312-13??
Jeanne V de Gueux 13??-1328
Marie I de Lions 1328-1347
Jeanne VI de Boucheville 1347-1349
Agnès III de Liège 1349-1357
Marie II de Gueux 1357-1369
Agnès IV La Chevrel 1369-1375
Jeanne VII de La Neuville 1375-1390
Laurence Jacob 1390-13??
Jeanne VIII de La Godicharde 13??-1402
Agnès V d'Issy 1402-1418
Jeanne IX des Essarts 1418-1437
Marie III de La Poterne 1437-1451
Marguerite I Gentianne 1451-1467
Jeanne X La Porchère 1467-1484
Jeanne XI Gerente 1484-1500
Jacqueline de Mailly 1500-1514
Jeanne XII de Hacqueville 1514-1532
Catherine I Picard 1532-15??
Jeanne XIII de Mailly 15??-1540
Georgette Cœur 1540-1550
Louise de Cerasme 1550-1559
Marie IV Lottin 1559-15??
Charlotte de La Chambre 15??-1567
Anne I de Fontaines 1567-1580
Jeanne XIV de Mailly 1580-1604
Françoise Potier 1604-1606
Bonne d'Amours 1606-1608
Catherine II Brûlart de Sillery 1608-1629
Claudine I Isabelle de Mailly 1629-1634
Isabelle II Mortier 1634-16??
Madeleine Placain 16??-1653
Catherine III de Bellièvre 1658-1668
Claudine II de Bellièvre 1668-1670
Claudine I Isabelle de Mailly 1670-1673
Catherine III Marie Dorat 1673-1676
Catherine-Elisabeth I de Gournay 1676-1679
Marguerite II Isabelle de Flecelles, 1679-1683
Catherine III Marie Dorat 1683-1685
Marie-Anne I Dorat 1685-1688
Anne-Marie de Bragelongne 1688-1691
Catherine III Marie Dorat 1691-1694
Marie-Anne I Dorat 1694-1697
Catherine III Marie Dorat 1697-1700
Marie-Anne I Dorat 1700-1700
Elisabeth-Henriette Guignard 1700-1703
Catherine III Marie Dorat 1703-1706
Marguerite III Agnès Nolet 1706-1709
Elisabeth-Henriette Guignard 1709-1712
Marguerite III Agnès Nolet 1712-1715
Catherine-Elisabeth II Le Cosquino 1715-1718
Marguerite III Agnès Nolet 1718-17??
Catherine-Elisabeth II Le Cosquino 17??-1721
Marie-Anne II Le Jau 1721-1724
Catherine-Elisabeth II Le Cosquino 1724-1730
Marie-Anne II Le Jau 1730-1733
Catherine-Elisabeth II Le Cosquino 1733-1737
Catherine IV Thérèse de Tourmont 1737-1740
Anne II Louise de Tourmont 1740-17??
Marie V Jeanne Jouy 17??-1790
Destruction
Longchamp Abbey underwent many vicissitudes. During the French Revolution, on 26 February 1790, the nuns were served with an order of expulsion; on 17 September 1792 the valuables and sacred objects were taken away from the chapel and by 12 October that year the nuns had left the abbey. In 1794 the empty building was offered for sale, but, as no one wished to purchase it, it was destroyed. In 1857 the remaining walls were pulled down, except for one tower, and the grounds were added to the Bois de Boulogne.
Depictions
Misbach, Vue de l'abbaye de Longchamp prise du pied du jardin de M. Lagarde, Bibliothèque nationale de France, Paris.
See also
Prix de l'Abbaye de Longchamp (a flat horse race, open to thoroughbreds aged two years or older, run at Longchamp Racecourse each year in early October).
References
Further reading
Gaston Duchesne, Histoire de l'abbaye royale de Longchamp, 1257–1789, Paris, 1904.
Gerturd Młynarczyk, Ein Franziskanerinnenkloster im XV. Jahrhundert, : Edition und Analyse von Besitzinventaren aus der Abtei Longchamp, Bonn, L. Röhrscheid, 1987.
Sean L. Field, Isabelle of France: Capetian Sanctity and Franciscan Identity in the Thirteenth Century (University of Notre Dame Press, 2006, .
Sean L. Field, ed. and trans., The Writings of Agnes of Harcourt: The Life of Isabelle of France and the Letter on Louis IX and Longchamp (University of Notre Dame Press, 2003).
Poor Clare monasteries in France
1255 establishments in Europe
1250s establishments in France | {
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{"url":"https:\/\/brilliant.org\/discussions\/thread\/create-a-math-problem-using-this-gif\/","text":"\u00d7\n\n# Create a math problem using this gif.\n\nThe SFH (Snow Flip Hat)\n\nNote by Llewellyn Sterling\n2\u00a0years, 3\u00a0months ago\n\n## Comments\n\nSort by:\n\nTop Newest\n\nWhat is the area under the path of the parabola!? \u00b7 2\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nDon't try this At Your Home,School or Anywhere...!! It could be injurious...!! \u00b7 2\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nThat's not a math question, but ok. \u00b7 2\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nAt what time must the hat be given a push so that it is a perfect SFH? \u00b7 2\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nWhy did the hat come upwards although it is pulled by gravity?? \u00b7 2\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nwat is the path followed by the hat? (level 1 ) \u00b7 2\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nSimple. It's Parabola. \u00b7 2\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nthats why its level 1 \u00b7 2\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\n\u00d7\n\nProblem Loading...\n\nNote Loading...\n\nSet Loading...","date":"2017-05-28 06:54:46","metadata":"{\"extraction_info\": {\"found_math\": false, \"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\": 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\": 0.9854234457015991, \"perplexity\": 13028.263933354598}, \"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-2017-22\/segments\/1495463609605.31\/warc\/CC-MAIN-20170528062748-20170528082748-00012.warc.gz\"}"} | null | null |
package com.github.pockethub.ui;
import android.content.DialogInterface.OnClickListener;
import android.os.Bundle;
import android.os.Parcelable;
import java.util.ArrayList;
/**
* Helper to display a single choice dialog
*/
public class SingleChoiceDialogFragment extends DialogFragmentHelper implements
OnClickListener {
/**
* Arguments key for the selected item
*/
public static final String ARG_SELECTED = "selected";
/**
* Choices arguments
*/
protected static final String ARG_CHOICES = "choices";
/**
* Selected choice argument
*/
protected static final String ARG_SELECTED_CHOICE = "selectedChoice";
/**
* Tag
*/
protected static final String TAG = "single_choice_dialog";
/**
* Confirm message and deliver callback to given activity
*
* @param activity
* @param requestCode
* @param title
* @param message
* @param choices
* @param selectedChoice
* @param helper
*/
protected static void show(final DialogFragmentActivity activity,
final int requestCode, final String title, final String message,
ArrayList<? extends Parcelable> choices, final int selectedChoice,
final DialogFragmentHelper helper) {
Bundle arguments = createArguments(title, message, requestCode);
arguments.putParcelableArrayList(ARG_CHOICES, choices);
arguments.putInt(ARG_SELECTED_CHOICE, selectedChoice);
show(activity, helper, arguments, TAG);
}
}
| {
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Bride slammed after blasting stylist who refused to do young bridesmaid's hair
By Karishma Sarkari| 5 months ago
A bride took to online forum Reddit to vent about her hair and make-up artist, who refused to do a young bridesmaid's hair on the big day.
In a rant also posted to Facebook, the bride blasted the woman for not doing the 10-year-old's hair just minutes before the start of the ceremony.
However, it was the bride who was slammed by users, leading to the Facebook post being deleted.
The online forum still exists and in the post the woman tells users the make-up artist (MUA) was running late and decided to leave the youngest bridesmaid (BM) till last, which she was fine with at first, till the stylist noticed something.
"The MUA was styling her hair when she noticed BM has head lice.
"She informed me that her policy is not to touch the client any further.
"She believes lice can jump from the child's head. This is not correct, they can't jump."
The bride shared her story on Reddit asking for advice on handling the situation moving forward (Reddit)
While understanding the make-up artist couldn't use her brushes to do the child's hair, she was upset the woman didn't use her hands to complete the 10-year-old's look.
"We felt that she could have done a simple plait style by hand. She would not touch the child to finish or change the hairstyle.
"As this occurred just before the ceremony there was no way for us to send for help from friends or family."
The bride says the unplanned disruption to the hair and make-up schedule caused her to miss a key moment at the start of the ceremony and left her young bridesmaid in tears.
"Our ceremony started late and the BM was in tears, she was so upset by it all. It was quite stressful."
The bride then asked for advice on handling the situation moving forward but got little-to-no sympathy from people online, who sided with the make-up artist.
" Lice are little monsters that spread like wildfires and hairstylists and groomers all have a 'Cant work with lice' policy because of that very reason ," one person commented.
(Reddit)
People sided with the make-up artist on Reddit (Reddit)
"I don't know a single stylist who would touch someone with lice," another wrote.
One person suggested: "Bride needs to get over herself and braid it herself if it's NBD [no big deal)]."
While one person guessed it would be a case of "damned if you do, damned if you don't" for the make-up artist.
"Well it's a good thing she was done last! Imagine the MUA using her lice infested brushes on the bride and the rest of the wedding party after doing the kid."
Responses on the now deleted Facebook post (Facebook)
"Also if the stylist kept doing the child's hair and spread it around to the rest of the party, they would have been posting about that too," a woman noted on the Facebook post.
While a former make-up artist weighed into the debate, saying: "What if the MUA had another gig after this one? What if she was infected and then infected the other wedding party? What if the other wedding party blamed her for lice and gave her a public review of her work and told everyone that she infected their girls with lice? Sorry but there is a lot that goes into this."
Property News: The colours to know before painting your house in 2020 - domain.com.au | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: How can I get video duration by using vimeo javascript API? I am having a premium channel on vimeo. I am trying to put a list of videos and their duration on my website. They have an example by using the event listeners. But as I am interested in simple text output of video duration. So How can I do it ?
A: Use getDuration() from the Vimeo player javascript API:
player.getDuration().then(function(duration) {
// duration = the duration of the video in seconds
}).catch(function(error) {
// an error occurred
});
Source: https://github.com/vimeo/player.js/blob/master/README.md#getduration-promisenumber-error
A: const iframe = document.querySelector('iframe');
const player = new Player(iframe);
player.loadVideo(123456).then(() => {
player.ready().then(() => {
player.getDuration().then((data) => console.log(data));
}).catch((err) => console.log(err));
})
A: (function () {
var vimeoPlayers = document.querySelectorAll('iframe'), player;
for (var i = 0, length = vimeoPlayers.length; i
| {
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\section{Introduction}
\label{sec:intro}
Model reduction techniques are often required for solving
challenging multiscale problems that have multiple scales and high contrast.
Many of these model
reduction techniques perform the discretization of the problem on a coarse
grid where coarse grid size is much larger than the fine-grid discretization.
The latter requires constructing reduced order models for the solution space
on a coarse grid. Some of these techniques involve upscaled models (e.g.,
\cite{dur91, weh02}) or multiscale methods (e.g., \cite{Arbogast_two_scale_04, Chu_Hou_MathComp_10,ee03,
egw10,eh09,ehg04, GhommemJCP2013,ReducedCon,MsDG,Wave,WaveGMsFEM}).
In this paper, we derive an a-posteriori error indicator for the Generalized
Multiscale Finite Element Method (GMsFEM) framework
\cite{egh12}.
This error indicator is further used
to develop an adaptive enrichment algorithm for
the linear elliptic equation with multiscale high-contrast coefficients.
GMsFEM is a flexible general
framework that generalizes the Multiscale Finite Element Method (MsFEM)
(\cite{hw97})
by systematically enriching the coarse spaces and taking into account
small scale information and
complex input spaces. This approach, as in many
multiscale model reduction techniques, divides the computation into
two stages: the offline and the online. In the offline stage,
a small dimensional space is constructed that can be
used in the online stage to construct multiscale basis functions.
These multiscale basis functions can be re-used for any input parameter
to solve the problem on a coarse grid. The main idea behind the construction
of offline and online spaces is the selection of local spectral problems
and the selection of the snapshot space.
In \cite{egh12}, we propose several general strategies. In this paper,
we investigate adaptive enrichment procedures.
In previous findings \cite{egw10, eglp13}, a-priori error bounds
for the GMsFEM are derived for linear elliptic equations.
It was shown that the convergence rate is proportional to
the inverse of the eigenvalue that corresponds to the eigenvector
which is not included in the coarse space. Thus, adding more basis
functions will improve the accuracy and
it is important to include the eigenvectors that correspond
to very small eigenvalues (\cite{egw10}).
Rigorous a-posteriori
error indicators are needed to perform an adaptive enrichment
which is a subject of this paper. We would like to point out that
there are many related activities in designing a-posteriori
error estimates \cite{ohl12, abdul_yun, dinh13, nguyen13, tonn11}
for global reduced models.
The main difference is that our error estimators are
based on special local eigenvalue problem and use the eigenstructure
of the offline space.
In the paper, we consider two kinds of error indicators where one is based on
the $L^2$-norm of the local residual
and the other is based on the weighted $H^{-1}$-norm (we will also call it
$H^{-1}$-norm based) of the local residual
where the weight is related to the coefficient of the elliptic equation.
We show that the use of weighted $H^{-1}$-norm residual
gives a more robust error indicator
which works well for cases with high contrast media.
The convergence analysis of the method is given.
In our analysis, we do not consider the error due to the fine-grid
discretization of local problems and only study the errors due
to the enrichment.
In this regard, we assume that the error is largely
due to coarse-grid discretization.
The fine-grid discretization error can be considered in general
(e.g., as in \cite{abdul_yun, ohl12}) and
this will give an additional error estimator.
The proposed error indicators allow adding multiscale
basis functions in the regions detected by the error indicator.
The multiscale basis functions are selected by choosing
next important eigenvectors (based on the increase of the eigenvalues)
from the offline space.
The convergence proof of our adaptive enrichment algorithm is based on
the techniques used for proving the convergence of adaptive refinement method
for classical conforming finite element methods for second order elliptic problems
\cite{BrennerScott,AdaptiveFEM}.
Contrary to \cite{AdaptiveFEM} where mesh refinement is considered,
we prove the convergence of our adaptive enrichment algorithm
as the approximation space is enriched for a fixed coarse mesh size.
The convergence is based on some
previously developed
spectral estimates. In particular, we use both stability
of the coarse-grid projection and the convergence of spectral interpolation.
Another key idea is that our error indicators are defined in a variational sense
instead of the pointwise residual of the differential equation.
By using this variational definition,
we avoid the use of the gradient of the multiscale coefficient.
Moreover, our convergence analysis does
not require that the gradient of the coefficient is bounded,
which is not the case for high-contrast multiscale flow problems.
In the proposed error indicators, we consider
the use of snapshot space in GMsFEM. In this case,
the residual contains an irreducible error due
to the difference between the snapshot solution
and the fine-grid solution. We consider the use of
snapshot space for approximating the residual error
in the case of
weighted $H^{-1}$-norm of the local residual.
We present several numerical tests by considering
two different high-contrast multiscale permeability fields.
We study both
error indicators based on
the $L^2$-norm of the local residual
and the weighted $H^{-1}$-norm of the local residual.
Our numerical results show that
the use of weighted $H^{-1}$-norm residual
gives a more robust error indicator
which works well for cases with high contrast media.
In our numerical results, we also compare the results obtained
by the proposed
indicators and the exact error indicator which is computed
by considering the energy norm of the difference between
the fine-scale solution and the offline solution.
Our numerical results show that the use of the exact error
indicator gives nearly similar results to the case of using
weighted $H^{-1}$ error indicator.
In our studies, we also consider
the errors between the fine-grid solution and the offline
solution as well as the snapshot solution and the offline solution.
All cases show that the proposed error indicator
is robust and can be used to detect
the regions where additional multiscale basis
functions are needed.
The paper is organized in the following way.
In the next section, we present Preliminaries.
The GMsFEM is presented in Section \ref{cgdgmsfem}.
In Section
\ref{sec:errorindicator},
we present the details of the error indicator and state our main
results. In Section \ref{sec:numresults}, numerical results are presented.
The proofs of our main results are presented in Section \ref{sec:analysis}.
The paper ends with a Conclusion.
\section{Preliminaries}
\label{prelim}
In this paper, we consider high-contrast flow problems of the form
\begin{equation} \label{eq:original}
-\mbox{div} \big( \kappa(x) \, \nabla u \big)=f \quad \text{in} \quad D,
\end{equation}
subject to the homogeneous Dirichlet boundary condition $u=0$ on $\partial D$,
where $D$ is the computational domain.
We assume that $\kappa(x)$ is a heterogeneous coefficient
with multiple scales and very high contrast.
To discretize (\ref{eq:original}), we
introduce the notion of fine and coarse grids.
We let $\mathcal{T}^H$ be a usual conforming partition of the computational domain $D$ into finite elements (triangles, quadrilaterals, tetrahedra, etc.). We refer to this partition as the coarse grid and assume that each coarse element is partitioned into a connected union of fine grid blocks. The fine grid partition will be denoted by $\mathcal{T}^h$, and is by definition a refinement of the coarse grid $\mathcal{T}^H$.
We use $\{x_i\}_{i=1}^{N}$ (where $N$ denotes the number of coarse nodes) to denote the vertices of
the coarse mesh $\mathcal{T}^H$, and define the neighborhood of the node $x_i$ by
\begin{equation} \label{neighborhood}
\omega_i=\bigcup\{ K_j\in\mathcal{T}^H; ~~~ x_i\in \overline{K}_j\}.
\end{equation}
See Figure~\ref{schematic} for an illustration of neighborhoods and elements subordinated to the coarse discretization. We emphasize the use of $\omega_i$ to denote a coarse neighborhood, and $K$ to denote a coarse element throughout the paper.
\begin{figure}[htb]
\centering
\includegraphics[width=0.65 \textwidth]{gridschematic}
\caption{Illustration of a coarse neighborhood and coarse element}
\label{schematic}
\end{figure}
Next, we briefly outline the GMsFEM.
We will consider the
continuous Galerkin (CG) formulation and we will use $\omega_i$ as the support of basis functions.
We
denote the basis functions by $\psi_k^{\omega_i}$, which is supported in $\omega_i$.
In particular, we note that the proposed approach will employ the use of multiple basis functions per coarse neighborhood,
and the index $k$ represents the numbering of these basis functions.
In turn, the CG solution will be sought as $u_{\text{ms}}(x)=\sum_{i,k} c_{k}^i \psi_{k}^{\omega_i}(x)$.
Once the basis functions are identified, the CG global coupling is given through the variational form
\begin{equation}
\label{eq:globalG} a(u_{\text{ms}},v)=(f,v), \quad \text{for all} \, \, v\in
V_{\text{off}},
\end{equation}
where $V_{\text{off}}$ is used to denote the space spanned by those basis functions
and $a(\cdot,\cdot)$ is a usual bilinear form corresponding to (\ref{eq:original}).
We also note that one can use
discontinuous Galerkin formulation (see e.g., \cite{Wave,WaveGMsFEM,eglmsMSDG}) to couple multiscale basis functions
defined on $K$.
Let $V$ be the conforming finite element space
with respect to the fine-scale partition $\mathcal{T}^h$.
We assume $u\in V$ is the fine-scale solution satisfying
\begin{equation*}
a(u,v) = (f,v), \quad v\in V.
\end{equation*}
\section{CG-based GMsFEM for high-contrast flow problems}
\label{cgdgmsfem}
In this section, we will give a brief description
of the GMsFEM for high contrast flow problems.
More details can be found in \cite{egh12, eglp13}.
In the following, we also give a general outline of the GMsFEM.
\begin{itemize}
\item[1.] Offline computations:
\begin{itemize}
\item 1.0. Coarse grid generation.
\item 1.1. Construction of snapshot space that will be used to compute an offline space.
\item 1.2. Construction of a small dimensional offline space by performing dimension reduction in the space of global snapshots.
\end{itemize}
\item[2.] Online computations:
\begin{itemize}
\item 2.1. For each input parameter, compute multiscale basis functions. (for parameter-dependent cases)
\item 2.2. Solution of a coarse-grid problem for any force term and boundary condition.
\item 2.3. Iterative solvers, if needed.
\end{itemize}
\end{itemize}
\subsection{Local basis functions}
\label{locbasis}
We now present the construction
of the basis functions
and the corresponding spectral problems
for obtaining a space reduction.
In the offline computation, we first construct a snapshot space $V_{\text{snap}}^{\omega}$.
The snapshot space can be the space of all fine-scale basis functions
or the solutions of some local problems with various choices of boundary conditions.
For example, we can use the following $\kappa$-harmonic extensions to form a snapshot space.
For each fine-grid function, $\delta_j^h(x)$,
which is defined by
$\delta_j^h(x)=\delta_{j,k},\,\forall j,k\in \textsl{J}_{h}(\omega_i)$, where $\textsl{J}_{h}(\omega_i)$ denotes the fine-grid boundary node on $\partial\omega_i$. \\
Given a fine-scale piecewise linear function defined on $\partial\omega$, we define $\psi_{j}^{\omega_i, \text{snap}}$ by
\begin{equation} \label{harmonic_ex}
-\text{div}(\kappa(x) \nabla \psi_{j}^{\omega_i, \text{snap}} ) = 0
\quad \text{in} \, \, \, \omega_i,
\end{equation}
where $\psi_{j}^{\omega_i, \text{snap}}=\delta_j^h(x)$ on $\partial\omega_i$.\\
For brevity of notation we now omit the superscript $\omega_i$, yet it is assumed throughout this section that the offline space computations are localized to respective coarse subdomains.
Let $W_i$ be the number of functions in the snapshot space in the region $\omega_i$, and
$$
V_{\text{snap}} = \text{span}\{ \psi_{j}^{ \text{snap}}:~~~ 1\leq l \leq W_i \},
$$
for each coarse subdomain $\omega_i$.
Denote
$$
R_{\text{snap}} = \left[ \psi_{1}^{\text{snap}}, \ldots, \psi_{W_i}^{\text{snap}} \right].
$$
In order to construct the offline space $V_{\text{off}}^\omega$, we perform a dimension reduction of the space of snapshots using an auxiliary spectral decomposition. The analysis in \cite{egw10} motivates the following eigenvalue problem in the space of snapshots:
\begin{equation} \label{offeig}
A^{\text{off}} \Psi_k^{\text{off}} = \lambda_k^{\text{off}} S^{\text{off}} \Psi_k^{\text{off}},
\end{equation}
where
\begin{equation*}
\displaystyle A^{\text{off}} = [a_{mn}^{\text{off}}] = \int_{\omega} \kappa(x) \nabla \psi_m^{\text{snap}} \cdot \nabla \psi_n^{\text{snap}} = R_{\text{snap}}^T A R_{\text{snap}}
\end{equation*}
%
\begin{center}
and
\end{center}
%
\begin{equation*}
\displaystyle S^{\text{off}} = [s_{mn}^{\text{off}}] = \int_\omega \widetilde{\kappa}(x)\psi_m^{\text{snap}} \psi_n^{\text{snap}} = R_{\text{snap}}^T S R_{\text{snap}},
\end{equation*}
where $A$ and $S$ denote analogous fine scale matrices as defined by
\begin{equation*}
A_{ij} = \int_{D} \kappa(x) \nabla \phi_i \cdot \nabla \phi_j \, dx
\quad
S_{ij} = \int_{D} \widetilde{\kappa}(x) \phi_i \phi_j \, dx
\end{equation*}
where $\phi_i$ is the fine-scale basis function. We will give the definition
of $\widetilde{\kappa}(x)$ later on.
To generate the offline space we then choose the smallest $M^{\omega}_{\text{off}}$ eigenvalues from Eq.~\eqref{offeig} and form the corresponding eigenvectors in the space of snapshots by setting
$\psi_k^{\text{off}} = \sum_j \Psi_{kj}^{\text{off}} \psi_j^{\text{snap}}$ (for $k=1,\ldots, M^{\omega}_{\text{off}}$), where $\Psi_{kj}^{\text{off}}$ are the coordinates of the vector $\Psi_{k}^{\text{off}}$.
\subsection{Global coupling}
\label{globcoupling}
In this section we create an appropriate solution space and variational formulation that for a continuous Galerkin approximation of Eq.~\eqref{eq:original}. We begin with an initial coarse space $V^{\text{init}}_0 = \text{span}\{ \chi_i \}_{i=1}^{N}$, where the $\chi_i$ are the standard multiscale partition of unity functions defined by
\begin{eqnarray} \label{pou}
-\text{div} \left( \kappa(x) \, \nabla \chi_i \right) = 0 \quad K \in \omega_i \\
\chi_i = g_i \quad \text{on} \, \, \, \partial K, \nonumber
\end{eqnarray}
for all $K \in \omega_i$, where $g_i$ is assumed to be linear. We note that the summed, pointwise energy $\widetilde{\kappa}$ required for the eigenvalue problems will be defined as
\begin{equation*}
\widetilde{\kappa} = \kappa \sum_{i=1}^{N} H^2 | \nabla \chi_i |^2,
\end{equation*}
where $H$ denotes the coarse mesh size.
We then multiply the partition of unity functions by the eigenfunctions in the offline space $V_{\text{off}}^{\omega_i}$ to construct the resulting basis functions
\begin{equation} \label{cgbasis}
\psi_{i,k} = \chi_i \psi_k^{\omega_i, \text{off}} \quad \text{for} \, \, \,
1 \leq i \leq N \, \, \, \text{and} \, \, \, 1 \leq k \leq M_{\text{off}}^{\omega_i},
\end{equation}
where $M_{\text{off}}^{\omega_i}$ denotes the number of offline eigenvectors that are chosen for each coarse node $i$. We note that the construction in Eq.~\eqref{cgbasis} yields continuous basis functions due to the multiplication of offline eigenvectors with the initial (continuous) partition of unity. Next, we define the continuous Galerkin spectral multiscale space as
\begin{equation} \label{cgspace}
V_{\text{off}} = \text{span} \{ \psi_{i,k} : \, \, 1 \leq i \leq N \, \, \, \text{and} \, \, \, 1 \leq k \leq M_{\text{off}}^{\omega_i} \}.
\end{equation}
Using a single index notation, we may write $V_{\text{off}} = \text{span} \{ \psi_{i} \}_{i=1}^{N_c}$, where $N_c$ denotes the total number of basis functions that are used in the coarse space construction. We also construct an operator matrix $R_0^T = \left[ \psi_1 , \ldots, \psi_{N_c} \right]$ (where $\psi_i$ are used to denote the nodal values of each basis function defined on the fine grid), for later use in this subsection.
We seek $u_{\text{ms}}(x) = \sum_i c_i \psi_i(x) \in V_{\text{off}}$ such that
\begin{equation} \label{cgvarform}
a(u_{\text{ms}}, v) = (f, v) \quad \text{for all} \,\,\, v \in V_{\text{off}},
\end{equation}
where
$ \displaystyle a(u, v) = \int_D \kappa(x) \nabla u \cdot \nabla v \, dx$, and $ \displaystyle (f,v) = \int_D f v \, dx$. We note that variational form in \eqref{cgvarform} yields the following linear algebraic system
\begin{equation}
A_0 U_0 = F_0,
\end{equation}
where $U_0$ denotes the nodal values of the discrete CG solution, and
\begin{equation*}
A_0 = [a_{IJ}] = \int_D \kappa(x) \, \nabla \psi_I \cdot \nabla \psi_J \, dx \quad \text{and} \quad F_0 = [f_I] = \int_D f \psi_I \, dx.
\end{equation*}
Using the operator matrix $R_0^T$, we may write $A_0 = R_0 A R_0^T$ and $F_0 = R_0 F$, where $A$ and $F$ are the standard, fine scale stiffness matrix and forcing vector corresponding to the form in Eq.~\eqref{cgvarform}. We also note that the operator matrix may be analogously used in order to project coarse scale solutions onto the fine grid.
\section{A-posteriori error estimate and adaptive enrichment}
\label{sec:errorindicator}
In this section, we will derive an a-posteriori error indicator
for the error $u-u_{\text{ms}}$ in energy norm.
We will then use the error indicator
to develop an adaptive enrichment algorithm.
The a-posteriori error indicator
gives an estimate of the local error on the coarse grid regions $\omega_i$,
and we can then add basis functions to improve the solution.
We will give two kinds of error indicators,
one is based on the $L^2$-norm of the local residual
and the other is based on the weighted
$H^{-1}$-norm of the local residual (for simplicity, we will also call it
$H^{-1}$-norm based indicator).
The $L^2$-norm residual is also used in the classical adaptive finite element method.
In our case, this type of error indicator works well when the coefficient
does not contain high contrast region.
We will provide a quantitative explanation for this in the next section.
On the other hand, the $H^{-1}$-norm based residual
gives a more robust error indicator
which works well for cases with high contrast media.
This section is devoted to the derivation of the a-posteriori error indicator
and the corresponding adaptive enrichment algorithm.
The convergence analysis of the method will be given in the next section.
Let $V$ be the fine scale finite element space. We recall that the fine scale solution $u$ satisfies
\begin{equation}
a(u, v) = (f, v) \quad \text{for all} \,\,\, v \in V
\label{eq:fine}
\end{equation}
and the multiscale solution $u_{\text{ms}}$ satisfies
\begin{equation}
a(u_{\text{ms}}, v) = (f, v) \quad \text{for all} \,\,\, v \in V_{\text{off}}.
\label{eq:coarse}
\end{equation}
We remark that
$V_\text{off} \subset V$.
Next we will give the definitions of the $L^2$-based and $H^{-1}$-based residuals.
{\bf $L^2$-based residual}:
Let $\omega_i$ be a coarse grid region. We define a linear functional $Q_i(v)$ on $L^2(\omega_i)$ by
\begin{equation}
Q_i(v) = \int_{\omega_i} fv\chi_i - \int_{\omega_i} a\nabla u_{\text{ms}}\cdot \nabla (v\chi_i).
\end{equation}
The norm of $Q_i$ is defined as
\begin{equation}
\| Q_i \| = \sup_{v\in L^2(\omega_i)} \frac{ |Q_i(v)| }{\| v\|_{L^2(\omega_i)}}.
\end{equation}
The norm $\|Q_i\|$ gives an estimate on the size of error.
{\bf $H^{-1}$-based residual}:
Let $\omega_i$ be a coarse grid region and let $V_i = H^1_0(\omega_i)$.
We define a linear functional $R_i(v)$ on $V_i$ by
\begin{equation}
R_i(v) = \int_{\omega_i} fv - \int_{\omega_i} a\nabla u_{\text{ms}}\cdot \nabla v.
\end{equation}
The norm of $R_i$ is defined as
\begin{equation}
\| R_i \|_{V_i^*} = \sup_{v\in V_i} \frac{ |R_i(v)| }{\| v\|_{V_i}}.
\end{equation}
where $\|v\|_{V_i} = a(v,v)^{\frac{1}{2}}$.
The norm $\|R_i\|_{V_i^*}$ gives an estimate on the size of error.
To simplify notations, we let $l_i = M_{\text{off}}^{\omega_i}$.
We recall that, for each $\omega_i$,
the eigenfunctions corresponding to $\lambda_1^{\omega_i}, \cdots, \lambda_{l_i}^{\omega_i}$
are used in the construction of $V_{\text{off}}$.
We also define $\widetilde{\kappa}_i = \min_{x\in \omega_i} \widetilde{\kappa}(x)$.
In the next section, we will prove the following theorem.
\begin{theorem}
Let $u$ and $u_{\text{ms}}$ be the solutions of \eqref{eq:fine} and \eqref{eq:coarse} respectively. Then
\begin{eqnarray}
\| u-u_{\text{ms}}\|_V^2 &\leq& C_{\text{err}}\sum_{i=1}^N \|Q_i\|^2 ( \widetilde{\kappa}_i \lambda^{\omega_i}_{l_i+1})^{-1}, \label{eq:res1} \\
\| u-u_{\text{ms}}\|_V^2 &\leq& C_{\text{err}} \sum_{i=1}^N \|R_i\|^2_{V_i^*} (\lambda^{\omega_i}_{l_i+1})^{-1}. \label{eq:res2}
\end{eqnarray}
\label{thm:post}
where $C_{\text{err}}$ is a uniform constant.
\end{theorem}
From \eqref{eq:res1} and \eqref{eq:res2}, we see that
the norms $\|Q_i\|$ and $\|R_i\|_{V_i^*}$
give indications on the size of the energy norm error $\| u-u_{\text{ms}}\|_V$.
Even though \eqref{eq:res1} and \eqref{eq:res2}
have the same form, we emphasize that
they give different convergence behavior
in the high contrast case.
We will now present the adaptive enrichment algorithm.
We use $m \geq 1$ to represent the enrichment level
and $V_{\text{off}}^m$ be the solution space at level $m$.
For each coarse region, we use $l_i^m$
be the number of eigenfunctions used at the enrichment level $m$
for the coarse region $\omega_i$.
{\bf Adaptive enrichment algorithm}: Choose $0 < \theta < 1$.
For each $m=1,2, \cdots$,
\begin{enumerate}
\item[Step 1:] Find the solution in the current space. That is,
find $u_{\text{ms}}^m \in V^m_{\text{off}}$ such that
\begin{equation}
a(u^m_{\text{ms}}, v) = (f, v) \quad \text{for all} \,\,\, v \in V^m_{\text{off}}.
\label{eq:solve}
\end{equation}
\item[Step 2:] Compute the local residual. For each coarse region $\omega_i$, we compute
\begin{equation*}
\eta^2_i =
\begin{cases}
& \|Q_i\|^2 (\widetilde{\kappa}_i \lambda^{\omega_i}_{l^m_i+1})^{-1},\quad \text{ for } L^2\text{-based residual} \\
& \|R_i\|^2_{V_i^*} (\lambda^{\omega_i}_{l^m_i+1})^{-1},\quad \text{ for } H^{-1}\text{-based residual}
\end{cases}
\end{equation*}
And we re-enumerate them in the decreasing order, that is, $\eta^2_1 \geq \eta^2_2 \geq \cdots \geq \eta^2_N$.
\item[Step 3:] Find the coarse region where enrichment is needed. We choose the smallest integer $k$ such that
\begin{equation}
\theta \sum_{i=1}^N \eta_i^2 \leq \sum_{i=1}^k \eta_i^2.
\label{eq:criteria}
\end{equation}
\item[Step 4:] Enrich the space. For each $i=1,2,\cdots, k$, we add basis function
for the region $\omega_i$ according to the following rule.
Let $s$ be the smallest positive integer such that
$\lambda_{l_i^m+s+1}$ is large enough (see the proof of Theorem \ref{thm:conv}) compared with $\lambda_{l_i^m+1}$.
Then
we include
the eigenfunctions $\Psi^{\text{off}}_{l_i^m+1}, \cdots, \Psi^{\text{off}}_{l_i^m+s}$
in the construction of the basis functions.
The resulting space is denoted as $V_{\text{off}}^{m+1}$.
\end{enumerate}
We remark that the choice of $s$ above will ensure the convergence of the enrichment algorithm,
and in practice, the value of $s$ is easy to obtain.
Moreover, contrary to classical adaptive refinement methods,
the total number of basis functions that we can add
is bounded by the dimension of the snapshot space.
Thus, the condition \eqref{eq:criteria} can be modified as follows.
We choose the smallest integer $k$ such that
\begin{equation*}
\theta \sum_{i=1}^N \eta_i^2 \leq \sum_{i\in I} \eta_i^2
\end{equation*}
where the index set $I$ is a subset of $\{ 1,2, \cdots, k\}$
and contains indices $j$ such that $l_j^m$
is less than the maximum number of eigenfunctions for the region $\omega_j$.
We now describe how the norms $\|Q_i\|$ and $\|R_i\|_{V_i^*}$ are computed.
Let $W_i$ be the diagonal matrix containing the nodal values of the fine grid cut-off function $\chi_i$ in the diagonal. Then
the norm $\|Q_i\|$ can be computed as
\begin{equation}
\label{eq:normQ}
\|Q_i\| = \| W_i AR_0^T U_0 \|.
\end{equation}
According to the Riez representation theorem, the norm $\|R_i\|_{V_i^*}$ can be computed as follows.
Let $z_i$ be the solution of
\begin{equation}\label{eq:loc_Dirichlet}
\int_{\omega_i} a\nabla z_i \cdot \nabla v = R_i(v), \quad \text{for all} \,\,\, v \in V_i.
\end{equation}
Then we have $\|R_i\|_{V_i^*} = \| z_i \|_{V_i}$.
Thus, to find the norm $\|R_i\|_{V_i^*}$,
we need to solve a local problem on each coarse region $\omega_i$.
Finally, we state the convergence theorem.
\begin{theorem}
There are positive constants $\tau, \delta, \rho, L_1$ and $L_2$ such that the following contracting property holds
\begin{equation*}
\| u-u_{\text{ms}}^{m+1}\|_V^2 + \frac{\tau}{1+\tau \delta L_2} \sum_{i=1}^N S_{m+1}(\omega_i)^2
\leq \varepsilon \Big( \|u-u_{\text{ms}}^m\|_V^2 + \frac{\tau}{1+\tau\delta L_2} \sum_{i=1}^N S_{m}(\omega_i)^2 \Big).
\end{equation*}
\label{thm:conv}
Note that $0 < \varepsilon < 1$ and
\begin{equation*}
\varepsilon = \max( 1- \frac{\theta^2}{L_1(1+\tau \delta L_2)}, \frac{2C_{\text{err}}}{\tau L_1}+\rho).
\end{equation*}
\end{theorem}
We remark that the precise definitions of the constants
$\tau, \delta, \rho, L_1$ and $L_2$
are given in Section \ref{sec:analysis}.
\section{Numerical Results}
\label{sec:numresults}
In this section, we will present some numerical experiments to show
the performance of the error indicators and the adaptive enrichment algorithm.
We take the domain $\Omega$ as a square,
set the forcing term $f=1$ and use a linear boundary condition for the problem \eqref{eq:original}.
In our numerical simulations, we use a $20 \times 20$ coarse grid,
and each coarse grid block is divided into $5\times 5$ fine grid blocks.
Thus, the whole computational domain is partitioned by a $100 \times 100$ fine grid.
We assume that the fine-scale solution is obtained
by discretizing \eqref{eq:original} by the classical conforming piecewise bilinear elements on the fine grid.
To test the performance of our algorithm, we consider two
permeability fields $\kappa$ as depicted in Figure \ref{fig:perms}.
We obtain similar numerical results for these cases, and therefore
we will only demonstrate the numerical results for the first permeability field (Figure \ref{fig:perm_HCC}).
Below, we list the indicators used in our simulations.
In particular, we will recall the definitions of the $L^2$-based and $H^{-1}$-based error indicators.
For comparison purpose, we also use an indicator computed by the exact error in energy norm.
We remark that the indicators are computed for each coarse neighborhood $\omega_i$ and are defined as follows.
\begin{itemize}
\item
The indicator constructed using the weighted $H^{-1}$-based residual is
\begin{equation}\label{eq:indicator_Numerical}
\eta^{\mbox{\scriptsize{En}}}_{\omega_i}= \|R_i\|^2_{V_i^*} (\lambda_{l^m_i+1}^{\omega_i})^{-1}
\end{equation}
and we name it the proposed indicator.
\item
The indicator constructed using the $L^{2}$-based residual is
\begin{equation}\label{eq:indicator_Numerical_l2}
\eta^{\mbox{\scriptsize{L2}}}_{\omega_i}= \|Q_i\|^2 (\widetilde{\kappa}_i \lambda^{\omega_i}_{l^m_i+1})^{-1}
\end{equation}
and we name it the $L^2$ indicator.
\item
The indicator constructed using the exact energy error is
\begin{align}\label{eq:indicator_cmp}
\eta^{\mbox{\scriptsize{Ex}}}_{\omega_i}=\norm{u-u_{\text{ms}}}_{V_i}^{2}
\end{align}
and name it the exact indicator.
\end{itemize}
We recall that, in the above definitions, the norms $\|Q_i\|$ and $\|R_i\|_{V_i^*}$
are computed in the way described in \eqref{eq:normQ} and \eqref{eq:loc_Dirichlet} respectively.
For each enrichment level, we will
compute the multiscale solution (Step 1) and the corresponding error indicators (Step 2).
The indicators
$\eta^{\mbox{\scriptsize{Ex}}}_{\omega_i}$, $\eta^{\mbox{\scriptsize{En}}}_{\omega_i}$ and $\eta^{\mbox{\scriptsize{L2}}}_{\omega_i}$
are then ordered in decreasing order.
To enrich the approximation space,
we select a few coarse neighborhoods such that (\ref{eq:criteria}) holds for a specific value of $\theta$ (Step 3).
In our simulations, we consider $\theta=0.7$ and $0.2$.
Finally, for selected coarse neighborhoods, we will enrich the offline space
by adding more basis functions (Step 4).
\begin{figure}[htb]
\centering
\subfigure[Permeability field 1]{\label{fig:perm_HCC}
\includegraphics[width = 0.40\textwidth, keepaspectratio = true]{perm_HCC.eps}
}
%
\subfigure[Permeability field 2]{\label{fig:newperm2_cross2}
\includegraphics[width = 0.40\textwidth, keepaspectratio = true]{newperm2_cross2.eps}
}
\caption{Permeability fields}
\label{fig:perms}
\end{figure}
We will consider two types of snapshot spaces,
namely the space spanned by all $\kappa$-harmonic extensions and
the space spanned by all fine-scale conforming piecewise bilinear functions.
The sequence of offline basis functions is then obtained by
solving the local spectral problem \eqref{offeig}
on the space of snapshots.
We will call the first type of basis functions as harmonic basis
and the second type of basis functions as spectral basis.
In addition, we use the notations $\eta^{\mbox{\scriptsize{H,En}}}_{\omega_i}$, $\eta^{\mbox{\scriptsize{H,L2}}}_{\omega_i}$
and $\eta^{\mbox{\scriptsize{H,Ex}}}_{\omega_i}$
to denote the $H^{-1}$-based, $L^2$-based and exact error indicators
for the case when the offline space is formed by harmonic basis.
Similarly, we use the notations $\eta^{\mbox{\scriptsize{U,En}}}_{\omega_i}$, $\eta^{\mbox{\scriptsize{U,L2}}}_{\omega_i}$
and $\eta^{\mbox{\scriptsize{U,Ex}}}_{\omega_i}$
to denote the $H^{-1}$-based, $L^2$-based and exact error indicators
for the case when the offline space is formed by spectral basis
(here, superscript $U$ stands for the fact that the snapshot space
consists of all fine-grid {\it unit} vectors).
In the following, we summarize the numerical examples we considered in this paper.
\begin{itemize}
\item {\bf Numerical results with harmonic basis (see Section \ref{sec:621}).}
We will present
numerical results to test the performance of the error indicator
$\eta^{\mbox{\scriptsize{H, En}}}_{\omega_i}$ and the adaptive enrichment algorithm
with $\theta=0.7$ and $\theta=0.2$. We also compare our results with the use of $\eta^{\mbox{\scriptsize{H,Ex}}}_{\omega_i}$
with $\theta=0.7$.
\item {\bf Numerical results with spectral basis (see Section \ref{sec:622}).}
We will present
numerical results to test the performance of the error indicator
$\eta^{\mbox{\scriptsize{U, En}}}_{\omega_i}$ and the adaptive enrichment algorithm
with $\theta=0.7$ and $\theta=0.2$. We also compare our results with the use of $\eta^{\mbox{\scriptsize{U,Ex}}}_{\omega_i}$
with $\theta=0.7$.
\item {\bf Numerical results with $L^2$ indicator (see Section \ref{sec:623}).}
We will present
numerical results to test the performance of the error indicator
$\eta^{\mbox{\scriptsize{H,L2}}}_{\omega_i}$ and the adaptive enrichment algorithm with $\theta=0.7$.
\item {\bf Numerical results when the proposed indicator is computed in the snapshot space (see Section \ref{sec:624}).}
We will present
numerical results to test the performance of the error indicator
$\eta^{\mbox{\scriptsize{H,En}}}_{\omega_i}$ and the adaptive enrichment algorithm with $\theta=0.7$.
In this case, the norm $\|R_i\|_{V_i^*}$ is computed in the snapshot space instead of the fine-grid space.
\end{itemize}
In the following,
we will give a brief summary of our conclusions before discussing the numerical results.
\begin{itemize}
\item The use of both $\eta^{\mbox{\scriptsize{H, En}}}_{\omega_i}$
and $\eta^{\mbox{\scriptsize{U, En}}}_{\omega_i}$
gives a convergent sequence of numerical solutions. This verfies
the convergence of our adaptive GMsFEM.
\item
The performance of the proposed indicators $\eta^{\mbox{\scriptsize{H, En}}}_{\omega_i}$
and $\eta^{\mbox{\scriptsize{U, En}}}_{\omega_i}$
is similar to that of the exact indicators $\eta^{\mbox{\scriptsize{H,Ex}}}_{\omega_i}$ and $\eta^{\mbox{\scriptsize{U,Ex}}}_{\omega_i}$.
Thus, the proposed indicator gives a good estimate of the exact error.
\item The performance of the weighted $H^{-1}$-based indicator
is much better than that of the $L^2$-based indicator for high-contrast
problems.
\item
The use of the snapshot space to compute $\eta^{\mbox{\scriptsize{H, En}}}_{\omega_i}$
and $\eta^{\mbox{\scriptsize{U, En}}}_{\omega_i}$
in \eqref{eq:loc_Dirichlet} gives similar results
compared to the use of local fine-grid solves.
Thus, the computations of $\eta^{\mbox{\scriptsize{H, En}}}_{\omega_i}$
and $\eta^{\mbox{\scriptsize{U, En}}}_{\omega_i}$
can be performed efficiently.
\item With the use of $\theta=0.2$, we obtain more accurate results for the same
dimensional offline spaces compared with $\theta=0.7$.
\end{itemize}
In the tables listed below, we recall that
$V_{\text{off}}$ denotes the offline space; $u$, $u_{\text{snap}}$ and $u_{\text{off}}$ denote the fine-scale, snapshot and offline solutions respectively.
Moreover, to compare the results, we will
compute the error $u-u_{\text{off}}$ using
the $L^2$ relative error and the energy relative error, which are defined as
\begin{equation}
\| u-u_{\text{off}}\|_{L^2_{\kappa}(D)} := \frac{ \| u - u_{\text{off}} \|_{L^2(V)} }{ \| u \|_{L^2(V)} }, \quad\quad
\| u-u_{\text{off}}\|_{H^1_{\kappa}(D)} := \frac{ a(u-u_{\text{off}}, u-u_{\text{off}})^{\frac{1}{2}} }{ a(u,u)^{\frac{1}{2}} }
\end{equation}
where the weighted $L^2$-norm is defined as $\|u\|_{L^2(V)} = \| \widetilde{\kappa}^{\frac{1}{2}} u \|_{L^2(D)}$.
We will also compute the error $u_{\text{snap}}-u_{\text{off}}$ using the same norms
\begin{equation}
\| u_{\text{snap}}-u_{\text{off}}\|_{L^2_{\kappa}(D)} := \frac{ \| u_{\text{snap}} - u_{\text{off}} \|_{L^2(V)} }{ \| u_{\text{snap}} \|_{L^2(V)} }, \quad\quad
\| u_{\text{snap}}-u_{\text{off}}\|_{H^1_{\kappa}(D)} := \frac{ a(u_{\text{snap}}-u_{\text{off}}, u-u_{\text{off}})^{\frac{1}{2}} }{ a(u_{\text{snap}},u_{\text{snap}})^{\frac{1}{2}} }.
\end{equation}
\subsection{Numerical results with harmonic basis}
\label{sec:621}
In this section, we present numerical examples to test the performance of the proposed indicator $\eta^{\mbox{\scriptsize{H,En}}}_{\omega_i}$
and the convergence of our adaptive enrichment algorithm with $\theta=0.7$ and $\theta=0.2$.
We will also compare our results with the use of the exact indicator $\eta^{\mbox{\scriptsize{H,Ex}}}_{\omega_i}$.
In the simulations, we take a snapshot space of dimension $7300$
giving errors of $0.05\%$ and $3.02\%$ in weighted $L^2$ and weighted $H^1$ norms, respectively.
Thus, the solution $u_{\text{snap}}$ is as good as the fine-scale solution $u$.
For the adaptive enrichment algorithm, the initial offline space has $4$
basis functions for each coarse grid node.
At each enrichment (Step 4), we will add one basis function for the coarse grid nodes
selected in Step 3.
We will terminate the iteration when the energy error $\| u - u_{\text{off}}\|_V$
is less than $5\%$ of $\| u - u_{\text{snap}}\|_V$.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$\text{dim}(V_{\text{off}})$} &
\multicolumn{2}{c|}{ $\|u-u_{\text{off}} \|$ (\%) } &
\multicolumn{2}{c|}{ $\|u_{\text{snap}}-u_{\text{off}} \|$ (\%) }\\
\cline{2-5} {}&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$\\
\hline\hline
$1524$ & $4.50$ & $31.29$ & $4.49$ & $34.14$ \\
\hline
$1711$ & $4.24$ & $27.37$ & $4.23$ & $27.19$\\
\hline
$2434$ & $2.34$ & $20.13$ & $2.36$ & $20.31$\\
\hline
$2637$ & $1.64$ & $15.43$ & $1.61$ & $15.13$\\
\hline
$3378$ & $0.54$ &$7.83$ & $0.51$ & $7.22$\\
\hline
\end{tabular}
\caption{Convergence history for harmonic basis with $\theta=0.7$ and
$18$ iterations. The snapshot space has dimension $7300$ giving $0.05\%$ and $3.02\%$ weighted $L^2$ and weighted energy errors. When using 12 basis per coarse inner node, the weighted $L^2$ and the weighted $H^1$ errors will be $2.34\%$ and $19.77\%$, respectively, and the dimension of offline space is 4412.}
\label{table:Cross_theta.7_Harmonic}
\end{table}
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$\text{dim}(V_{\text{off}})$} &
\multicolumn{2}{c|}{ $\|u-u_{\text{off}} \|$ (\%) } &
\multicolumn{2}{c|}{ $\|u_{\text{snap}}-u_{\text{off}} \|$ (\%) } \\
\cline{2-5} {}&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$
\\
\hline\hline
$1524$ & $4.50$ & $31.29$ & $4.49$ & $34.14$ \\
\hline
$1646$ & $4.05$ & $26.80$ & $4.04$ & $26.62$\\
\hline
$1864$ & $3.09$ & $20.34$ & $3.07$ & $20.11$\\
\hline
$2220$ & $1.24$ & $14.43$ & $1.20$ & $14.11$\\
\hline
$3135$ & $0.48$ &$7.98$ & $0.45$ & $7.39$\\
\hline
\end{tabular}
\caption{Convergence history for harmonic basis with $\theta=0.7$ and
$66$ iterations.
The number of iterations is $19$. The snapshot space has dimension $7300$ giving $0.05\%$ and $3.02\%$ weighted $L^2$ and weighted energy errors.
When using 12 basis per coarse inner node, the weighted $L^2$ and the weighted $H^1$ errors will be $2.34\%$ and $19.77\%$, respectively, and the dimension of offline space is 4412.}
\label{table:Cross_theta.2_Harmonic}
\end{table}
In Table \ref{table:Cross_theta.7_Harmonic} and Table \ref{table:Cross_theta.2_Harmonic},
we present the convergence history of the adaptive enrichment algorithm
for $\theta=0.7$ and $\theta=0.2$ respectively.
For both cases, we see a convergence of the algorithm.
For the case $\theta=0.7$, the algorithm requires $18$ iterations to achieve the desired accuracy.
The dimension of the corresponding offline space is $3378$.
Moreover, the error $u-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.54\%$ and $7.83\%$ respectively, while the
error $u_{\text{snap}}-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.51\%$ and $7.22\%$ respectively.
And we see the similarity of the errors $u-u_{\text{off}}$ and $u_{\text{snap}}-u_{\text{off}}$.
For the case $\theta=0.2$, the algorithm requires $66$ iterations to achieve the desired accuracy.
The dimension of the corresponding offline space is $3135$.
Moreover, the error $u-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.48\%$ and $7.98\%$ respectively, while the
error $u_{\text{snap}}-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.45\%$ and $7.39\%$ respectively.
Furthermore, we observe that the use of $\theta=0.2$
gives the same level of error for a smaller offline space compared with $\theta=0.7$.
Thus, we conclude that a smaller value of $\theta$ will give a more economical offline space.
To show that our adaptive enrichment algorithm gives a more efficient scheme,
we report some computational results with uniform enrichment.
In this case, we use $12$ basis functions for each interior coarse grid node giving
an offline space of dimension $4412$.
The relative weighted $L^2$ and energy errors are $2.32\%$ and $19.53\%$ respectively.
From this result, we see that our adaptive enrichment algorithm
gives a smaller offline space and at the same time a better accuracy
than a scheme with uniform number of basis functions.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$\text{dim}(V_{\text{off}})$} &
\multicolumn{2}{c|}{ $\|u-u^{\text{off}} \|$ (\%) } &
\multicolumn{2}{c|}{ $\|u^{\text{snap}}-u^{\text{off}} \|$ (\%) } \\
\cline{2-5} {}&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$
\\
\hline\hline
$1524$ & $4.50$ & $31.29$ & $4.49$ & $34.14$ \\
\hline
$1762$ & $3.96$ & $27.09$ & $3.95$ & $26.91$\\
\hline
$2333$ & $2.07$ & $19.00$ & $2.04$ & $18.75$\\
\hline
$2522$ & $1.38$ & $15.12$ & $1.36$ & $14.81$\\
\hline
$3466$ & $0.46$ &$7.52$ & $0.44$ & $6.89$\\
\hline
\end{tabular}
\caption{Convergence history for harmonic basis with $\theta=0.7$ and the exact indicator.
The number of iterations is $23$. The snapshot space has dimension $7300$ giving $0.05\%$ and $3.02\%$ weighted $L^2$ and weighted energy errors.
}
\label{table:Cross_theta.7_Harmonic_Uniform}
\end{table}
To test the reliability and efficiency of the proposed indicator, we apply the adaptive enrichment algorithm
with the exact energy error as indicator and $\theta=0.7$. The results are shown
in Table \ref{table:Cross_theta.7_Harmonic_Uniform}.
In particular, the algorithm requires $19$ iterations to achieve the desired accuracy.
The dimension of the corresponding offline space is $3466$.
Moreover, the error $u-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.46\%$ and $7.52\%$ respectively,
while the
error $u_{\text{snap}}-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.44\%$ and $6.89\%$ respectively.
Comparing the results in Table \ref{table:Cross_theta.7_Harmonic} and Table \ref{table:Cross_theta.7_Harmonic_Uniform}
for the use of the proposed and the exact indicator respectively,
we see that both indicators give similar convergence behavior and offline space dimensions.
\begin{figure}[htb]
\centering
\subfigure[Proposed indicator with $\theta=0.7$]{\label{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross}
\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{HarmonicBasis_NoOfBasisPerCoarseNode_Cross.eps}
}
\subfigure[Proposed indicator with $\theta=0.2$]{\label{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross_theta.2}
\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{HarmonicBasis_NoOfBasisPerCoarseNode_Cross_theta.2.eps}
}
\subfigure[Exact indicator with $\theta=0.7$]{\label{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross_Uniform}
\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{HarmonicBasis_NoOfBasisPerCoarseNode_Cross_Uniform.eps}
}
\caption{Dimension distributions of the last offline space for harmonic basis with permeability field \ref{fig:newperm2_cross2}.}
\label{fig:BasisPerNode_Harmonic_Cross}
\end{figure}
In Figure \ref{fig:BasisPerNode_Harmonic_Cross}, we display the number of basis functions for each coarse grid node
of the last offline spaces
for the proposed indicator with $\theta=0.7$, the proposed indicator with $\theta=0.2$ and the exact indicator with $\theta=0.7$.
From Figures \ref{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross} and \ref{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross_theta.2},
we observe a similar dimension distribution for the use of the proposed indicator with $\theta=0.7$ and $\theta=0.2$,
and the case $\theta=0.2$ gives a smaller number of basis functions.
For the case with the exact indicator, we see from Figure \ref{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross_Uniform}
that the dimension distribution follows a similar pattern, but with regions that contain larger number of basis functions.
\begin{figure}[htb]
\centering
\subfigure[Proposed indicator with the last offline space]{\label{fig:Hm_ErrorDistri_Cross_larger}\includegraphics[width = 0.22\textwidth, keepaspectratio = true]{Hm_ErrorDistri_Cross_larger.eps}}
%
\subfigure[Proposed indicator with an intermediate offline space]{\label{fig:Hm_ErrorDistri_Cross_smaller}\includegraphics[width = 0.22\textwidth, keepaspectratio = true]{Hm_ErrorDistri_Cross_smaller.eps}}
%
\subfigure[Exact indicator with the last offline space]{\label{fig:Hm_ErrorDistri_Cross_U_larger}\includegraphics[width = 0.22\textwidth, keepaspectratio = true]{Hm_ErrorDistri_Cross_U_larger.eps}}
%
\subfigure[Exact indicator with an intermediate offline space]{\label{fig:Hm_ErrorDistri_Cross_U_smaller}\includegraphics[width = 0.22\textwidth, keepaspectratio = true]{Hm_ErrorDistri_Cross_U_smaller.eps}}
%
\caption{Coarse-grid energy error distribution using harmonic basis with permeability field \ref{fig:newperm2_cross2}.}
\label{fig:ErrorDistri_Hm_Cross}
\end{figure}
Finally,
we present the energy errors on coarse neighborhoods
for $\theta=0.7$ for an intermediate offline space and the last offline space of the
proposed indicator $\eta^{\mbox{\scriptsize{H,En}}}_{\omega_i}$ and the exact indicator $\eta^{\mbox{\scriptsize{H,Ex}}}_{\omega_i}$.
In Figures \ref{fig:Hm_ErrorDistri_Cross_larger} and \ref{fig:Hm_ErrorDistri_Cross_smaller},
the energy error distributions for the last offline spaces and an intermediate offline space
obtained by the proposed indicator are shown respectively.
We see how the energy error is reduced by enriching the space
from an intermediate step to the final step.
A similar situation is also seen from Figures \ref{fig:Hm_ErrorDistri_Cross_U_larger} and \ref{fig:Hm_ErrorDistri_Cross_U_smaller}
for the case with the exact indicator.
\subsection{Numerical results with spectral basis}
\label{sec:622}
In this section,
we repeat the above tests using the spectral snapshot space instead of the harmonic snapshot space with the proposed indicator $\eta^{\mbox{\scriptsize{U,En}}}_{\omega_i}$ and the exact indicator $\eta^{\mbox{\scriptsize{U,Ex}}}_{\omega_i}$. The results are presented in Tables \ref{table:Cross_theta.7_Spectral}, \ref{table:Cross_theta.2_Spectral} and \ref{table:Cross_theta.7_Spectral_Uniform}.
In the simulations, we take a snapshot space of dimension $3690$
giving errors of $0.01\%$ and $2.84\%$ in weighted $L^2$ and energy norms respectively.
Thus, the solution $u_{\text{snap}}$ is as good as the fine-scale solution $u$.
For the adaptive enrichment algorithm, the initial offline space has $2$
basis functions for each coarse grid node.
At each enrichment (Step 4), we will add one basis function for the coarse grid nodes
selected in Step 3.
We will terminate the iteration when the energy error $\| u - u_{\text{off}}\|_V$
is less than $5\%$ of $\| u - u_{\text{snap}}\|_V$.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$\text{dim}(V_{\text{off}})$} &
\multicolumn{2}{c|}{ $\|u-u^{\text{off}} \|$ (\%) } &
\multicolumn{2}{c|}{ $\|u^{\text{snap}}-u^{\text{off}} \|$ (\%) } \\
\cline{2-5} {}&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$
\\
\hline\hline
$802$ & $0.87$ & $20.15$ & $0.87$ & $19.94$ \\
\hline
$868$ & $0.83$ & $16.51$ & $0.83$ & $16.26$\\
\hline
$979$ & $0.33$ & $12.62$ & $0.33$ & $12.30$ \\
\hline
$1106$ & $0.32$ & $10.44$ & $0.32$ & $10.05$\\
\hline
$1410$ & $0.10$ & $7.43$ & $0.10$ & $6.87$ \\
\hline
\end{tabular}
\caption{Convergence history for spectral basis with $\theta=0.7$ and $5$ iterations.
The snapshot space has dimension $3690$ giving $0.01\%$ and $2.84\%$ weighted $L^2$ and weighted energy errors. When using $5$ basis per interior coarse node, the weighted $L^2$ and the weighted energy errors will be $0.09\%$ and $7.40\%$, respectively, and the dimension of offline space is $1885$.}
\label{table:Cross_theta.7_Spectral}
\end{table}
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$\text{dim}(V_{\text{off}})$} &
\multicolumn{2}{c|}{ $\|u-u^{\text{off}} \|$ (\%) } &
\multicolumn{2}{c|}{ $\|u^{\text{snap}}-u^{\text{off}} \|$ (\%) } \\
\cline{2-5} {}&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$
\\
\hline\hline
$802$ & $0.87$ & $20.15$ & $0.87$ & $19.94$ \\
\hline
$856$ & $0.83$ & $16.25$ & $0.82$ & $16.00$\\
\hline
$960$ & $0.34$ & $12.58$ & $0.33$ & $12.26$ \\
\hline
$1116$ & $0.32$ & $10.27$ & $0.32$ & $9.87$\\
\hline
$1334$ & $0.09$ & $7.55$ & $0.09$ & $6.99$ \\
\hline
\end{tabular}
\caption{Convergence history for spectral basis with $\theta=0.7$ and $19$ iterations.
The snapshot space has dimension $3690$ giving $0.01\%$ and $2.84\%$ weighted $L^2$ and weighted energy errors. When using $5$ basis per interior coarse node, the weighted $L^2$ and the weighted energy errors will be $0.09\%$ and $7.40\%$, respectively, and the dimension of offline space is $1885$.}
\label{table:Cross_theta.2_Spectral}
\end{table}
In Table \ref{table:Cross_theta.7_Spectral} and Table \ref{table:Cross_theta.2_Spectral},
we present the convergence history of the adaptive enrichment algorithm
for $\theta=0.7$ and $\theta=0.2$ respectively.
For both cases, we see a clear convergence of the algorithm.
For the case $\theta=0.7$, the algorithm requires $5$ iterations to achieve the desired accuracy.
The dimension of the corresponding offline space is $1410$.
Moreover, the error $u-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.10\%$ and $7.43\%$ respectively, while the
error $u_{\text{snap}}-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.10\%$ and $6.87\%$ respectively.
For the case $\theta=0.2$, the algorithm requires $19$ iterations to achieve the desired accuracy.
The dimension of the corresponding offline space is $1334$.
Moreover, the error $u-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.09\%$ and $7.55\%$ respectively, while the
error $u_{\text{snap}}-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.09\%$ and $6.99\%$ respectively.
Furthermore, we observe that the use of $\theta=0.2$
gives the same level of error for a smaller offline space compared with $\theta=0.2$.
Thus, we conclude that a smaller value of $\theta$ will give a more economical offline space.
To show that our adaptive enrichment algorithm gives a more efficient scheme,
we report some computational results with uniform enrichment.
In this case, we use $5$ basis functions for each interior coarse grid node giving
an offline space of dimension $1885$.
The relative weighted $L^2$ and energy errors are $0.09\%$ and $7.40\%$ respectively.
From this result, we see that our adaptive enrichment algorithm
gives a smaller offline space and at the same time a better accuracy
than a scheme with uniform number of basis functions.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$\text{dim}(V_{\text{off}})$} &
\multicolumn{2}{c|}{ $\|u-u^{\text{off}} \|$ (\%) } &
\multicolumn{2}{c|}{ $\|u^{\text{snap}}-u^{\text{off}} \|$ (\%) } \\
\cline{2-5} {}&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$
\\
\hline\hline
$802$ & $0.87$ & $20.15$ & $0.87$ & $19.94$ \\
\hline
$884$ & $0.42$ & $14.73$ & $0.42$ & $14.45$ \\
\hline
$1000$ & $0.18$ & $12.25$ & $0.18$ & $11.91$\\
\hline
$1119$ & $0.17$ & $9.83$ & $0.17$ & $9.41$\\
\hline
$1392$ & $0.10$ & $7.12$ & $0.10$ & $6.53$ \\
\hline
\end{tabular}
\caption{Convergence history for spectral basis with $\theta=0.7$ and the exact indicator.
The number of iteration is $6$.
The snapshot space has dimension $3690$ giving $0.01\%$ and $2.84\%$ weighted $L^2$ and weighted energy errors. When using $5$ basis per interior coarse node, the weighted $L^2$ and the weighted energy errors will be $0.09\%$ and $7.40\%$, respectively, and the dimension of offline space is $1885$.
}
\label{table:Cross_theta.7_Spectral_Uniform}
\end{table}
To test the reliability and efficiency of the proposed indicator, we apply the adaptive enrichment algorithm
with the exact energy error as indicator and $\theta=0.7$. The results are shown
in Table \ref{table:Cross_theta.7_Spectral_Uniform}.
In particular, the algorithm requires $6$ iterations to achieve the desired accuracy.
The dimension of the corresponding offline space is $1392$.
Moreover, the error $u-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.10\%$ and $7.12\%$ respectively,
while the
error $u_{\text{snap}}-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.10\%$ and $6.53\%$ respectively.
Comparing the results in Table \ref{table:Cross_theta.7_Spectral} and Table \ref{table:Cross_theta.7_Spectral_Uniform}
for the use of the proposed and the exact indicator respectively,
we see that both indicators give similar convergence behavior and offline space dimensions.
We also observe that the exact indicator performs better
in this case.
\begin{figure}[htb]
\centering
\subfigure[Proposed indicator with $\theta=0.7$]{\label{fig:SpectralBasis_NoOfBasisPerCoarseNode_Cross}\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{SpectralBasis_NoOfBasisPerCoarseNode_Cross.eps}}
\subfigure[Proposed indicator with $\theta=0.2$]{\label{fig:SpectralBasis_NoOfBasisPerCoarseNode_Cross_theta.2}\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{SpectralBasis_NoOfBasisPerCoarseNode_Cross_theta.2.eps}}
\subfigure[Exact indicator with $\theta=0.7$]{\label{fig:SpectralBasis_NoOfBasisPerCoarseNode_Cross_Uniform}\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{SpectralBasis_NoOfBasisPerCoarseNode_Cross_Uniform.eps}}
\caption{Dimension distributions of the last offline space for spectral basis
with permeability field \ref{fig:newperm2_cross2}.}
\label{fig:BasisPerNode_spectral_Cross}
\end{figure}
In Figure \ref{fig:BasisPerNode_spectral_Cross}, we display the number of basis functions for each coarse grid node
of the last offline spaces
for the proposed indicator with $\theta=0.7$, the proposed indicator with $\theta=0.2$ and the exact indicator with $\theta=0.7$.
From Figures \ref{fig:SpectralBasis_NoOfBasisPerCoarseNode_Cross} and \ref{fig:SpectralBasis_NoOfBasisPerCoarseNode_Cross_theta.2},
we observe a similar dimension distribution for the use of the proposed indicator with $\theta=0.7$ and $\theta=0.2$,
and the case $\theta=0.2$ gives a smaller number of basis functions.
For the case with the exact indicator, we see from Figure \ref{fig:SpectralBasis_NoOfBasisPerCoarseNode_Cross_Uniform}
that the dimension distribution follows a similar pattern, but with regions that contain larger number of basis functions.
\begin{figure}[htb!]
\centering
\subfigure[Proposed indicator with the last offline space]{\label{fig:Spectral_ErrorDistri_Cross_larger}
\includegraphics[width = 0.22\textwidth, keepaspectratio = true]{Spectral_ErrorDistri_Cross_larger.eps}}
%
\subfigure[Proposed indicator with an intermediate offline space]{\label{fig:Spectral_ErrorDistri_Cross_smaller}\includegraphics[width = 0.22\textwidth, keepaspectratio = true]{Spectral_ErrorDistri_Cross_smaller.eps}}
%
\subfigure[Exact indicator with the last offline space]{\label{fig:Spectral_ErrorDistri_Cross_U_larger}
\includegraphics[width = 0.22\textwidth, keepaspectratio = true]{Spectral_ErrorDistri_Cross_U_larger.eps}}
%
\subfigure[Exact indicator with an intermediate offline space]{\label{fig:Spectral_ErrorDistri_Cross_U_smaller}
\includegraphics[width = 0.22\textwidth, keepaspectratio = true]{Spectral_ErrorDistri_Cross_U_smaller.eps}}
%
\caption{Coarse-grid energy error distribution using spectral basis with permeability field \ref{fig:newperm2_cross2}.}
\label{fig:ErrorDistri_Spectral_Cross}
\end{figure}
Finally,
we present the energy errors on coarse neighborhoods
for $\theta=0.7$ for an intermediate offline space and the last offline space of the
proposed indicator $\eta^{\mbox{\scriptsize{H,En}}}_{\omega_i}$ and the exact indicator $\eta^{\mbox{\scriptsize{H,Ex}}}_{\omega_i}$.
In Figures \ref{fig:Spectral_ErrorDistri_Cross_larger} and \ref{fig:Spectral_ErrorDistri_Cross_smaller},
the energy error distributions for the last offline spaces and an intermediate offline space
obtained by the proposed indicator are shown respectively.
We see clearly that how the energy error is reduced by enriching the space
from an intermediate step to the final step.
A similar situation is also seen from Figures \ref{fig:Spectral_ErrorDistri_Cross_U_larger} and \ref{fig:Spectral_ErrorDistri_Cross_U_smaller}
for the case with the exact indicator.
\subsection{Numerical results with the $L^2$ indicator}
\label{sec:623}
In this section, we present some numerical simulations to test the performance of the $L^2$ indicator.
We note that this is the most natural error indicator, as it is more efficient to compute and is
widely used for classical adaptive finite element methods \cite{AdaptiveFEM}.
However, this indicator does not work well for high contrast coefficients.
In the simulation, we will conduct the same test as in Section \ref{sec:621}
with the indicator replaced by $\eta^{\mbox{\scriptsize{H,L2}}}_{\omega_i}$.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$\text{dim}(V_{\text{off}})$} &
\multicolumn{2}{c|}{ $\|u-u^{\text{off}} \|$ (\%) } &
\multicolumn{2}{c|}{ $\|u^{\text{snap}}-u^{\text{off}} \|$ (\%) } \\
\cline{2-5} {}&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$
\\
\hline\hline
$1524$ & $4.50$ & $31.29$ & $4.49$ & $31.14$ \\
\hline
$1913$ & $3.59$ & $26.88$ & $3.57$ & $26.69$\\
\hline
$2513$ & $2.43$ & $21.46$ & $2.43$ & $20.89$\\
\hline
$3006$ & $1.11$ & $17.11$ & $1.12$ & $16.83$\\
\hline
$4509$ & $0.06$ &$7.97$ & $0.04$ & $7.37$\\
\hline
\end{tabular}
\caption{Convergence history for harmonic basis using the $L^2$ indicator with $\theta=0.7$ and $94$ iterations.
The snapshot space has dimension $7300$ giving $0.05\%$ and $3.02\%$ weighted $L^2$ and weighted energy errors.
When using $12$ basis per interior coarse node, the weighted $L^2$ and the weighted energy errors will be $2.34\%$ and $19.77\%$, respectively, and the dimension of offline space is $4412$.
}
\label{table:HCC_theta.7_Harmonic_nv}
\end{table}
In Table \ref{table:HCC_theta.7_Harmonic_nv},
we present the convergence history of the adaptive enrichment algorithm
for $\theta=0.7$, and
we observe a clear convergence of the algorithm.
Notice that, the algorithm requires $94$ iterations to achieve the desired accuracy.
The dimension of the corresponding offline space is $4509$.
If we compare these results to the case with the proposed indicator,
we see that the $L^2$ indicator
will give a much larger offline space and a larger number of iterations,
in order to achieve a similar accuracy.
\begin{figure}[htb]
\centering
\subfigure[Basis distribution for $L^2$ indicator]{\label{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross_nv}
\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{HarmonicBasis_NoOfBasisPerCoarseNode_Cross_nv.eps}
}
\subfigure[Energy error with the last offline space]{\label{fig:Hm_ErrorDistri_Cross_tem_larger_nv}
\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{Hm_ErrorDistri_Cross_tem_larger_nv.eps}
}
\subfigure[Energy error with an intermediate offline space]{\label{fig:Hm_ErrorDistri_Cross_tem_smaller_nv}
\includegraphics[width = 0.30\textwidth, keepaspectratio = true]{Hm_ErrorDistri_Cross_tem_smaller_nv.eps}
}
\caption{Basis distribution and error distribution for harmonic basis with $L^2$ indicator.}
\label{fig:BasisPerNode_Harmonic_Cross_nv}
\end{figure}
Finally we will compare the basis function and error distributions
for the $L^2$ indicator with those for the proposed indicator.
In Figure \ref{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross_nv},
the number of basis functions for each coarse node is shown.
We observe that the distribution is similar to the case with the proposed indicator
shown in Figure \ref{fig:HarmonicBasis_NoOfBasisPerCoarseNode_Cross}.
We also observe that the number of basis functions for the $L^2$ indicator
is much larger than that for the proposed indicator.
In Figures \ref{fig:Hm_ErrorDistri_Cross_tem_larger_nv} and \ref{fig:Hm_ErrorDistri_Cross_tem_smaller_nv},
the energy error distributions for the last offline spaces and an intermediate offline space
obtained by the $L^2$ indicator are shown respectively.
We see clearly that how the energy error is reduced by enriching the space
from an intermediate step to the final step.
However, we also see a very slow decay in energy error
for the $L^2$ indicator.
\subsection{Numerical results using snapshot solutions for the proposed indicator}
\label{sec:624}
In this section, we present numerical tests to show that
our adaptive method is equally good when the proposed indicator $\eta^{\mbox{\scriptsize{H,En}}}_{\omega_i}$
is computed in the snapshot space.
In particular, we will solve the local problems \eqref{eq:loc_Dirichlet} in the space of snapshots
instead of the fine scale space, in order to reduce the computational costs.
We will again repeat the same test as in Section \ref{sec:621}.
In Table \ref{table:HCC_theta.7_Harmonic_loc_snapshot}
we present the convergence history of the adaptive enrichment algorithm
with $\theta=0.7$, and observe
a clear convergence of the algorithm.
Moreover, the algorithm requires $22$ iterations to achieve the desired accuracy.
The dimension of the corresponding offline space is $3688$.
In addition, the error $u-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.17\%$ and $7.83\%$ respectively, while the
error $u_{\text{snap}}-u_{\text{off}}$ in relative weighted $L^2$ and energy norms
are $0.14\%$ and $7.26\%$ respectively.
If we compare these results with those for the proposed indicator (see Table \ref{table:Cross_theta.7_Harmonic}),
we see the use of snapshot space to compute the error indicator
will give a similar offline space and accuracy,
but with a larger number of iterations.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$\text{dim}(V_{\text{off}})$} &
\multicolumn{2}{c|}{ $\|u-u^{\text{off}} \|$ (\%) } &
\multicolumn{2}{c|}{ $\|u^{\text{snap}}-u^{\text{off}} \|$ (\%) } \\
\cline{2-5} {}&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$&
$\hspace*{0.8cm} L^{2}_\kappa(D) \hspace*{0.8cm}$ &
$\hspace*{0.8cm} H^{1}_\kappa(D) \hspace*{0.8cm}$
\\
\hline\hline
$1524$ & $7.60$ & $50.86$ & $7.59$ & $50.75$ \\
\hline
$1772$ & $4.18$ & $27.08$ & $4.18$ & $26.90$\\
\hline
$2398$ & $2.41$ & $20.59$ & $2.39$ & $20.36$\\
\hline
$2824$ & $1.28$ & $13.98$ & $1.25$ & $13.64$\\
\hline
$3688$ & $0.17$ &$7.83$ & $0.14$ & $7.26$\\
\hline
\end{tabular}
\caption{Convergence history for harmonic basis using snapshot space to compute the proposed indicator.
We take $\theta=0.7$ and the algorithm converges in $22$ iterations.
}
\label{table:HCC_theta.7_Harmonic_loc_snapshot}
\end{table}
\section{Convergence analysis}
\label{sec:analysis}
In this section, we will give the proofs for
the a-posteriori error estimates \eqref{eq:res1}-\eqref{eq:res2}
and the convergence of the adaptive enrichment algorithm.
For each $i=1,2,\cdots, N$, we let $P_i: V \rightarrow \text{span}\{ \psi_k^{\omega_i,\text{off}} \}$
be the projection defined by
\begin{equation*}
P_i v = \sum_{k=1}^{l_i} \Big( \int_{\omega_i} \widetilde{\kappa} v \psi_k^{\omega_i,\text{off}} \Big) \psi_k^{\omega_i,\text{off}}.
\end{equation*}
The projection $P_i$ has following stability bound
\begin{equation}
\| \chi_i (P_i v) \|_{V_i} \leq C_{\text{stab}}^{\omega_i} \| v\|_{V_i}
\label{eq:eigenstab}
\end{equation}
where the constant $C_{\text{stab}}^{\omega_i} = \max ( 1, H^{-1} (\lambda_{l_i+1}^{\omega_i})^{-\frac{1}{2}} )$.
Moreover the following
convergence result holds
\begin{equation}
\| \chi_i ( v - P_i v) \|_{V_i} \leq C_{\text{conv}}^{\omega_i} (\lambda_{l_i+1}^{\omega_i})^{-\frac{1}{2}} \| v\|_{V_i}
\label{eq:eigenbound}
\end{equation}
where $C_{\text{conv}}^{\omega_i}$ is a uniform constant.
We also define the projection $\Pi: V \rightarrow V_{\text{off}}$ by $\Pi v = \sum_{i=1}^N \chi_i (P_i v)$.
For the analysis below, we let $C_{\text{stab}} = \max_{1\leq i\leq N} C_{\text{stab}}^{\omega_i}$
and $C_{\text{conv}} = \max_{1\leq i\leq N} C_{\text{conv}}^{\omega_i}$.
\subsection{Proof of Theorem \ref{thm:post}}
Let $v\in V$ be an arbitrary function in the space $V$. Using \eqref{eq:fine}, we have
\begin{equation*}
a(u-u_{\text{ms}},v) = a(u,v) - a(u_{\text{ms}},v) = (f,v) - a(u_{\text{ms}},v).
\end{equation*}
Then
\begin{equation*}
a(u-u_{\text{ms}},v) = (f,v) - a(u_{\text{ms}},v)
= (f,v-\Pi v) + (f,\Pi v) - a(u_{\text{ms}},\Pi v) - a(u_{\text{ms}},v-\Pi v).
\end{equation*}
Thus, using \eqref{eq:coarse}, we have
\begin{equation}
a(u-u_{\text{ms}},v) = (f,v-\Pi v) - a(u_{\text{ms}},v-\Pi v).
\label{eq:err1}
\end{equation}
Writing (\ref{eq:err1}) as a sum over coarse regions,
\begin{equation}
a(u-u_{\text{ms}},v) = \sum_{i=1}^N \Big( \int_{\omega_i} f (v-P_i v) \chi_i
- \int_{\omega_i} a\nabla u_{\text{ms}} \cdot \nabla ( (v-P_i v)\chi_i ) \Big).
\label{eq:err2}
\end{equation}
Using the definition of $Q_i$, we see that \eqref{eq:err2} can be written as
\begin{equation*}
a(u-u_{\text{ms}},v) = \sum_{i=1}^N Q_i (v - P_i v).
\end{equation*}
Thus, we have
\begin{equation*}
a(u-u_{\text{ms}},v) \leq \sum_{i=1}^N \| Q_i \| \| v-P_i v\|_{L^2(\omega_i)}.
\end{equation*}
Using the definition of $\widetilde{\kappa}_i$, we have
\begin{equation*}
a(u-u_{\text{ms}},v) \leq \sum_{i=1}^N ( \widetilde{\kappa}_i)^{-\frac{1}{2}} \| Q_i \| \| \widetilde{\kappa}^{\frac{1}{2}} (v-P_i v)\|_{L^2(\omega_i)}.
\end{equation*}
Thus, by the definition of the eigenvalue problem \eqref{offeig},
\begin{equation*}
a(u-u_{\text{ms}},v) \leq \sum_{i=1}^N ( \widetilde{\kappa}_i)^{-\frac{1}{2}} (\lambda_{l_i+1}^{\omega_i})^{-\frac{1}{2}} \| Q_i \| \| v\|_{V_i}
\end{equation*}
The inequality \eqref{eq:res1} is then followed by taking $v = u-u_{\text{ms}}$
and $\sum_{i=1}^N \| v\|^2_{V_i} \leq C \|v\|^2_V$.
Using the definition of $R_i$, we see that \eqref{eq:err2} can be written as
\begin{equation*}
a(u-u_{\text{ms}},v) = \sum_{i=1}^N R_i ( \chi_i(v - P_i v) ).
\end{equation*}
Thus, we have
\begin{equation*}
a(u-u_{\text{ms}},v) \leq \sum_{i=1}^N \| R_i \|_{V_i^*} \|\chi_i( v-P_i v)\|_{V_i}.
\end{equation*}
Using \eqref{eq:eigenbound},
\begin{equation*}
a(u-u_{\text{ms}},v) \leq C_{\text{conv}} \sum_{i=1}^N \| R_i \|_{V_i^*} (\lambda_{l_i+1}^{\omega_i})^{-\frac{1}{2}} \| v\|_{V_i}.
\end{equation*}
The inequality \eqref{eq:res2} is then followed by taking $v = u-u_{\text{ms}}$
and $\sum_{i=1}^N \| v\|^2_{V_i} \leq C \|v\|^2_V$.
\subsection{Some auxiliary lemmas}
In this section, we will prove some auxiliary lemmas which are required for the proof of
the convergence of the adaptive enrichment algorithm
stated in Theorem \ref{thm:conv}.
We use the notation $P_i^m$
to denote the projection operator $P_i$ at the enrichment level $m$.
Specifically, we define
\begin{equation*}
P_i^m v = \sum_{k=1}^{l_i^m} \Big( \int_{\omega_i} \widetilde{\kappa} v \psi_k^{\omega_i,\text{off}} \Big) \psi_k^{\omega_i,\text{off}}.
\end{equation*}
In Theorem \ref{thm:post}, we see that $\|R_i\|_{V_i^*}$
gives an upper bound of the energy error $\|u-u_{\text{ms}}\|_V$.
We will first show that, $\|R_i\|_{V_i^*}$
is also a lower bound up to a correction term.
To state this precisely, we define
\begin{equation}
S_m(\omega_i) = (\lambda_{l^m_i+1}^{\omega_i})^{-\frac{1}{2}} \sup_{v\in V_i} \frac{ |R_i(v - (P_i^{m+1} v)\chi_i) |}{\|v\|_{V_i}}
\label{eq:defSR}
\end{equation}
which is a measure on how small $(v-\chi_i P_i^{m+1}v)$ is.
Notice that the residual $R_i$ is computed using the solution $u_{\text{ms}}^m$ obtained at enrichment level $m$.
We omit the index $m$ in $R_i$ to simplify notations. Next,
we will prove the following lemma.
\begin{lemma}
\label{lem:R1}
We have
\begin{equation}
\| R_i \|^2_{V_i^*} (\lambda_{l^m_i+1}^{\omega_i})^{-1}
\leq 2(C_{\text{stab}}^{\omega_i})^2 (\lambda_{l^m_i+1}^{\omega_i})^{-1} \|u_{\text{ms}}^{m+1}-u_{\text{ms}}^m\|_{V_i}^2 + 2S_m(\omega_i)^2
\label{eq:Rbound}
\end{equation}
\end{lemma}
\begin{proof}
By linearity
\begin{equation*}
R_i(v) = R_i( \chi_i (P_i^{m+1} v) ) + R_i(v - \chi_i(P_i^{m+1} v)).
\end{equation*}
Since $\chi_i(P_i^{m+1} v)$ is a test function in the space $V_{\text{off}}^{m+1}$, by the definition of $R_i$ and \eqref{eq:solve}, we have
\begin{equation*}
\begin{split}
R_i(\chi_i(P_i^{m+1} v)) &= \int_{\omega_i} f (P_i^{m+1} v) \chi_i - \int_{\omega_i} a\nabla u^m_{\text{ms}}\cdot \nabla ( (P_i^{m+1} v)\chi_i) \\
&= \int_{\omega_i} a\nabla u^{m+1}_{\text{ms}} \cdot \nabla ( (P_i^{m+1} v)\chi_i) - \int_{\omega_i} a\nabla u^m_{\text{ms}}\cdot \nabla ( (P_i^{m+1}v) \chi_i).
\end{split}
\end{equation*}
Using the stability estimate \eqref{eq:eigenstab},
\begin{equation*}
R_i( \chi_i(P_i^{m+1} v )) \leq \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i} \| (P_i^{m+1} v)\chi_i \|_{V_i}
\leq C_{\text{stab}}^{\omega_i} \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i} \|v\|_{V_i}.
\end{equation*}
Thus, we obtain
\begin{equation}
\| R_i\|_{V_i^*} \leq
C_{\text{stab}}^{\omega_i} \|u_{\text{ms}}^{m+1}-u_{\text{ms}}^m\|_{V_i} + \sup_{v\in V_i} \frac{ |R_i(v - (P_i^{m+1} v)\chi_i) |}{\|v\|_{V_i}}
\label{eq:Rbound1}
\end{equation}
The inequality \eqref{eq:Rbound} follows from the definition of $S_m(\omega_i)$.
\end{proof}
We remark that one can replace $u_{\text{ms}}^{m+1}$
by $u_{\text{snap}}$
and $P^{m+1}_i$ by $P^{\text{snap}}_i$,
where $P^{\text{snap}}_i$ is the projection onto the snapshot space defined by
\begin{equation*}
P_i^{\text{snap}} v = \sum_{k=1}^{W_i} \Big( \int_{\omega_i} \widetilde{\kappa} v \psi_k^{\omega_i,\text{off}} \Big) \psi_k^{\omega_i,\text{off}}.
\end{equation*}
We also define $S(\omega_i)$ by
\begin{equation*}
S(\omega_i) = (\lambda_{l^m_i+1}^{\omega_i})^{-\frac{1}{2}} \sup_{v\in V_i} \frac{ |R_i(v - (P_i^{\text{snap}} v)\chi_i) |}{\|v\|_{V_i}}.
\end{equation*}
Following the proof of the above lemma, we get
\begin{equation*}
\| R_i \|^2_{V_i^*} (\lambda_{l^m_i+1}^{\omega_i})^{-1}
\leq 2 (C_{\text{stab}}^{\omega_i})^2 (\lambda_{l^m_i+1}^{\omega_i})^{-1} \|u_{\text{snap}}-u_{\text{ms}}^m\|_{V_i}^2 + 2 S(\omega_i)^2
\end{equation*}
which suggests that $\| R_i \|^2_{V_i^*} (\lambda_{l^m_i+1}^{\omega_i})^{-1}$ gives a lower bound of
the error $\|u_{\text{snap}}-u_{\text{ms}}^m\|_{V_i}^2$
up to a correction term $S(\omega_i)^2$.
To prove Theorem \ref{thm:conv}, we will
need the following recursive properties for $S_m(\omega_i)$.
\begin{lemma}
\label{lem:recurR}
For any $\alpha_R >0$, we have
\begin{equation}
S_{m+1}(\omega_i)^2 \leq (1+\alpha_R) C_R \frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}} S_m(\omega_i)^2
+ (1+\alpha_R^{-1}) D_R (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-1} \|u_\text{ms}^{m+1}-u_{\text{ms}}^m\|_{V_i}^2
\label{eq:Sbound}
\end{equation}
where the enrichment level dependent constants $C_R$ and $D_R$ are defined by
\begin{equation*}
C_R = (1 + 2 C_{\text{conv}}^{\omega_i} (\lambda_{l_i^{m}+1}^{\omega_i})^{-\frac{1}{2}} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}})^2
\quad\text{and}\quad
D_R = (C_{\text{stab}}^{\omega_i})^2 (1 + 2 C_{\text{conv}}^{\omega_i} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}})^2.
\end{equation*}
\end{lemma}
\begin{proof}
By direct calculation, we have
\begin{equation}
\begin{split}
&\: \int_{\omega_i} f (v - (P_i^{m+2} v) \chi_i) - \int_{\omega_i} a\nabla u_{\text{ms}}^{m+1} \cdot \nabla ( v- (P_i^{m+2}v)\chi_i ) \\
= &\: \int_{\omega_i} f (v - (P_i^{m+1} v )\chi_i) - \int_{\omega_i} a\nabla u_{\text{ms}}^{m} \cdot \nabla ( v- (P_i^{m+1}v)\chi_i ) \\
&\: - \int_{\omega_i} a \nabla (u_{\text{ms}}^{m+1}-u_{\text{ms}}^m) \cdot \nabla (v - (P_i^{m+2}v)\chi_i) \\
&\: + \int_{\omega_i} f (P_i^{m+1}v - P_i^{m+2}v)\chi_i - \int_{\omega_i} a\nabla u_{\text{ms}}^m \cdot \nabla ( (P_i^{m+1}v-P_i^{m+2}v)\chi_i).
\end{split}
\label{err}
\end{equation}
By definition of $S_m(\omega_i)$, we have
\begin{equation}
S_m(\omega_i) = (\lambda_{l_i^m+1}^{\omega_i})^{-\frac{1}{2}}
\sup_{v\in V_i} \frac{ | \int_{\omega_i} f (v - (P_i^{m+1} v )\chi_i) - \int_{\omega_i} a\nabla u_{\text{ms}}^{m} \cdot \nabla ( v- (P_i^{m+1}v)\chi_i ) | }{\| v\|_{V_i}}.
\end{equation}
Multiplying \eqref{err} by $(\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \|v\|_{V_i}^{-1}$ and taking supremum with respect to $v$, we have
\begin{equation}
S_{m+1}(\omega_i) \leq (\frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}})^{\frac{1}{2}} S_m(\omega_i)
+ I_1 + I_2
\label{err1}
\end{equation}
where
\begin{equation*}
I_1 = (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \sup_{v\in V_i} \frac{ |- \int_{\omega_i} a \nabla (u_{\text{ms}}^{m+1}-u_{\text{ms}}^m) \cdot \nabla (v - (P_i^{m+2}v)\chi_i)| }{\|v\|_{V_i}}
\end{equation*}
and
\begin{equation*}
I_2 = (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \sup_{v\in V_i} \frac{ | \int_{\omega_i} f (P_i^{m+1}v - P_i^{m+2}v)\chi_i - \int_{\omega_i} a\nabla u_{\text{ms}}^m \cdot \nabla ( (P_i^{m+1}v-P_i^{m+2}v)\chi_i) |}{\|v\|_{V_i}}.
\end{equation*}
To estimate $I_1$, we use the stability estimate \eqref{eq:eigenstab} to obtain
\begin{equation*}
\begin{split}
&\int_{\omega_i} a \nabla (u_{\text{ms}}^{m+1}-u_{\text{ms}}^m) \cdot \nabla (v - (P_i^{m+2}v)\chi_i) \\
=& \int_{\omega_i} a \nabla (u_{\text{ms}}^{m+1}-u_{\text{ms}}^m) \cdot \nabla v
- \int_{\omega_i} a \nabla (u_{\text{ms}}^{m+1}-u_{\text{ms}}^m) \cdot \nabla ( (P_i^{m+2}v)\chi_i )\\
\leq &C_{\text{stab}}^{\omega_i} \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m\|_{V_i} \|v\|_{V_i}
\end{split}
\end{equation*}
which implies
\begin{equation*}
I_1 \leq C_{\text{stab}}^{\omega_i} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i}.
\end{equation*}
To estimate $I_2$, we use the definition of $R_i$ to obtain
\begin{equation*}
I_2 \leq (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \| R_i \|_{V_i^*} \sup_{v\in V_i} \frac{ \| \chi_i(P_i^{m+1}v - P_i^{m+2}v) \|_{V_i}}{ \|v\|_{V_i}}.
\end{equation*}
By the convergence bound \eqref{eq:eigenbound} and the fact that
$\lambda_{l_i^{m+1}+1}^{\omega_i} < \lambda_{l_i^{m+2}+1}^{\omega_i}$, we have
\begin{equation*}
\| \chi_i(P_i^{m+1}v - P_i^{m+2}v) \|_{V_i}
\leq \| \chi_i(P_i^{m+1}v - v) \|_{V_i} + \| \chi_i(v - P_i^{m+2}v) \|_{V_i} \leq 2 C_{\text{conv}}^{\omega_i} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \|v\||_{V_i}
\end{equation*}
which implies
\begin{equation*}
I_2 \leq 2 C_{\text{conv}}^{\omega_i} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-1} \|R_i\|_{V_i^*}.
\end{equation*}
Combining results and using \eqref{err1}, we get
\begin{equation*}
S_{m+1}(\omega_i) \leq
(\frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}})^{\frac{1}{2}} S_m(\omega_i)
+ C_{\text{stab}}^{\omega_i} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i}
+ 2 C_{\text{conv}}^{\omega_i} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-1} \|R_i\|_{V_i^*}.
\end{equation*}
Using \eqref{eq:Rbound1} and the definition of $S_m(\omega_i)$,
\begin{equation*}
\begin{split}
S_{m+1}(\omega_i) \leq
& (1 + 2 C_{\text{conv}}^{\omega_i} (\lambda_{l_i^{m}+1}^{\omega_i})^{-\frac{1}{2}} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}}) (\frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}})^{\frac{1}{2}} S_m(\omega_i) \\
&+ C_{\text{stab}}^{\omega_i} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} (1 + 2 C_{\text{conv}}^{\omega_i} (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}})
\| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i}.
\end{split}
\end{equation*}
Hence, \eqref{eq:Sbound} is proved.
\end{proof}
Next, we consider the $L^2$-based residual $Q_i$
and prove similar inequalities.
We define
\begin{equation}
S_m(\omega_i) = (\widetilde{\kappa}_i \lambda_{l^m_i+1}^{\omega_i})^{-\frac{1}{2}} \sup_{v\in L^2(\omega_i)} \frac{ |Q_i( v - P_i^{m+1} v) |}{\|v\|_{L^2(\omega_i)}}
\label{eq:defSQ}
\end{equation}
which is a measure on how small $(v- P_i^{m+1}v)$ is.
Notice that the residual $Q_i$ is computed using the solution $u_{\text{ms}}^m$ obtained at enrichment level $m$.
We omit the index $m$ in $Q_i$ to simplify notations.
We also note that we have used the same notation $S_m(\omega_i)$ as the case for the $H^{-1}$-based residual
to again simplify notations.
It will be clear which residual we are referring to when the notation $S_m(\omega_i)$ appears in the text.
We define the jump of the coefficient in each coarse region by
\begin{equation*}
\beta_i = \frac{ \max_{x\in\omega_i} \kappa(x)}{ \min_{x\in\omega_i} \kappa(x) }.
\end{equation*}
Next,
we will prove the following lemma.
\begin{lemma}
\label{lem:Q1}
We have
\begin{equation}
\| Q_i \|^2 ( \widetilde{\kappa}_i \lambda_{l^m_i+1}^{\omega_i})^{-1}
\leq 2 (C_{\text{inv}} \beta_i^{\frac{1}{2}} h^{-1})^2 ( \lambda_{l^m_i+1}^{\omega_i})^{-1} \|u_{\text{ms}}^{m+1}-u_{\text{ms}}^m\|_{V_i}^2 + 2 S_m(\omega_i)^2
\label{eq:Qbound}
\end{equation}
\end{lemma}
\begin{proof}
By linearity
\begin{equation*}
Q_i(v) = Q_i( P_i^{m+1} v ) + Q_i(v - P_i^{m+1} v).
\end{equation*}
By the definition of $Q_i$ and \eqref{eq:solve}, we have
\begin{equation*}
\begin{split}
Q_i( P_i^{m+1} v) &= \int_{\omega_i} f (P_i^{m+1} v) \chi_i - \int_{\omega_i} a\nabla u^m_{\text{ms}}\cdot \nabla ( (P_i^{m+1} v)\chi_i) \\
&= \int_{\omega_i} a\nabla u^{m+1}_{\text{ms}} \cdot \nabla ( (P_i^{m+1} v)\chi_i) - \int_{\omega_i} a\nabla u^m_{\text{ms}}\cdot \nabla ( (P_i^{m+1}v) \chi_i)
\end{split}
\end{equation*}
which implies
\begin{equation*}
Q_i( P_i^{m+1} v ) \leq \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i} \| (P_i^{m+1} v)\chi_i \|_{V_i}.
\end{equation*}
Using the inverse inequality,
\begin{equation*}
\| (P_i^{m+1} v)\chi_i \|_{V_i} \leq C_{\text{inv}} h^{-1} \| \widetilde{\kappa}^{\frac{1}{2}} P_i^{m+1} v\|_{L^2(\omega_i)}
\leq C_{\text{inv}} h^{-1} \| \widetilde{\kappa}^{\frac{1}{2}} v\|_{L^2(\omega_i)}
\end{equation*}
where $C_{\text{inv}}$ is independent of the mesh size.
Thus, we obtain
\begin{equation}
(\widetilde{\kappa}_i)^{-\frac{1}{2}} \| Q_i\|_{V_i^*} \leq
C_{\text{inv}} \beta_i^{\frac{1}{2}} h^{-1} \|u_{\text{ms}}^{m+1}-u_{\text{ms}}^m\|_{V_i}
+ (\widetilde{\kappa}_i)^{-\frac{1}{2}} \sup_{v\in L^2(\omega_i)} \frac{ |Q_i(v - P_i^{m+1} v) |}{\|v\|_{L^2(\omega_i)}}.
\label{eq:Qbound1}
\end{equation}
The inequality \eqref{eq:Qbound} follows from the definition of $S_m(\omega_i)$.
\end{proof}
Next we will prove the following recursive property for $S_m(\omega_i)$.
The proof follows from the same lines as Lemma \ref{lem:recurR}.
\begin{lemma}
\label{lem:recurQ}
For any $\alpha_Q >0$, we have
\begin{equation}
S_{m+1}(\omega_i)^2 \leq (1+\alpha_Q) C_Q \frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}} S_m(\omega_i)^2
+ (1+\alpha_Q^{-1}) D_Q (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-1} \|u_\text{ms}^{m+1}-u_{\text{ms}}^m\|_{V_i}^2
\label{eq:SQbound}
\end{equation}
where the enrichment level dependent constants $C_R$ and $D_R$ are defined by
\begin{equation*}
C_Q = (1 + \beta_i^{\frac{1}{2}})^2
\quad\text{and}\quad
D_Q = C_{\text{inv}} \beta_i^{\frac{1}{2}} h^{-1} (2\widetilde{\kappa}_i + \beta_i^{\frac{1}{2}} ).
\end{equation*}
\end{lemma}
\begin{proof}
By direct calculation, we have
\begin{equation}
\begin{split}
&\: \int_{\omega_i} f (v - P_i^{m+2} v) \chi_i - \int_{\omega_i} a\nabla u_{\text{ms}}^{m+1} \cdot \nabla ( (v- P_i^{m+2}v)\chi_i ) \\
= &\: \int_{\omega_i} f (v - P_i^{m+1} v )\chi_i - \int_{\omega_i} a\nabla u_{\text{ms}}^{m} \cdot \nabla ( (v- P_i^{m+1}v)\chi_i ) \\
&\: - \int_{\omega_i} a \nabla (u_{\text{ms}}^{m+1}-u_{\text{ms}}^m) \cdot \nabla ((v - P_i^{m+2}v)\chi_i) \\
&\: + \int_{\omega_i} f (P_i^{m+1}v - P_i^{m+2}v)\chi_i - \int_{\omega_i} a\nabla u_{\text{ms}}^m \cdot \nabla ( (P_i^{m+1}v-P_i^{m+2}v)\chi_i).
\end{split}
\label{errQ1}
\end{equation}
By definition of $S_m(\omega_i)$, we have
\begin{equation}
S_m(\omega_i) = ( \widetilde{\kappa}_i \lambda_{l_i^m+1}^{\omega_i})^{-\frac{1}{2}}
\sup_{v\in L^2(\omega_i)} \frac{ | \int_{\omega_i} f (v - P_i^{m+1} v )\chi_i - \int_{\omega_i} a\nabla u_{\text{ms}}^{m} \cdot \nabla ( (v- P_i^{m+1}v)\chi_i ) | }{\| v\|_{L^2(\omega_i)}}.
\end{equation}
Multiplying \eqref{errQ1} by $( \widetilde{\kappa}_i \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \|v\|_{L^2(\omega_i)}^{-1}$ and taking supremum with respect to $v$, we have
\begin{equation}
S_{m+1}(\omega_i) \leq (\frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}})^{\frac{1}{2}} S_m(\omega_i)
+ I_1 + I_2
\label{errQ2}
\end{equation}
where
\begin{equation*}
I_1 = ( \widetilde{\kappa}_i \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \sup_{v\in L^2(\omega_i)} \frac{ |- \int_{\omega_i} a \nabla (u_{\text{ms}}^{m+1}-u_{\text{ms}}^m) \cdot \nabla ((v - P_i^{m+2}v)\chi_i)| }{\|v\|_{L^2(\omega_i)}}
\end{equation*}
and
\begin{equation*}
I_2 = ( \widetilde{\kappa} \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \sup_{v\in L^2(\omega_i)} \frac{ | \int_{\omega_i} f (P_i^{m+1}v - P_i^{m+2}v)\chi_i - \int_{\omega_i} a\nabla u_{\text{ms}}^m \cdot \nabla ( (P_i^{m+1}v-P_i^{m+2}v)\chi_i) |}{\|v\|_{L^2(\omega_i)}}.
\end{equation*}
To estimate $I_1$, we use the inverse inequality to obtain
\begin{equation*}
\int_{\omega_i} a \nabla (u_{\text{ms}}^{m+1}-u_{\text{ms}}^m) \cdot \nabla (v - (P_i^{m+2}v)\chi_i)
\leq 2 C_{\text{inv}} h^{-1} \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m\|_{V_i} \| \widetilde{\kappa}^{\frac{1}{2}} v\|_{L^2(\omega_i)}
\end{equation*}
which implies
\begin{equation*}
I_1 \leq 2 C_{\text{inv}} ( \widetilde{\kappa}_i \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \beta_i^{\frac{1}{2}} h^{-1} \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i}.
\end{equation*}
To estimate $I_2$, we use the definition of $Q_i$ to obtain
\begin{equation*}
I_2 \leq (\widetilde{\kappa}_i \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \| Q_i \| \sup_{v\in L^2(\omega_i)} \frac{ \| P_i^{m+1}v - P_i^{m+2}v \|}{ \|v\|_{L^2(\omega_i)}}
\end{equation*}
which implies
\begin{equation*}
I_2 \leq (\widetilde{\kappa}_i \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \beta_i^{\frac{1}{2}} \| Q_i \|.
\end{equation*}
Combining results and using \eqref{errQ2}, we get
\begin{equation*}
S_{m+1}(\omega_i) \leq
(\frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}})^{\frac{1}{2}} S_m(\omega_i)
+ 2 C_{\text{inv}} ( \widetilde{\kappa}_i \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \beta_i^{\frac{1}{2}} h^{-1} \| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i}
+ (\widetilde{\kappa}_i \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \beta_i^{\frac{1}{2}} \|Q_i\|.
\end{equation*}
Using \eqref{eq:Qbound1},
\begin{equation*}
S_{m+1}(\omega_i) \leq
(1 + \beta_i^{\frac{1}{2}}) (\frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}})^{\frac{1}{2}} S_m(\omega_i)
+ C_{\text{inv}} ( \lambda_{l_i^{m+1}+1}^{\omega_i})^{-\frac{1}{2}} \beta_i^{\frac{1}{2}} h^{-1} (2\widetilde{\kappa}_i + \beta_i^{\frac{1}{2}} )
\| u_{\text{ms}}^{m+1} - u_{\text{ms}}^m \|_{V_i}
\end{equation*}
Hence, \eqref{eq:SQbound} is proved.
\end{proof}
\subsection{Proof of Theorem \ref{thm:conv}}
In this section, we prove the convergence of the adaptive enrichment algorithm.
We will give a unified proof for both the $L^2$-based and $H^{-1}$-based residuals.
First of all, we use $\eta_i$ as a unified notation for the residuals, namely,
\begin{equation*}
\eta^2_i =
\begin{cases}
& \|Q_i\|^2 (\widetilde{\kappa}_i \lambda^{\omega_i}_{l^m_i+1})^{-1},\quad \text{ for } L^2\text{-based residual}, \\
& \|R_i\|^2_{V_i^*} (\lambda^{\omega_i}_{l^m_i+1})^{-1},\quad \text{ for } H^{-1}\text{-based residual}.
\end{cases}
\end{equation*}
Then Lemma \ref{lem:R1} and Lemma \ref{lem:Q1} can be written as
\begin{equation}
\eta_i^2
\leq B_i ( \lambda_{l^m_i+1}^{\omega_i})^{-1} \|u_{\text{ms}}^{m+1}-u_{\text{ms}}^m\|_{V_i}^2 + 2 S_m(\omega_i)^2
\label{eq:Uibound}
\end{equation}
where the constant $B_i$ is given by
\begin{equation*}
B_i =
\begin{cases}
& 2(C_{\text{inv}} \beta_i^{\frac{1}{2}} h^{-1})^2,\quad \text{ for } L^2\text{-based residual} \\
& 2 (C_{\text{stab}}^{\omega_i,m+1})^2,\quad \text{ for } H^{-1}\text{-based residual}
\end{cases}
\end{equation*}
We remark that the definitions of $S_m(\omega_i)$ are given in \eqref{eq:defSQ} and \eqref{eq:defSR}
for the $L^2$-based and $H^{-1}$-based residuals respectively.
Moreover, Lemma \ref{lem:recurR} and Lemma \ref{lem:recurQ} can be unified as
\begin{equation}
S_{m+1}(\omega_i)^2 \leq (1+\alpha_S) C_S \frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}} S_m(\omega_i)^2
+ (1+\alpha_S^{-1}) D_S (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-1} \|u_\text{ms}^{m+1}-u_{\text{ms}}^m\|_{V_i}^2
\label{eq:Uibound1}
\end{equation}
where $\alpha_S = \alpha_Q, C_S=C_Q$ and $D_S=D_Q$ for the $L^2$-based residual
while $\alpha_S = \alpha_R, C_S=C_R$ and $D_S=D_R$ for the $H^{-1}$-based residual.
Notice that $\alpha_S>0$ is a constant defined uniformly over coarse regions and is
to be determined.
The convergence proof is based on \eqref{eq:Uibound} and \eqref{eq:Uibound1}.
Let $0 < \theta < 1$. We choose an index set $I$ so that
\begin{equation}
\theta^2 \sum_{i=1}^N \eta_i^2
\leq \sum_{i\in I} \eta_i^2.
\label{eq:indicator}
\end{equation}
We also assume there is a real number $\gamma$ with
$0 < \gamma < 1$ satisfies
\begin{equation}
\gamma^2 \sum_{i=1}^n S_m(\omega_i)^2 \leq \sum_{i\in I} S_m(\omega_i)^2.
\end{equation}
We will then add basis function for those $\omega_i$ with $i\in I$.
Then, using Theorem \ref{thm:post} and \eqref{eq:indicator}, we have
\begin{equation*}
\theta^2 \| u-u_{\text{ms}}^m\|_V^2 \leq \theta^2 C_{\text{err}} \sum_{i=1}^N \eta_i^2
\leq C_{\text{err}} \sum_{i\in I} \eta_i^2.
\end{equation*}
By (\ref{eq:Uibound}),
\begin{equation*}
\theta^2 \| u-u_{\text{ms}}^m\|_V^2 \leq 2 C_{\text{err}} \sum_{i=1}^N S_m(\omega_i)^2 + L_1 \| u_H^{m+1}-u_H^m\|_V^2
\end{equation*}
where
\begin{equation}
L_1 = C_{\text{err}} \max_{1\leq i\leq N} \Big( B_i (\lambda_{l_i^m+1}^{\omega_i})^{-1} \Big).
\label{eq:assumeL}
\end{equation}
Note that, by Galerkin orthogonality, we have
\begin{equation*}
\| u_{\text{ms}}^{m+1}-u_{\text{ms}}^m\|_V^2 = \|u-u_{\text{ms}}^m\|_V^2 - \|u-u_{\text{ms}}^{m+1}\|_V^2.
\end{equation*}
So, we have
\begin{equation*}
\theta^2 \| u-u_{\text{ms}}^m\|_V^2 \leq 2 C_{\text{err}} \sum_{i=1}^N S_m(\omega_i)^2 + L_1 (\|u-u_H^m\|_V^2 - \|u-u_H^{m+1}\|_V^2)
\end{equation*}
which implies
\begin{equation}
\| u-u_{\text{ms}}^{m+1} \|_V^2 \leq ( 1-\frac{\theta^2}{L_1} ) \| u-u_{\text{ms}}^m\|_V^2 + \frac{2C_{\text{err}}}{L_1} \sum_{i=1}^N S_m(\omega_i)^2.
\label{eq:conv1}
\end{equation}
On the other hand,
\begin{equation*}
\sum_{i=1}^N S_{m+1}(\omega_i)^2
= \sum_{i\in I} S_{m+1}(\omega_i)^2 + \sum_{i\ne I} S_{m+1}(\omega_i)^2
\end{equation*}
By (\ref{eq:Uibound1}) and that $S_{m+1}(\omega_i) = S_m(\omega_i)$ for $i\ne I$, we have
\begin{equation*}
\sum_{i=1}^N S_{m+1}(\omega_i)^2
\leq
\sum_{i\in I} \Big( (1+\alpha_S) C_S \frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}} S_m(\omega_i)^2
+ (1+\alpha_S^{-1}) D_S (\lambda_{l_i^{m+1}+1}^{\omega_i})^{-1} \|u_\text{ms}^{m+1}-u_{\text{ms}}^m\|_{V_i}^2 \Big) +
\sum_{i\ne I} S_{m}(\omega_i)^2.
\end{equation*}
We assume the enrichment is obtained so that
\begin{equation*}
\delta = C_S \max_{1\leq i\leq N} \frac{\lambda_{l_i^m+1}^{\omega_i}}{\lambda_{l_i^{m+1}+1}^{\omega_i}} < 1.
\end{equation*}
We then have
\begin{equation*}
\sum_{i=1}^N S_{m+1}(\omega_i)^2
\leq (1+\alpha_S) \sum_{i=1}^N S_{m}(\omega_i)^2 - (1+\alpha_S) (1-\delta) \sum_{i\in I} S_m(\omega_i)^2 + \delta L_2 \|u_{\text{ms}}^{m+1}-u_{\text{ms}}^m\|_V^2
\end{equation*}
where
\begin{equation}
L_2 = (1+\alpha_S^{-1}) \max_{1\leq i\leq N} \Big( D_S C_S^{-1} (\lambda_{l_i^m+1}^{\omega_i})^{-1} \Big).
\label{eq:assumeL2}
\end{equation}
By assumption on $\gamma$,
\begin{equation*}
\sum_{i=1}^N S_{m+1}(\omega_i)^2
\leq (1+\alpha_S) \sum_{i=1}^N S_{m}(\omega_i)^2 - (1+\alpha_S) (1-\delta) \gamma^2 \sum_{i=1}^N S_m(\omega_i)^2 + \delta L_2 \|u_{\text{ms}}^{m+1}-u_{\text{ms}}^m\|_V^2.
\end{equation*}
Let $\rho = (1+\alpha_S)( 1 - (1-\delta) \gamma^2)$. We choose $\alpha_S>0$ small so that
$0<\rho<1$. The above is then written as
\begin{equation}
\sum_{i=1}^N S_{m+1}(\omega_i)^2 \leq \rho \sum_{i=1}^N S_{m}(\omega_i)^2
+ \delta L_2 ( \|u-u_{\text{ms}}^m\|_V^2 - \|u-u_{\text{ms}}^{m+1}\|_V^2 ).
\label{eq:conv2}
\end{equation}
Next, we take a constant $\tau$ so that
\begin{equation*}
\tau > 0, \quad \frac{2C_{\text{err}}}{\tau L_1} + \rho < 1.
\end{equation*}
Finally, we combine \eqref{eq:conv1} and \eqref{eq:conv2} to obtain the following
\begin{equation*}
\begin{split}
&\: \| u - u_{\text{ms}}^{m+1}\|_V^2 + \tau \sum_{i=1}^N S_{m+1}(\omega_i)^2 \\
\leq &\: ( 1-\frac{\theta^2}{L_1} ) \| u-u_{\text{ms}}^m\|_V^2 + \frac{2C_{\text{err}}}{L_1} \sum_{i=1}^N S_m(\omega_i)^2
+ \tau \rho \sum_{i=1}^N S_{m}(\omega_i)^2 + \tau \delta L_2 ( \|u-u_{\text{ms}}^m\|_V^2 - \|u-u_{\text{ms}}^{m+1}\|_V^2 ).
\end{split}
\end{equation*}
Rearranging the terms, we have
\begin{equation*}
(1+\tau \delta L_2) \|u-u_{\text{ms}}^{m+1}\|_V^2 + \tau \sum_{i=1}^N S_{m+1}(\omega_i)^2
\leq ( 1-\frac{\theta^2}{L_1} + \tau \delta L_2 ) \| u-u_{\text{ms}}^m\|_V^2 + (\frac{2C_{\text{err}}}{L_1}+\tau\rho) \sum_{i=1}^N S_m(\omega_i)^2.
\end{equation*}
Hence we obtain
\begin{equation*}
\| u-u_{\text{ms}}^{m+1}\|_V^2 + \frac{\tau}{1+\tau \delta L_2} \sum_{i=1}^N S_{m+1}(\omega_i)^2
\leq ( 1- \frac{\theta^2}{L_1(1+\tau \delta L_2)} ) \|u-u_{\text{ms}}^m\|_V^2 + \frac{\tau}{1+\tau \delta L_2} (\frac{2C_{\text{err}}}{\tau L_1}+\rho) \sum_{i=1}^N S_{m}(\omega_i)^2.
\end{equation*}
Therefore, Theorem \ref{thm:conv} is proved.
\section{Conclusions}
In this paper, we derive an a-posteriori error indicator
for the Generalized
Multiscale Finite Element Method (GMsFEM).
In particular,
we study an adaptive spectral enrichment procedure and derive an
error indicator which gives an estimate of the local error
over coarse grid regions.
We consider two kinds of error indicators where one is based on
the $L^2$-norm of the local residual
and the other is based on the weighted $H^{-1}$-norm of the local residual
where the weight is related to the coefficient of the elliptic equation.
We show that the use of weighted $H^{-1}$-norm residual
gives a more robust error indicator
which works well for cases with high contrast multiscale problems. The convergence analysis of the method is given. Numerical results are presented that demonstrate the robustness
of the proposed error indicators. We
show the convergence of the proposed indicators and their similarities to
the ones when exact solution is used in the indicator.
We compare the performance of
the weighted $H^{-1}$-based indicator with
that of the $L^2$-based indicator for high-contrast
problems. Our numerical results show that the former is more appropriate
for high-contrast multiscale problems.
Although the results presented in this paper are encouraging, there
is scope for further exploration.
As our intent here was to derive and demonstrate
the robustness of error indicators for challenging high-contrast
multiscale problems, we did not consider
the fine-grid discretization error and assumed that the coarse-grid
error is the main contributor, and thus assuming that the fine-grid
solution is the desired quantity. In general when solving continuous
PDEs, one can also add fine-grid discretization errors due to basis
computations. This will be a subject of our future research.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,640 |
Q: rewrite directory on url with .htaccess? On my website I have a /profile.php.
I also have a directory named /profile/.
I am wondering if I could use .htaccess to rewrite
"/profile/[username]"
to be
"/profile?name=[username]"
Thanks in advance!
A: Since you have a file /profile.php I believe you want to redirect it to /profile.php?name=[username] instead of /profile?name=[username]
Enable mod_rewrite and .htaccess through httpd.conf and then put this code in your .htaccess under DOCUMENT_ROOT directory:
Options +FollowSymLinks -MultiViews
# Turn mod_rewrite on
RewriteEngine On
RewriteBase /
RewriteRule ^profile/([^/]+)/?$ /profile.php?name=$1 [L,QSA,NC]
In case you still want to redirect to /profile?name=[username] you can change above rule to /profile?name=$1
A: Give this a try:
Options +FollowSymlinks
RewriteEngine on
RewriteRule ^profile/([a-z0-9-_]+)/?$ /profile?name=$1 [QSA,NC]
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,129 |
Posted on September 2, 2016 by Doc
I really don't know what has gotten into Kevin Smith.
The fan favorite director who manned such films as Clerks, Chasing Amy and Dogma has taken the film community into a stranger and more bizarre place than he did with Tusk. And that was a movie about a man who turned into a walrus.
Yet, I cannot say that there wasn't a perverse amount of enjoyment to be had in the world of cartoon Canada, where mini, one-foot tall Nazi made out of bratwurst (aka Bratzis) who had sauerkraut for blood ran around killing people for… you know…reasons.
The film was pretty stupid. Still, I found it fun. Kind of like the fun of watching a ten car pile up on I-80. You just can't look away.
Kevin Smith took two characters from the film Tusk and spun them off into their own film. He further connected Yoga Hosers to Tusk by returning Guy Lapointe (Johnny Depp) to appear in the film once again.
Colleen C (Lily-Rose Depp) and Colleen M (Harley Quinn Smith) are 15-year old girls who are always on their phone and work, begrudgingly, at the Eh-2-Zed convenience store in Manitoba, Canada. They are all "aboot" their own lives and can barely be considered clerks. When they are invited to a senior party, they are out of their minds, until they get stuck working. Unfortunately for them, the Bratzis have awakened and are ready to re-establish the Nazi Party in Canada.
There is just no way around it. This movie is a full-on, ridiculous B movie in the spirit of Plan 9 from Outer Space or Piranha. It is the kind of film that would be a perfect fit for the Rifftrax boys. Still, there is some kind of joy here, maybe because Kevin Smith is so in on it. He full admits that this film may not be for everyone, and that he made it because he thought it was funny. It is a passion project for Smith that is unceasingly brave in taking that next ridiculous step.
Lily-Rose Depp and Harley Quinn Smith, daughters of Johnny Depp and Kevin Smith respectfully, are actually quite good in this. They are real life friends and that camaraderie plays through with the two Colleens. They feel natural (to borrow a line from Smith). They are very lovely and command attention on screen. Yes, the material around them is strange and, conceivably, dumb, but they go past it.
Now, some of this movie does seem amateurish. The music sometimes is too loud to hear dialogue (especially Harley Quinn Smith, who is too soft spoken at times) and some of the shots are very weird. Since Kevin Smith has been a successful director before, I suspect that some of this manner of creating film was intentional. I think he intentionally made a film that would be seen as a low-level B film (if not even lower) and embraced the quirks of it. If he purposefully made a movie bad, does that make it a better movie?
I was able to see this on a special premiere night as a Fathom Event and that meant that Kevin Smith introduced the film. When Kevin Smith introduces something, that means he is going to talk. Kevin Smith is known for being a talker, and I think he is legitimately one of the best storytellers that we have today. The 20 minutes or so that Kevin Smith talked about the making of Tusk and, eventually, the spin off of Yoga Hosers, really helped the film take on a new life for me. I do not know what I would have thought if I did not have that initial story surrounding how the two daughters ended up in Tusk, how Johnny Depp wound up with a "dick" on his nose and working on Tusk, how Depp loved the Guy accent despite everyone in his life hating it, and how Yoga Hosers came about because Kevin Smith felt bad for taking his daughter to all of these male dominated super hero movies. The introduction by Kevin Smith really put me in a proper mindset and I do not know what I would have thought of Yoga Hosers without it. I have a feeling that I would not have as much of a positive feeling as I do without it.
The movie is not good, and there is no denying that. But I found myself entertained by the sheer brazen silliness and outright camp of the film. I had fun that night, even though I could barely believe what I was seeing. I have always been a fan of Kevin Smith so I probably came in with a predisposition toward liking Yoga Hosers. That helped.
Kevin Smith said he made this a child's movie even though the target audience wouldn't be able to see it. I commend him for doing something so out there that I had a lot of trouble deciding what I thought of it afterwards. I wasn't sure how I was to score this film. I did have a lot of fun, but the film was not a good one. Smith even admits that. So in the end, as a critic, I am giving Yoga Hosers a….
However, if you are in the proper mood, and don't mind some seriously weird and warped storytelling, including a bizarre performance from Johnny Depp, then maybe you should check out this movie. If you approach it with the proper mindset, you may even laugh a few times.
← Hell or High Water
Morgan → | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,556 |
Biskopskors, även kallat pektoralkors (latin: crux pectorale, av pectus, 'bröst'), är ett ämbetstecken buret av biskopar inom vissa kyrkor.
Korsen bärs i en kedja eller snodd runt halsen. Inom romersk-katolska kyrkan har bröstkors i vissa fall även utdelats till abbotar. Inom protestantiska kyrkor upphörde bruket av biskopskors generellt i samband med reformationen men har delvis återupptagits under 1900-talet.
Inom Svenska kyrkan avskaffades biskopskorsen i samband med reformationen, för att återinföras 1805 av kung Gustav IV Adolf. De svenska biskopskorsen utgörs av ett latinskt kors av guld, buret i en guldkedja. Ärkebiskopens guldkors är dessutom i Sverige och Finland försett med strålar utgående från korsets mitt.
Källor
Bengt Andersson, "Biskopskorset 150 år" i Växjö stifts hembygdskalender 46 (1955), s. 152–157.
Bengt O.T. Sjögren, "Biskopskorset 200 år" i Medlemsblad för Skara Stiftshistoriska Sällskap, 2005:1
Bengt Stolt, Kyrklig skrud enligt svensk tradition, 1964, s. 23–24.
Yrsa Lindroos, Spåren till ett ärkebiskopskors, Hembygden 5/2010
Kors | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,828 |
// Copyright 2014 The Bazel Authors. All rights reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
package com.google.devtools.build.lib.runtime.commands;
import com.google.devtools.build.lib.events.ExtendedEventHandler.Postable;
/**
* An event describing a project file which has been parsed.
*/
public class GotProjectFileEvent implements Postable {
private final String projectFile;
/**
* Construct the event.
* @param projectFile The workspace-relative path of the project file.
*/
public GotProjectFileEvent(String projectFile) {
this.projectFile = projectFile;
}
/** Returns the project file that was parsed. */
public String getProjectFile() {
return projectFile;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,822 |
<div class="modal" ng-controller="DocumentModalEdgeController" role="dialog" aria-hidden="true">
<div class="modal-dialog modal-lg">
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<div class="modal-header">
<button type="button" class="close" data-dismiss="modal" ng-click="$hide();cancelSave()" aria-hidden="true">×
</button>
<h5>New Edge from {{source["@rid"]}} to {{target["@rid"]}}</h5>
</div>
<div ng-if="!showEdgeForm()">
<div class="modal-body document">
<ng-include src="'views/database/record.html'">
</ng-include>
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<button class="btn btn-primary btn-sm" ng-click="modal.saved=true;$hide();save()">Create Edge
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<div ng-if="showEdgeForm()">
<div class="modal-body document">
<form class="form-horizontal" role="form">
<div class="form-group">
<label class="control-label col-md-2">Class</label>
<div class="col-md-10">
<select ng-model="selectedClass" class="form-control" ng-options="p.toString() for p in listClasses">
<option value=""></option>
</select>
</div>
</div>
<div class="form-group">
<label class="control-label col-md-2">Lightweight</label>
<div class="col-md-10">
<input type="checkbox" ng-model="lightweight">
</div>
</div>
</form>
</div>
<div class="modal-footer">
<div class="btn-toolbar pull-right">
<div class="btn-group">
<button type="button" class="btn btn-sm" ng-click="$hide();cancelSave()"> Cancel
</button>
</div>
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<button type="button" ng-show="lightweight" ng-disabled="!selectedClass" class="btn btn-sm" ng-click="createLightEdge(selectedClass)"> Create Edge
</button>
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<button type="button" ng-show="!lightweight" ng-disabled="!selectedClass" class="btn btn-sm" ng-click="setSelectClass(selectedClass)"> Next
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</div> | {
"redpajama_set_name": "RedPajamaGithub"
} | 582 |
\section{Introduction}
\label{sec:intro}
The setting of this paper is studies of associations between exposures and time-to-event outcomes, such as disease diagnosis or death, analysed using Cox regression \cite{Cox:1972,Cox:1975}. Missing data in explanatory variables are common and the impact of ignoring the missing data and performing a `complete-case' analysis on the subset of individuals with no missing data are loss of efficiency and, depending on the missing data mechanism, biased estimates. Multiple imputation (MI) is a widely used approach for handling missing data that involves generating multiple plausible values for the missing data to create multiple imputed datasets. The multiply imputed datasets are each analysed to obtain estimates of interest and corresponding standard errors, which are then combined using rules developed by Rubin (1987) \cite{Rubin:1987}. The way in which the plausible values for missing data are obtained is important, and in general use of a mis-specified imputation model results in invalid inferences. In general it is desirable that the imputation model is compatible with the chosen substantive model. There exist a range of methods for performing MI covering different substantive model types -- see Carpenter and Kenward (2013) \cite{Carpenter:2013} for an overview. Two MI approaches have been described for imputation of missing data on covariates in Cox regression. White and Royston (2009) \cite{White:2009} outlined an approximately compatible method which can be implemented in standard software, and Bartlett et al. (2015) \cite{Bartlett:2015} described an alternative `substantive model compatible' approach which does not require approximations.
In time-to-event analyses it is often of interest to study whether the association of certain covariates with the hazard changes over time. Furthermore, assessment of whether the covariate effect changes over time is the basis of a test of the proportional hazards assumption, which is important aspect of model assessment in Cox regression. Ignoring time-varying effects (TVE) and estimating an `average' hazard ratio can result in misleading conclusions \cite{Schemper:1992}. Cox \cite{Cox:1972} described an extended version of his model to incorporate time-varying effects (TVE) of covariates and there is a large literature on methods for estimating and testing for TVEs in Cox regression: Therneau and Grambsch (2013) \cite{Therneau:2013} (Chapter 6) summarise some of the more popular methods. There is also a more recent literature on model building in Cox regression incorporating TVEs \cite{Buchholz:2011,Abrahamowicz:2007,Wynant:2014,Sauerbrei:2007,Binquet:2008}.
The existing imputation methods for handling missing data in Cox regression \cite{White:2009,Bartlett:2015} do not account for TVEs of covariates, which could result in invalid inferences. In this paper we extend the methods of White and Royston (2009) \cite{White:2009} and Bartlett et al. (2015) \cite{Bartlett:2015} to accommodate imputation of covariates modelled with TVEs in the Cox regression model. The methods are presented for a general form for a TVE. Specific details are given for TVEs modelled using restricted cubic splines, which are flexible and do not require a form for the TVE to be pre-specified. We also present a model selection algorithm which incorporates imputation of missing data into a procedure for testing for proportional hazards, and selecting a flexible functional form for TVEs. Throughout, we make the assumption that data are `missing at random' (MAR) \cite{Rubin:1987,Seaman:2013}. Although the term `time-varying effect' is used, we note that a hazard ratio changing over time does not necessarily correspond to a covariate's causal effect changing over time, but may instead occur when the association between a baseline covariate and the hazard becomes weaker (for example) over time, or due to time-varying confounding.
The paper is organised as follows. In Section \ref{sec:general.MI} we describe extensions to the methods of White and Royston (2009) \cite{White:2009} and Bartlett et al. (2015) \cite{Bartlett:2015} to accommodate TVEs, for a general functional form for the TVEs. Use of restricted cubic splines to model the TVEs is described in Section \ref{sec:func.form}. In Section \ref{sec:model.selection} we discuss testing the proportional hazards assumption and present a model selection algorithm. The proposed methods are investigated using simulation studies, described in Section \ref{sec:sim}, in which several underlying functional forms for the TVEs are considered. The methods are illustrated using data from the Rotterdam Breast Cancer Study in Section \ref{sec:example} and we conclude with a discussion in Section \ref{sec:discussion}. Supplementary Materials provide additional details and R code for implementation of the methods.
\section{MI for Cox regression with time-varying effects (TVE)}
\label{sec:general.MI}
\subsection{Preliminaries}
Let $T$ denote an event or censoring time and $D$ be an indicator of whether an individual had the event ($D=1$) or was censored ($D=0$). For simplicity we focus on a single covariate $X_{1}$ with missing data and a fully observed covariate, $X_{2}$. Extensions to missingness in several variables are described in the Supplementary Materials (Section S2). Under the extended Cox model that allows TVEs of covariates \cite{Cox:1972,Therneau:2013}, the hazard function can be written in the general form
\begin{equation}
h(t|X_{1},X_{2})=h_{0}(t)\exp\left\{f_{X_{1}}(t;\bm{\beta}_{X1})X_{1}+f_{X_{2}}(t;\bm{\beta}_{X2})X_{2}\right\}
\label{eq:gen.haz}
\end{equation}
where $h_{0}(t)$ is the baseline hazard and the potential TVEs for $X_{1}$ and $X_{2}$ are represented respectively by the functions $f_{X_{1}}(t;\bm{\beta}_{X1})$ and $f_{X_{2}}(t;\bm{\beta}_{X2})$. Under the standard Cox proportional hazards model, i.e. with no TVEs, $f_{X_{1}}(t;\bm{\beta}_{X1})=\beta_{X1}$ and $f_{X_{2}}(t;\bm{\beta}_{X2})=\beta_{X2}$.
\subsection{MI overview}
Using MI, the general procedure for obtaining estimates of the model parameters $\bm{\beta}_{X1}$ and $\bm{\beta}_{X2}$ is as follows (\cite{Carpenter:2013}, p. 39).
A model $p(X_{1}|T,D,X_{2}; \alpha)$ with non-informative prior on parameters $\alpha$ is specified for $p(X_{1}|T,D,X_{2})$, the distribution of $X_{1}$ given $T$, $D$ and $X_{2}$. Then, for $m=1, \ldots, M$,
\begin{enumerate}
\item
A value $\alpha^{(m)}$ is drawn from its posterior distribution given the observed data.
\item
For each individual $i$ with missing $X_{1i}$, a value $X_{1i}^{(m)}$ is drawn from $p(X_{1i} | T_i, D_i, X_{2i}; \alpha^{(m)})$, giving an `imputed' data set in which there are no missing values.
\item The substantive model, here the Cox regression model, is fitted to this imputed data set to give estimates $(\hat{\bm{\beta}}^{(m)}_{X1}, \hat{\bm{\beta}}^{(m)}_{X2})$ of $(\bm{\beta}_{X1}, \bm{\beta}_{X2})$, and a corresponding estimate $\hat{\Sigma}^{(m)}$ of ${\rm Var} (\hat{\bm{\beta}}^{(m)}_{X1}, \hat{\bm{\beta}}^{(m)}_{X2})$.
\end{enumerate}
Estimates $(\hat{\bm{\beta}}^{(m)}_{X1}, \hat{\bm{\beta}}^{(m)}_{X2})$ ($m=1, \ldots, M$) and $\hat{\Sigma}^{(m)}$ are combined using `Rubin's rules' \cite{Rubin:1987} to give an overall estimate of $(\bm{\beta}_{X1} ,\bm{\beta}_{X2})$ and of ${\rm Var} (\bm{\beta}_{X1} ,\bm{\beta}_{X2})$.
The main difficulty which arises when the substantive model is a Cox regression is that $p(X_{1}|T,D,X_{2})$ is a non-standard distribution which is only semi-parametrized since $h_{0}(t)$ is non-parametric; therefore we cannot easily draw values from the distribution $p(X_{1}|T,D,X_{2})$ to obtain the imputations. Although in principle any model $p(X_{1}|T,D,X_{2}; \alpha)$ could be used, potentially serious (asymptotic) bias in the estimators of $(\bm{\beta}_{X1} ,\bm{\beta}_{X2})$ and ${\rm Var} (\bm{\beta}_{X1} ,\bm{\beta}_{X2})$ could arise if this model is misspecified. In particular, assuming the substantive model is correctly specified, if the imputation model is not compatible with the substantive model, under certain conditions this implies the imputation model is misspecified \cite{Bartlett:2015}. Consequently, it is desirable that these two models be compatible (or approximately compatible), i.e.\ that there exists a model for the joint distribution $(X_{1}, T, D | X_{2})$ that implies as submodels the model $p(X_{1}|T,D,X_{2}; \alpha)$ used for imputation and the Cox model used for analysis. Two different approaches to using a compatible, or approximately compatible, imputation model have been described by White and Royston (2009) \cite{White:2009} and Bartlett et al. (2015) \cite{Bartlett:2015}, which we refer to respectively as MI-Approx and MI-SMC. In the next two subsections we describe extensions of these imputation methods to accommodate TVEs in the Cox regression model.
\subsection{MI-TVE-Approx}
For the standard Cox proportional hazards model assuming no TVEs ($f_{X1}(t;\bm{\beta}_{X1})=\beta_{X1}$ and $f_{X2}(t;\bm{\beta}_{X2})=\beta_{X2}$), White and Royston (2009) \cite{White:2009} showed that an approximately compatible imputation model for $X_{1}$ is a logistic regression (for binary $X_{1}$) or linear regression (for continuous $X_{1}$) with linear predictor including main effects of $D$, $X_{2}$, $\widehat{H}(t)$ and the interaction between $X_{2}$ and $\widehat{H}(t)$, where $\widehat{H}(t)$ is the Nelson-Aalen estimate of the cumulative hazard. Investigations have found, in the settings examined, that the interaction term adds little \cite{White:2009,Borgan:2015}.
When the substantive model is the extended Cox model with TVEs in \ref{eq:gen.haz}, we can show that an approximately compatible imputation model for $X_{1}$ is a logistic regression (for binary $X_{1}$) or linear regression (for continuous $X_{1}$) with linear predictor including main effects of $X_{2}$, $D f_{X1}(T)$, $\widehat{H}(T)$, $\widehat{H}^{(1)}(T)$ and interactions of $X_{2}$ with $\widehat{H}(T)$ and $\widehat{H}^{(1)}(T)$, where $\widehat{H}^{(1)}(T)$ is the Nelson-Aalen-type estimator $\widehat{H}^{(1)}(T)=\sum_{t\leq T}\frac{td(t)}{n(t)}$ ($d(t)$ and $n(t)$ denote the number of events and number at risk at time $t$). The details of the derivation are given in the Supplementary Materials (Section S1). We refer to the resulting approach as MI-TVE-Approx. In the simulations we will investigate whether it is important to include the higher order cumulative hazard term $\widehat{H}^{(1)}(T)$ and/or the interaction terms $X_{2}\times \widehat{H}(T)$ and $X_{2}\times \widehat{H}^{(1)}(T)$. The imputation procedure is as follows.
\begin{enumerate}
\item Fit the imputation model to the subset of individuals with complete data. For continuous $X_{1}$ this is
{\small
$$
X=\alpha_{0}+\alpha_{1}X_{2}+\alpha_{2}^{\prime}D f_{X1}(T)+\alpha_{3}\widehat{H}(T)+\alpha_{4}\widehat{H}^{(1)}(T)+\alpha_{5}X_{2}\widehat{H}(T)+\alpha_{6}X_{2}\widehat{H}^{(1)}(T)+\epsilon,
\label{eq:imp.model.step1.linear}
$$}
and for binary $X_{2}$
{\small
$$
\mbox{logit }\mbox{Pr}(X_{1}=1|T,D,X_{2})=\alpha_{0}+\alpha_{1}X_{2}+\alpha_{2}^{\prime}D f_{X1}(T)+\alpha_{3}\widehat{H}(T)+\alpha_{4}\widehat{H}^{(1)}(T)+\alpha_{5}X_{2}\widehat{H}(T)+\alpha_{6}X_{2}\widehat{H}^{(1)}(T).
\label{eq:imp.model.step1/logistic}
$$}
\item Take $M$ random draws values of the parameters from their approximate posterior distribution (we refer to \cite{White:2011} for details), denoted $\alpha^{(m)}_{j}$ $(j=0,1\ldots,6)$ (binary and continuous $X_{1}$, and additionally $\sigma^{2(m)}_{\epsilon}$ for continuous $X_{1}$
\item The imputed value of $X_{1i}$ in the $m$th imputed data set is given (for continuous $X_{1}$) by $X_{1i}^{(m)}=\alpha^{(m)}_{0}+\alpha^{(m)}_{1}X_{2}+\alpha^{(m)\prime}_{2}D f_{X1}(T)+\alpha^{(m)}_{3}\widehat{H}(T)+\alpha^{(m)}_{4}\widehat{H}^{(1)}(T)+\alpha^{(m)}_{5}X_{2}\widehat{H}(T)+\alpha^{(m)}_{6}X_{2}\widehat{H}^{(1)}(T)+\epsilon^{(m)}.$, where $\epsilon^{(m)}$ is a random draw from a normal distribution with mean 0 and variance $\sigma^{2(m)}_{\epsilon}$. For binary $X_{1}$, the imputed value is a draw from a Bernoulli distribution with $\mbox{logit } \mbox{Pr}(X_{1}=1|T,D,X_{2})=\alpha^{(m)}_{0}+\alpha^{(m)}_{1}X_{2}+\alpha^{(m)\prime}_{2}D f_{X1}(T)+\alpha^{(m)}_{3}\widehat{H}(T)+\alpha^{(m)}_{4}\widehat{H}^{(1)}(T)+\alpha^{(m)}_{5}X_{2}\widehat{H}(T)+\alpha^{(m)}_{6}X_{2}\widehat{H}^{(1)}(T)$.
\end{enumerate}
\subsection{MI-TVE-SMC}
In the context of the standard Cox proportional hazards model without TVEs, MI-Approx has been found to work well in a range of circumstances \cite{White:2009,Carpenter:2013}. However, the approximation can perform badly in some `extreme' situations, including when there are large effect sizes and when the event rate is high \cite{White:2009}. Bartlett et al (2015) \cite{Bartlett:2015} described an approach, referred to here as MI-SMC, which ensures the imputation model is compatible with the user's chosen substantive model, here a Cox regression (`substantive model compatible' -- SMC). However, they did not accommodate TVEs. We outline this extension, which was first described by Bartlett \cite{Bartlett:2010} in the context of time-dependent covariates, and refer to the resulting method as MI-TVE-SMC. The MI-TVE-SMC imputation procedure is as follows.
First, a model $p(X_{1}|X_{2};\gamma_{X1})$ is specified. For binary $X_{1}$ this may be a logistic regression model and for continuous $X_{1}$ a linear regression model. The steps used to obtain the $m$th imputed data set are then:
\begin{enumerate}
\item Fill in the missing variables with arbitrary starting values, to create a complete data set.
\item Fit the Cox regression model of interest, including TVEs, to the current complete data set to obtain estimates $(\bm{\hat{\beta}}_{X1},\bm{\hat{\beta}}_{X2})$ and their estimated variance $\widehat{\Sigma}$.
Draw values $\bm{\beta}^{(m)}_{X1},\bm{\beta}^{(m)}_{X2}$ from a joint normal distribution with mean $(\hat{\beta}_{X1},\hat{\beta}_{X2})$ and variance $\widehat{\Sigma}$.
\item Calculate Breslow's estimate \cite{Breslow:1972}, denoted $H^{(m)}_{0}(t)$, of the baseline cumulative hazard $H_{0}(t)$ using parameter values $\bm{\beta}^{(m)}_{X1},\bm{\beta}^{(m)}_{X2}$ and current imputations of $X_{1}$.
\item Estimate parameters $\gamma_{X1}$ and their variance by fitting the assumed regression model for $X_{1}$ on $X_{2}$ to the current complete data set. Draw a value $\gamma_{X1}^{*}$ from the approximate joint posterior distribution of $\gamma_{X1}$ \cite{White:2011}.
\item For each individual for whom $X_{1}$ is missing, (a) draw a value $X_{1}^{*}$ from the distribution $p(X_{1}|X_{2};\gamma_{X1}^{*})$, and (b) draw a value $U$ from a uniform distribution on $[0,1]$. Accept the value $X_{1}^{*}$ if
{\footnotesize
$$
\left\{
\begin{array}{ll}
U\leq \exp \left[- \sum_{j:t_{j} \leq T} \Delta H^{(m)}_{0}(t_{j}) \exp\left\{f_{X1}\left(t_{j};\bm{\beta}^{(m)}_{X1}\right)X_{1}^{*} +f_{X2}\left(t_{j};\bm{\beta}^{(m)}_{X2}\right) X_{2}\right\} \right]&\mbox{if }D=0\\
U\leq \Delta H^{(m)}_{0}(T)\exp\left\{1+f_{X1}\left(T;\bm{\beta}^{(m)}_{X1}\right)X_{1}^{*} +f_{X2}\left(T;\bm{\beta}^{(m)}_{X2}\right) X_{2}- \sum_{j:t_{j} \leq T} \Delta H^{(m)}_{0}(t_{j})e^{f_{X1}\left(t_{j};\bm{\beta}^{(m)}_{X1}\right)X_{1}^{*} +f_{X2}\left(t_{j};\bm{\beta}^{(m)}_{X2}\right) X_{2}}\right\}&\mbox{if }D=1
\end{array}\right.
$$}
where $\Delta H^{(m)}_{0}(t)$ denotes the increment in $H^{(m)}_{0}(t)$ at time $t$ and $t_{1},..,t_{k}$ denote the unique failure times. Repeat (a) and (b) until a value $X_{1}^{*}$ is accepted.
\item Return to steps 2--5 until the imputed $X_{1}$ values have converged to a stationary distribution. These are then the imputed values in the $m$th imputed data set.
\end{enumerate}
The difference between the MI-SMC approach, which does not accommodate TVEs, and the MI-TVE-SMC approach is in the terms used for the rejection in step 5, and the fact that a Cox model with TVEs is fitted in Step 2.
We have outlined the MI-TVE-Approx and MI-TVE-SMC approaches for the simple setting of missing data in a single covariate $X_{1}$ with a TVE. Both methods extend to handle missingness in several covariates using the fully conditional specification (FCS) approach (also referred to as multiple imputation by chained equations), in which an imputation model is specified for each partially missing covariate conditional on all the other covariates and an iterative approach is used to fit the imputation models \cite{vanBuuren:2007,White:2011}. Details are provided in the Supplementary Materials (Section S2).
\section{Functional form of time-varying effects (TVE)}
\label{sec:func.form}
In the preceding section the MI methods were described for a general functional form for the TVEs, $f_{X}(t;\bm{\beta}_{X})$. Approaches to modelling TVEs include use of pre-specified parametric functional forms \cite{Quantin:1999} (e.g. $f_{X}(t;\bm{\beta}_{X})=\beta_{0}+\beta_{1}t$), step-functions \cite{Gore:1984,Moreau:1985,Quantin:1999}, fractional polynomials \cite{Sauerbrei:2007}, and splines \cite{Hess:1994,Heinzl:1996,Abrahamowicz:1996,Abrahamowicz:2007,Wynant:2014,Hastie:1993,Gray:1992,Kooperberg1995}. In this paper we focus on using a restricted cubic spline form \cite{Hess:1994, Heinzl:1996} for the TVE function because they allow a flexible form for the TVE with relatively few parameters. Under a restricted cubic spline with $L$ knots at $u_{1},\ldots,u_{L}$ the TVE function for a covariate $X$ is
{\small
\begin{equation}
f_{X}(t;\bm{\beta}_{X})=\beta_{X0}+\beta_{X1}t+\sum_{i=1}^{L-2}\theta_{Xi}\left\{(t-u_{i})^{3}_{{\tiny +}}-\left(\frac{(t-u_{L-1})^{3}_{{\tiny +}}(u_{L}-u_{i})}{(u_{L}-u_{L-1})}\right)+\left(\frac{(t-u_{L})^{3}_{{\tiny +}}(u_{L-1}-u_{i})}{(u_{L}-u_{L-1})}\right)\right\}
\label{eq:rcs}
\end{equation}}
where $(t-u_{i})_{{\tiny +}}$ takes value $(t-u_{i})$ if $(t-u_{i})>0$ and 0 otherwise.
The number of knots used, and the position of the knots, has to be decided by the user and there is no formal theoretical basis for the decision. Hess (1994) \cite{Hess:1994} noted empirical evidence that three to five knots are usually adequate and the fit is not greatly altered by altering the knot positions. Stone (1986) \cite{Stone:1986} also recommended using 5 knots in restricted cubic splines. Hess (1994) \cite{Hess:1994} suggested placing knots at quantiles of the observed follow-up times (including both event times and censoring times); including the outer knots near the extremes; and placing the knots approximately uniformly over the quantiles of the distribution of the follow-up times. Similar recommendations were given by Durrleman and Simon (1989) \cite{Durrleman:1989} in the context of restricted cubic splines for functional forms of covariates in survival analyses. In the simulations we consider using restricted cubic splines with 5 knots placed at percentiles (5th, 25th, 50th, 75th, 95th) of the event time distribution.
In the case of a restricted cubic spline with $L=5$ knots, the MI-TVE-Approx imputation model for $X_{1}$ should include $X_{2}$, $D$, the interaction between $D$ and $T$, interactions of $D$ with \\$\left\{(T-u_{i})^{3}_{{\tiny +}}-\left(\frac{(T-u_{L-1})^{3}_{{\tiny +}}(u_{L}-u_{i})}{(u_{L}-u_{L-1})}\right)+\left(\frac{(T-u_{L})^{3}_{{\tiny +}}(u_{L-1}-u_{i})}{(u_{L}-u_{L-1})}\right)\right\}$ (for $i=1,2,3$), $\widehat{H}(T)$, $\widehat{H}^{(1)}(T)$ and interactions of $X_{2}$ with $\widehat{H}(T)$ and $\widehat{H}^{(1)}(T)$.
\section{Testing the proportional hazards assumption and model selection}
\label{sec:model.selection}
In most contexts, when using Cox regression modelling it is important to assess whether covariates have TVEs, that is, to perform tests of the proportional hazards assumption. TVEs can then be included for covariates for which the proportional-hazards assumption appears not to hold. Tests of proportional hazards based on TVEs modelled using splines have been previously described by Abrahamowicz et al.\ \cite{Abrahamowicz:1996}. With fully observed data on covariates, the proportional hazards assumption can be tested using a likelihood ratio test, comparing a model including TVEs to the model without TVEs. A joint Wald test of the TVE parameters is asymptotically equivalent: assuming a restricted cubic spline for the TVE for $X$ (equation \ref{eq:rcs}, this is a joint test of $\beta_{X1}=\theta_{X1}=\ldots =\theta_{X,L-1}=0$. Wood et al.\ \cite{Wood:2008} described the use of Wald tests for model selection using multiply imputed data, and this was further evaluated by Morris et al.\ \cite{Morris:2015} in the context of covariate transformations based on fractional polynomials. We suggest this approach for tests of TVEs. The joint Wald test of the parameters of interest (null hypothesis $\beta_{X1}=\theta_{X1}=\ldots =\theta_{X,L-1}=0$) is performed using the parameter estimates and the corresponding variance covariance matrix obtained from Rubin's rules. For the purposes of testing the proportional hazards assumption as part of a model assessment and selection procedure, we recommend allowing TVEs for all variables at the imputation stage of the analysis; the importance of doing so for valid tests of the proportional hazards assumption is investigated in the simulation studies.
Finally, we propose an algorithm (the MI-MTVE algorithm) which provides a model selection procedure for identifying TVEs using multiply imputed data. Several authors have proposed algorithms for model selection involving both TVEs and transformation of covariates \cite{Abrahamowicz:2007,Wynant:2014,Sauerbrei:2007}, though all assume fully observed datasets. The MI-MTVE algorithm is an adaptation of the MFPT algorithm of Sauerbrei et al.\ \cite{Sauerbrei:2007}, which uses fractional polynomial transformations of covariates and fractional polynomial forms for TVEs. Our adaptation employs restricted cubic spline transformations, rather than fractional polynomials, for TVEs, and is similar to a procedure advocated by Wynant and Abrahamowicz (2014) \cite{Wynant:2014}. Forwards selection is used to accommodate investigation of TVEs in multiple covariates and selection of a functional form for TVEs using restricted cubic splines with up to 5 knots.
\textbf{MI-MTVE algorithm}
\begin{description}
\item[Step 1] Perform MI-TVE-Approx or MI-TVE-SMC, assuming a restricted cubic spline with 5 knots for the TVE for each covariate, to obtain $M$ imputed data sets.
\item[Step 2] In each imputed data set, fit the model with no TVEs of any covariate (denoted model $\mathcal{M}_0$). Denote the set of covariates by $C$. For each $c~(c \in C)$, fit four TVE models of increasing complexity (indexed by $j$) to each imputed data set: linear form ($f_{X}(t;\bm{\beta}_{X})=\beta_{X0}+\beta_{X1}t$), and restricted cubic splines with 3, 4 and 5 knots.
\item[Step 3] For each covariate $c$, test for TVEs in each model $j$ using joint Wald tests of the TVE parameters based on Rubin's rules. Select the combination of covariate ($c$) and TVE model ($j$) which returns the smallest p-value in the test for TVEs. If no combination of $c$ and $j$ gives a p-value less than a chosen level $\alpha$, stop; the working model without TVEs $\mathcal{M}_0$ is adequate. Otherwise, update the working model $\mathcal{M}_0$ to include TVEs for the covariate $c$ and TVE model $j$ which returned the smallest p-value. Call this new working model $\mathcal{M}_1$.
\item[Step 4] Repeat steps 2--3 with updated working models until there are no remaining covariates $c$ not in the current working model that have a significant TVE (at level $\alpha$) under any TVE model $j$. Stop; this working model is the final selected model.
\end{description}
The estimates of the parameters of the final selected model, and corresponding estimated covariance matrix, are those obtained by applying Rubin's rules to the results from fitting the final model to each imputed data set. The MI is performed only in Step 1 and is based on a TVE for each covariate of the most complex form that we consider in this paper (a restricted cubic spline with 5 knots). This means that a restriction of the imputation model should be compatible with the model selected by the algorithm (termed `semi-compatibility' \cite{liu14,Bartlett:2015,Morris:2015}). The imputation model may include some redundant parameters, but this will not impact on the validity of MI inference.
\section{Simulation study} \label{sec:sim}
We now present a simulation study which was designed to evaluate the MI methods across a variety of data-generating scenarios.
\subsection{Data-generating mechanisms} \label{subs:dgms}
Data were generated for a cohort of 2,000 individuals. Two covariates, $X_{1}$ and $X_{2}$, are considered. Event times $T_{E}$ were generated according to the exponential hazard model
\begin{equation}
h(t|X_{1},X_{2})=\lambda_{E}\exp\left\{f_{X1}(t;\bm{\beta}_{X1})X_{1}+f_{X2}(t;\bm{\beta}_{X2}) X_{2}\right\}
\label{eq:sim.model}
\end{equation}
We consider five forms for the TVEs. These are listed in the table at the top of Figure \ref{fig:tves}. In scenario 1, neither covariate has a time-varying effect. In scenarios 2--5, $X_1$ has TVE but $X_2$ does not. Figure \ref{fig:tves} depicts the form of the TVEs. These forms include examples previously used by Buchholz \cite{Buchholz:2010} and Buchholz and Sauerbrei \cite{Buchholz:2011}.
\begin{figure}
\caption{Time varying effect functions used in simulation studies.}
\label{fig:tves}
\begin{center}
\begin{tabular}{ccll}
\hline
Scenario & TVE & $f_{X_1}(t)$ & $f_{X_2}(t)$ \\
\hline
1 & -- & $0.5$ & $0.5$ \\
2 & $X_1$ & $0.1+0.2t$ & $0.5$ \\
3 & $X_1$ & $0.1+0.8t^{0.3}$ & $0.5$ \\
4 & $X_1$ & $0.32+1.42e^{-t}-0.02t^{0.7}$ & $0.5$ \\
5 & $X_1$ & $\frac{4}{1+e^{1.2(t+0.5)}}+\frac{4}{3(1.1+e^{10-t})}+0.02$ & $0.5$ \\
\hline
\end{tabular} \bigskip \\
\begin{tikzpicture}[baseline]
\begin{axis}[
samples=300,
domain=0:10,
height=8cm,
width=12cm,
restrict y to domain=-.5:2.1,
restrict x to domain=0:10,
xlabel=Follow-up time $t$, ylabel=log-hazard ratio,
ymin=-.4,ymax=2.3,
]
\addplot[black,semithick,opacity=0.5 ] plot(\x,{.5});
\addplot[black,semithick,opacity=0.5 ] plot(\x,{.1+.2*\x});
\addplot[black,semithick,opacity=0.5 ] plot(\x,{.1 + .8*(\x^.3)});
\addplot[black,semithick,opacity=0.5 ] plot(\x,{.32 + (1.42*exp(-\x)) -(.02*(\x^.7)});
\addplot[black,semithick,opacity=0.5 ] plot(\x,{(4/(1 + exp(1.2*(\x+0.5)))) + (4/(3*(1.1+exp(10-\x)))) + 0.02)});
\node[align=center] at(800,100) {\small Scenario 1};
\node[align=center] at(660,150) {\small Scenario 2};
\node[align=center] at(450,190) {\small Scenario 3};
\node[align=center] at(100,220) {\small Scenario 4};
\node[align=center] at(500,30) {\small Scenario 5};
\end{axis}
\end{tikzpicture}
\end{center}
\end{figure}
Random drop out times, $T_{C}$, were generated according to an exponential distribution with rate $\lambda_{C}$, and administrative censoring was imposed after 10 years of follow-up. The observed time for each individual was calculated as $T=\mathrm{min}(T_{E},T_{C},10)$. Values for $\lambda_{E}$ and $\lambda_{C}$ were chosen such that 10\% of individuals have the event of interest and 50\% are censored due to random drop out, with the remainder being administratively censored.
Both binary and continuous covariates are considered. Binary $X_{1}$ was generated from a binomial distribution such that $P(X_{1}=1)=0.2$ and binary $X_{2}$ was generated using $\mbox{logit}\{P(X_{2}=1|X_{1})\}=X_{1}$. Continuous $X_{1}$ and $X_{2}$ were generated from a bivariate normal distribution with means 0, variances 1 and correlation 0.5.
Non-monotone missing data were generated in $X_{1}$ and $X_{2}$ according to a MAR mechanism in which the probability of missingness in $X_{1}$ depends on observed values of $X_{2}$, and vice versa (see Supplementary Materials Section S4). In this, $X_{1}$ is missing for 30\% of individuals and $X_{2}$ for 30\% of individuals, resulting in approximately 50\% of individuals missing at least one of the measurements.
There are 10 main simulation scenarios: five different scenarios for TVEs of $X_1$, and binary and continuous $X_{1}$ and $X_{2}$. Five hundred simulated data sets were generated under each scenario (justified in the Supplementary Materials Section S5). In Section \ref{sec:sims.extra} we present results from additional sensitivity scenarios with a higher event rate, lower level of missingness and a MAR mechanism in which the missingness in $X_{1}$ and $X_{2}$ additionally depends on the outcome $D$.
\subsection{Methods compared}
The methods we investigate are the proposed MI-TVE-Approx and MI-TVE-SMC approaches, and for comparison the corresponding approaches which do not incorporate TVEs (MI-Approx and MI-SMC). We also performed a complete-data analysis (before missing data is introduced) and a complete-case analysis, which uses only the subset with no missing data. In MI-Approx we omitted the interaction terms between covariates and $\widehat{H}(T)$, because their inclusion was not found to result in material differences in the results. For the same reason, in MI-TVE-Approx we omitted the interaction terms and terms involving $\widehat{H}_{1}(T)$. The MI-TVE-Approx imputation model recommended for $X_{1}$ therefore includes $X_{2}$, $D$, the interaction between $D$ and $T$, interactions of $D$ with $\left\{(T-u_{i})^{3}_{{\tiny +}}-\left(\frac{(T-u_{L-1})^{3}_{{\tiny +}}(u_{L}-u_{i})}{(u_{L}-u_{L-1})}\right)+\left(\frac{(T-u_{L})^{3}_{{\tiny +}}(u_{L-1}-u_{i})}{(u_{L}-u_{L-1})}\right)\right\}$ for $i=1,2,3$, and $\widehat{H}(T)$. The recommended imputation model for $X_{2}$ is the same but with $X_{2}$ replaced by $X_{1}$.
In all Cox regression analyses TVE for $X_{1}$ and $X_{2}$ are modelled using a restricted cubic spline with 5 knots placed at percentiles (5th, 25th, 50th, 75th, 95th) of the distribution of the observed event times. This includes a TVE model for $X_{2}$ ($f_{X2}(t;\beta_{X2})$) even though in the data generating process there is no TVE of $X_{2}$. In MI-TVE-Approx and MI-TVE-SMC the TVE was incorporated based on the same functional form.
In the MI analyses we used 10 imputed data sets. For the analysis of studies in practice, we recommend the rule of thumb suggested by White, Royston and Wood (2011) \cite{White:2011} to set the number of imputations to be approximately the same as the percentage of missing data, with a larger number chosen if numerical reproducibility of estimates is desired.
\subsection{Performance measures} \label{subs:perfmeas}
The performance of methods was assessed in a number of ways, described below. Each assessment was performed for both $X_{1}$ and $X_{2}$.
\begin{itemize}
\item Curve-wise estimate of the TVE, presented visually over the follow-up time and averaged over simulation runs.
\item Bias in the estimated curve at 1, 5 and 9 years, and corresponding 95\% Monte Carlo confidence intervals. The bias from the MI methods and the complete-case analysis was calculated relative to the complete-data results, i.e. as the difference between the MI or complete-case estimates and the mean of the complete-data estimates. This was done because the true data generating mechanism is not a restricted cubic spline and therefore we do not necessarily expect to get completely unbiased estimates from the complete-data analysis.
\item Coverage of confidence intervals, estimated at 1, 5 and 9 years, defined as the proportion of simulated data sets for which the true curve lies within the 95\% confidence intervals at time $t$.
\item Rejection fractions for the test of the proportional hazards assumption. For scenario 1, this corresponds to a type I error rate for the TVEs of both $X_1$ and $X_{2}$. For all other scenarios, this corresponds to power for the TVE of $X_1$ and type I error rate for the TVE of $X_2$. The proportional hazards assumption is assessed using a joint Wald test of the TVE parameters.
\end{itemize}
We generated 500 estimated data sets under each scenario. Justification for this, referring to Monte Carlo errors in the bias and coverage, are given in the Supplementary Materials (Section S5).
All simulations and analyses were performed using R. The substantive model was fitted using \texttt{coxph} in the \texttt{survival} package. MI-Approx and MI-TVE-Approx were implemented using \texttt{mice} \cite{vanBuuren:2011}, and MI-SMC using \texttt{smcfcs} (\url{https://github.com/jwb133/smcfcs}). We extended the \texttt{smcfcs} code to implement MI-TVE-SMC. Example code for all methods, and an example simulated data set, are given in the Supplementary Materials (Section S6 and additional files).
\subsection{Simulation results}
\subsubsection{Curve-wise estimates and bias}
Figures 2 and 3 show the curve-wise TVE estimates for covariate $X_{1}$ in the binary and continuous covariates settings. Figures 4 and 5 show the bias in the estimated TVE curves at three time points (1, 5, 9), corresponding to the difference between the mean curves shown in Figures 2 and 3 and the true curve. Similar plots for $X_{2}$, which always has a time-constant effect, are shown in the Supplementary Materials (Figures S1 and S2).
The complete-data and complete-case analyses give approximately unbiased TVE estimates, except for some bias in the complete-case analysis in the extremes of some curves. As noted earlier, the complete-data analysis could give estimates with some slight bias because the data were not generated under the restricted cubic spline model which is used in the analysis. Note that we expect the complete-case analysis to give an approximately unbiased result because the missingness does not depend on the outcome. The MI methods which accommodate TVEs, MI-TVE-Approx and MI-TVE-SMC, perform similarly for binary $X_{1}$ and give estimated TVE estimates similar to that from the complete-data analysis. However, for continuous $X_{1}$ only MI-TVE-SMC performs well in general, with MI-TVE-Approx giving clearly biased estimates in scenarios 2 and 3 at times where the TVE is quite large. MI-TVE-Approx requires additional approximations for continuous covariates and the approximation does not perform well in these scenarios. Poor performance of MI-Approx in scenarios with continuous covariates and large effect sizes has been found previously in the setting without TVEs \cite{White:2009}. The results show that failing to account for the TVE in the imputation, as in MI-Approx and MI-SMC, results in a biased estimate of the TVE curve. The bias is such that the TVE appears attenuated.
Tables of coverages of the estimated TVE curves at three time points are shown in Supplementary Tables S1 and S2. The coverages tend to be higher than the nominal 95\% level and many are 100\%, including in the complete-data analyses. Coverage not at the nominal level has been previously observed for spline-based models \cite{Cummins:2001}.
\subsubsection{Tests of the proportional hazards assumption}
Tables 1 and 2 show the percentage of simulations in which the proportional hazards assumption (based on joint Wald tests) was rejected at the 5\% level for $X_{1}$ and $X_{2}$, in the binary and continuous covariates settings. In Scenario 1, where neither $X_1$ nor $X_{2}$ has a TVE, the percentage of simulations in which the null hypothesis of proportional hazards was rejected was close to 5\% in the complete-data and complete-case analyses, indicating approximately correct Type I errors. The Type I error rates from MI-TVE-SMC were slightly inflated in some scenarios.
In scenario 2-5 with TVEs for $X_{1}$, the power to reject the PH null hypothesis varied under the complete-data analysis, from 100\% (continuous covariates, scenario 2) to 33\% (binary covariates, scenario 3). Power was generally lower in the setting with binary covariates. Power was reduced under the complete-case analysis, for example in scenario 4 with continuous covariates the power from the complete-case analysis was 77\% compared to 99\% in the complete-data analysis, and in scenario 4 with binary covariates the power from the complete-case analysis was 17\% compared to 56\% in the complete-data analysis. For binary covariates the power under MI-TVE-Approx and MI-TVE-SMC was much increased relative to the complete-case analysis and was highest for MI-TVE SMC. Power using MI-TVE-SMC was also high in the setting with continuous covariates, but lower for MI-TVE-Approx, and in scenarios 2 and 3 lower than that from the complete-case analysis; this is partly a consequence of the bias observed using MI-TVE-Approx for continuous covariates. The results show that if the TVEs are ignored in the imputation (MI-Approx and MI-SMC) there is a large loss of power to reject the null hypothesis of proportional hazards across all scenarios, and power from these methods was lower than that from the complete-case analysis.
\subsection{Additional simulation investigations}
\label{sec:sims.extra}
We investigated the performance of the methods in three additional situations:
\begin{itemize}
\item[(i)] Missingness depends on the outcome (see Supplementary Materials Section S4). Missingness depending on the outcome is plausible if there is an underlying latent feature which is associated with the subsequent outcome and with missingness.
\item[(ii)] 50\% of individuals have the event. This may not be a common situation but is relevant for certain clinical studies.
\item[(iii)] A lower percentage of individuals with missing covariate data. The percentage of individuals missing $X_{1}$ and missing $X_{2}$ was reduced to 10\% (see Supplementary Materials Section S4), which results in approximately 20\% of individuals missing at least of the measurements.
\end{itemize}
Other aspects of the simulations were as described above. For additional simulations (i) and (ii) results are presented for scenario 4 (Figure \ref{fig:tves}) with binary covariates, representing a situation in which the association between $X_{1}$ and the hazard becomes weaker over time. For additional simulation (iii) we focused on scenario 2 (Figure \ref{fig:tves}) with continuous covariates, for which we found biased estimates using MI-TVE-Approx in the earlier simulation results.
When the missingness depends on the outcome the complete case analysis gives biased estimates (\ref{fig:sim.MARd}). The results show that the proposed MI methods continue to perform well, while ignoring TVEs in the imputation still results in bias, as we would expect based on our earlier results. The results in Figure \ref{fig:sim.50pcevent} show that the proposed methods continue to perform well in a situation in which 50\% of individuals have the event. When the proportion of individuals with missing data is reduced the bias from MI-TVE-Approx in scenario 2 with continuous covariates is reduced (Figure \ref{fig:sim.10pcmiss}), but still evident when the time-varying effects is large.
\section{Illustration: Rotterdam Breast Cancer Study} \label{sec:example}
The methods were illustrated using data on 2,982 individuals with primary breast cancer from the Rotterdam tumour bank. This data set is freely available (we used the data set provided at
http://portal.uni-freiburg.de/imbi/Royston-Sauerbrei-book/index.html\#datasets) and was used by Sauerbrei et al (2007) \cite{Sauerbrei:2007} and Royston and Sauerbrei (2008) \cite{Royston:2008} to illustrate time-varying exposure effects. Individuals were followed-up from the time of breast cancer diagnosis to a composite event of the first of disease recurrence or death due to breast cancer. Over the course of follow-up, which ranged from 1 to 231 months, 1,518 individuals (51\%) had the outcome of interest and the remainder were censored. In this illustration we focus on eight variables used by Sauerbrei et al.\ (2007) \cite{Sauerbrei:2007} and Royston and Sauerbrei (2008) \cite{Royston:2008}: age, tumour size 1 ($\leq$ 20mm, $>$ 20mm), tumour size 2 ($\leq$ 50mm, $>$ 50mm), tumour grade (grade 2 or 3 versus grade 1), squared transformed number of positive lymph nodes ($\mathrm{enodes}=\exp(-2\times0.12\times\mathrm{nodes})$), treatment with hormonal therapy (yes vs.\ no), treatment with chemotherapy (yes vs.\ no), and transformed progesterone receptors (pmol/l) ($\log(\mathrm{pgr}+1)$). Sauerbrei et al (2007) \cite{Sauerbrei:2007} and Royston and Sauerbrei (2008) \cite{Royston:2008} detected time-varying effects for two of the variables, tumour size 1 and $\log(\mathrm{pgr}+1)$, via interactions with log time ($f_{X}(t;\bm{\beta}_{X})=\beta_{X0}+\beta_{X1}\log t$ in our notation).
For this illustration we generated missing data at random (MAR) in five variables (tumour grade, $\mathrm{enodes}$, hormonal therapy, chemotherapy, and $\log(\mathrm{pgr}+1)$) with the probability of missingness depending on age and tumour size ($e^{-9+0.1\times \mbox{age}-\mbox{tumour size 2}}/(1+e^{-9+0.1\times \mbox{age}-\mbox{tumour size 2}})$). The missing data were generated conditionally independently for each variable such that approximately 5\% of individuals have missing data in any given variable. This resulted in approximately 20\% of individuals having missing data on at least one variable.
We performed the following analyses: a complete-data analysis before missingness was introduced; a complete-case analysis on the subset with no missing data; MI-Approx; MI-SMC; MI-TVE-Approx; MI-TVE-SMC. The imputations allowing TVEs assumed restricted cubic splines with 5 knots for all covariates. In each analysis the substantive model was fitted first assuming no TVEs. A test of proportional hazards was performed for each covariate in turn (using joint Wald tests), based on TVE models of four forms (linear form, and restricted cubic splines with 3, 4 and 5 knots). The TVE form giving the smallest p-value was selected. This corresponds to the first step of the MI-MTVE algorithm. The MI-MTVE algorithm was then applied to arrive at a final model. For the complete data and complete-case analyses the algorithm was applied using the single complete-data or complete-case data set. A p-value cut-off of 0.01 was used in the model selection. In the MI analyses we used 20 imputations.
The results are shown in Table \ref{table:rotterdam} and Figure \ref{fig:rotterdam}. In tests of the proportional hazards assumption for individual covariates, all methods identified strong evidence for a TVE for all variables except hormone therapy (Table \ref{table:rotterdam}(a)). However, the functional form for the TVE which gave the smallest p-value in the test differed across methods. Covariate $\log (\mathrm{pgr}+1)$ was selected to the final model with a TVE under all methods, and tumour size 1 was selected to the final model with a TVE in all analyses except the complete-case analysis. The enodes covariate was identified as having a TVE in the final model using the MI analyses, but not the complete-data or complete-case analyses. Age was identified as having a TVE in the final model using the complete-case analysis but not the other methods. The MI methods gave similar estimated TVE forms for $\log (\mathrm{pgr}+1)$, tumour size 1 and enodes (Figure \ref{fig:rotterdam}). Using MI-TVE-SMC gave very wide confidence bounds for the $\log (\mathrm{pgr}+1)$ estimates, while the other MI methods performed better, giving narrower confidence bounds than under the complete-case analysis. The figure showing results for tumour size 1, which was identified to have a TVE under all methods apart from the complete-case analysis, illustrates that ignoring the missing data could result in qualitatively different conclusions about the nature of the association of this variable with the outcome. For covariates without TVEs, all methods gave similar estimated hazard ratios, while the standard errors from the MI analyses were smaller than those from the complete-case analysis, illustrating the loss of efficiency from the complete-case analysis (Table \ref{table:rotterdam}(b)).
\section{Discussion} \label{sec:discussion}
In this article we have introduced two multiple imputation methods allowing for time-varying effects (TVEs) to be included in Cox regression models. In the absence of TVEs, the methods of White and Royston \cite{White:2009} (MI-Approx) and Bartlett et al \cite{Bartlett:2015} (MI-SMC) can be used. MI-Approx is conceptually simpler, more convenient to code and faster to run, while MI-SMC method has better statistical properties. Our two proposals are extensions of these methods. The methods were described for a general functional form for TVEs. Researchers use different approaches to modelling TVEs. The correct functional form for a TVE is typically not known in advance, and so it is desirable to allow a flexible form. We therefore focused on a situation in which TVEs are modelled using restricted cubic splines. In some studies it may be desirable for the TVE to be a simply step function and we provided details on this in the Supplementary Materials (Section S3). In simulation studies, we used imputation model assuming a 5-knot restricted cubic spline for the TVE. For binary covariates with missing data, both of our proposed methods performed well. However, the performance of the approximate method (MI-TVE-Approx) was slightly disappointing for continuous covariates when the effect size is large, though it still outperformed complete case analysis in all but scenario 2, and the observed bias was found to be smaller when the proportion with missing data is lower. The SMC method (MI-TVE-SMC) performed well across all scenarios, minimising bias, retaining the size of tests for non-proportional hazards and maximising power compared to other methods across all scenarios. Our results showed that ignoring TVEs in the imputation model will result in incorrect type I errors in the test for non-proportional hazards when the null hypothesis of proportional hazards is true, and in a large loss of power to detect a TVE when one exists.
In practice, TVEs will often appear in the context of model building, including tests of the proportional hazards assumption. We therefore proposed the MI-MTVE model selection algorithm, an adaptation the `MFPT' algorithm of Sauerbrei et al. \cite{Sauerbrei:2007}, for such settings. We applied our proposed methods to the analysis of the Rotterdam breast cancer study data, followed by the MI-MTVE model selection algorithm. The methods led to different results, demonstrating that the choice of method will impact on substantive conclusions. Our algorithm draws on earlier work by Wood et al. \cite{Wood:2008} on variable selection methods using multiply imputed data. There is a sizeable literature on model selection incorporating estimation of TVEs, without including treatment of missing data. Berger et al (2003) \cite{Berger:2003} proposed the use of fractional polynomials \cite{Royston:1994} to select parsimonious forms for TVEs in Cox regression. Sauerbrei et al (2007) \cite{Sauerbrei:2007} proposed a model selection algorithm for use in Cox regression in which both the functional form for continuous covariates and the functional form for TVEs of covariates are modelled using fractional polynomials (the MFPT algorithm). See also Royston and Sauerbrei (2008) \cite{Royston:2008} (Chapter 11). Abrahamowicz et al (2007) \cite{Abrahamowicz:2007} and Wynant and Abrahamowicz (2014) (\cite{Wynant:2014} also described methods for joint estimation of time-varying and non-linear effects based on splines. Areas for further work include the extension of the methods proposed in this paper to a setting in which TVEs are modelled using fractional polynomials, and to allow selection of functional forms for continuous variables and covariate interactions. This will build on the work Morris et al (2015) \cite{Morris:2015} on how to incorporate MI into a fractional polynomial model building procedure for explanatory variables. The MI-TVE-SMC approach is particularly suitable for extensions involving transformed covariates. Finally, further work is needed to investigate the validity of inferences following data-dependent model selection processes in the missing data context.
In the Supplementary Materials we provide example R code which can be used to implement the proposed imputation models. MI-TVE-Approx is straightforward to apply in standard software, and although we provide example R code, this method can also be easily applied in Stata (\texttt{mi impute}) or SAS (\texttt{PROC MI}), for example. MI-SMC (not incorporating TVEs) can be applied using the \texttt{smcfcs} package in R and Stata \cite{BartlettMorris:2015}. We have also provided an adaptation of this code for implementing MI-TVE-SMC in the setting with two covariates with TVEs, as in the simulation studies, and work is underway to make a more general version available.
There are of course limitations to this work. In particular, in some settings complete case analysis will be unbiased and MI biased. It follows that our methods are only applicable to settings in which MI is judged to be the best approach. When using our proposed methods, other forms of mis-specification of the imputation model could result in bias and this should be borne in mind, as in any MI analysis, especially for partially missing variables which are continuous for which the normality assumption may not hold. Further work is also need to investigate the performance of different approaches to model selection in this context. The general results given for MI-TVE-Approx and MI-TVE-SMC assumed that any censoring occurs independently of covariates with missing data. In MI-TVE-Approx censoring depending on covariates with missing data can be accommodated by adding a further term, $\widehat{H}_{C}(T)$, into the imputation model, which denotes the Nelson-Aalen estimate of the cumulative hazard for the censoring \cite{Borgan:2015}. MI-SMC has been extended to allow competing risks \cite{BartlettTaylor:2016} and this can be used to handle dependence of right-censoring on variables with missing data by modelling the censoring as a competing event. MI-TVE-SMC can be extended in the same way. In both cases, it is assumed that the association between covariates and the hazard for censoring is not time-varying. Event times are also commonly subject to left-truncation. Using MI-TVE-Approx, it can be shown that left-truncation can be accommodated by replacing $\widehat{H}(T)$ by $\widehat{H}(T)-\widehat{H}(T_{L})$ (and $\widehat{H}^{(1)}(T)$ by $\widehat{H}^{(1)}(T)-\widehat{H}^{(1)}(T_{L})$) in the imputation model, and also $\widehat{H}_{C}(T)$ by $\widehat{H}_{C}(T)-\widehat{H}_{C}(T_{L})$ if the censoring is suspected to depend on partially missing covariates. MI-TVE-SMC can also be extended to accommodate left-truncation, however further work is needed on this topic, including to implement the methods in the software.
We have focused on estimation of TVEs by modelling these in the Cox regression model. There exist other methods for estimating and testing for TVEs in Cox regression. Schoenfeld residual plots can be used to visually assess the proportional hazards assumption \cite{Schoenfeld:1982} and smoothed residuals can be used to estimate TVEs \cite{Grambsch:1994,Winnett:2003}. Scheike and Martinussen (2004) \cite{Scheike:2004} outlined tests for proportional hazards based on an iterative procedure to estimate cumulative regression coefficients, which does not require specification of the functional form for time-varying effects. Ng'andu (1997) \cite{Ngandu:1997} summarized and compared several tests for the proportional hazards assumption. Other methods for estimating TVEs include those based on a kernel-weighted local partial likelihood \cite{Tian:2005} and penalized partial likelihoods \cite{Zucker:1990,Verweij:1995,Yan:2012}. Buchholz and Sauerbrei (2011) \cite{Buchholz:2011} proposed a measure for use in choosing between different models for the time-varying effect to discover which is closest to the true shape.
In summary, for settings in which MI is judged to be appropriate and TVEs are a feature of the analysis, the approaches we have described should be used. Where computational time is not too large, the MI-TVE-SMC approach is recommended, though MI-TVE-Approx should also perform well if all covariates with missingness are binary or if effect sizes are small. Ignoring TVEs in the imputation may result in biased estimates and misleading conclusions.
\textbf{Acknowledgements} The authors are grateful to Dr Ian White (MRC Clinical Trials Unit at UCL, UK), Professor Mike Kenward (Department of Medical Statistics, London School of Hygiene and Tropical Medicine, UK) and Dr Jonathan Bartlett (Statistical Innovation Group, AstraZeneca, UK) for comments on this work and to Professor Patrick Royston (MRC Clinical Trials Unit at UCL, UK) for advice on the example.
Ruth Keogh is funded by a Medical Research Council Methodology Fellowship (MR/M014827/1).
\clearpage
\begin{figure}\caption{Curve-wise estimates of TVEs for covariate $X_{1}$ in the setting with binary covariates $X_{1}$ and $X_{2}$. The thick dotted black line indicates the true curve.}
\begin{center}
\includegraphics[width=.8\textwidth]{paper_plots_binary_X1.jpg}
\end{center}
\end{figure}
\begin{figure}\caption{Curve-wise estimates of TVEs for covariate $X_{1}$ in the setting with continuous covariates $X_{1}$ and $X_{2}$. The thick dotted black line indicates the true curve.}
\begin{center}
\includegraphics[width=.8\textwidth]{paper_plots_conts_X1.jpg}
\end{center}
\end{figure}
\begin{figure}\caption{Bias in the estimated TVE curve at three time points for covariate $X_{1}$ in the setting with binary covariates $X_{1}$ (black) and $X_{2}$ (grey). The point indicates the bias and the bar indicates the 95\% confidence interval.}
\begin{center}
\includegraphics[scale=1]{paper_biasplots_binary_X1.jpg}
\end{center}
\end{figure}
\begin{figure}\caption{Bias in the estimated TVE curve at three time points for covariate $X_{1}$ in the setting with continuous covariates $X_{1}$ (black) and $X_{2}$ (grey). The point indicates the bias and the bar indicates the 95\% confidence interval.}
\begin{center}
\includegraphics[scale=1]{paper_biasplots_conts_X1.jpg}
\end{center}
\end{figure}
\begin{table}[ht]
\caption{Percentage of simulations in which the null hypotheses of proportional hazards for $X_{1}$ and $X_{2}$ were rejected using joint Wald tests for binary (left columns) and continuous (right columns) exposures. For a given rejection percentage $\pi$, the Monte Carlo SE is $\sqrt{\tfrac{\pi(100-\pi)}{500}\times \tfrac{1}{100}}$}
\centering
\begin{tabular}{l|rl|rl|rl|rl|rl}
\hline
& \multicolumn{2}{c}{Scenario 1} & \multicolumn{2}{c}{Scenario 2} & \multicolumn{2}{c}{Scenario 3} & \multicolumn{2}{c}{Scenario 4} & \multicolumn{2}{c}{Scenario 5} \\
& $X_1$ & $X_2$ & $X_1$ & $X_2$ & $X_1$ & $X_2$ & $X_1$ & $X_2$ & $X_1$ & $X_2$ \\
\hline
\textit{Binary $X1,X2$} & & & & & & & & & & \\
Complete data & 3 & 3 & 89 & 3 & 33 & 3 & 56 & 6 & 45 & 4 \\
Complete case & 2 & 3 & 42 & 3 & 14 & 2 & 17 & 2 & 14 & 3 \\
MI-Approx & 0 & 0 & 21 & 0 & 3 & 0 & 4 & 0 & 2 & 0 \\
MI-SMC & 0 & 0 & 17 & 0 & 2 & 0 & 3 & 0 & 1 & 0 \\
MI-TVE-Approx & 2 & 3 & 67 & 3 & 16 & 2 & 27 & 2 & 21 & 2 \\
MI-TVE-SMC & 3 & 4 & 68 & 6 & 24 & 6 & 34 & 5 & 27 & 5 \\
\hline
\textit{Continuous $X1,X2$} & & & & & & & & & & \\
Complete data & 7 & 5 & 100& 3 & 79 & 5 & 99 & 3 & 96 & 4 \\
Complete case & 6 & 6 & 94 & 5 & 42 & 4 & 77 & 4 & 60 & 5 \\
MI-Approx & 0 & 0 & 68 & 0 & 6 & 0 & 43 & 0 & 27 & 0 \\
MI-SMC & 0 & 0 & 82 & 0 & 12 & 0 & 46 & 0 & 29 & 0 \\
MI-TVE-Approx & 5 & 5 & 90 & 3 & 26 & 4 & 86 & 4 & 71 & 4 \\
MI-TVE-SMC & 10 & 9 & 99 & 8 & 57 & 6 & 89 & 6 & 78 & 8 \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\caption{Results from additional simulations in which the probability of missingness in $X_{1}$ and $X_{2}$ depends on $D$. Results are from scenario 4 in the situation with binary covariates. The upper-left plot shows the curve-wise estimates of TVEs for covariate $X_{1}$. The thick dotted black line indicates the true curve. The upper-right table shows the percentage of simulations in which the null hypotheses of proportional hazards for $X_{1}$ and $X_{2}$ were rejected using joint Wald tests. The lower plot shows the bias in the estimated TVE curve at three time points for $X_{1}$ (in black) and $X_{2}$ (grey)}\label{fig:sim.MARd}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.65]{scen4_binary_X1_MARd2.jpeg}
\end{minipage}
\begin{minipage}[b]{0.35\textwidth}
\centering
\begin{tabular}{rrr}
\hline
& X1 & X2 \\
\hline
Complete data & 58 & 5 \\
Complete case & 14 & 2 \\
MI-Approx & 2 & 0 \\
MI-SMC & 1 & 0 \\
MI-TVE-Approx & 30 & 4 \\
MI-TVE-SMC & 38 & 0 \\
\hline
\end{tabular}
\vspace{1.2cm}
\end{minipage}
\includegraphics[scale=0.8]{biasplot_scen4_binaryX_MARd2_relcompdata.jpeg}
\end{figure}
\begin{figure}
\caption{Results from additional simulations in which 50\% of individuals had the event. Results are from scenario 4 in the situation with binary covariates. The upper-left plot shows the curve-wise estimates of TVEs for covariate $X_{1}$. The thick dotted black line indicates the true curve. The upper-right table shows the percentage of simulations in which the null hypotheses of proportional hazards for $X_{1}$ and $X_{2}$ were rejected using joint Wald tests. The lower plot shows the bias in the estimated TVE curve at three time points for $X_{1}$ (in black) and $X_{2}$ (grey)}\label{fig:sim.50pcevent}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.65]{scen4_binary_X1_50pcevent.jpeg}
\end{minipage}
\begin{minipage}[b]{0.35\textwidth}
\centering
\begin{tabular}{rrr}
\hline
& X1 & X2 \\
\hline
Complete data & 100 & 3 \\
Complete case & 96 & 5 \\
MI-Approx & 90 & 0 \\
MI-SMC & 86 & 0 \\
MI-TVE-Approx & 99 & 3 \\
MI-TVE-SMC & 100 & 9 \\
\hline
\end{tabular}
\vspace{1.2cm}
\end{minipage}
\includegraphics[scale=0.8]{biasplot_scen4_binaryX_50pcevent_relcompdata.jpeg}
\end{figure}
\begin{figure}
\caption{Results from additional simulations in which the proportion of individuals missing $X_{1}$ or $X_{2}$ was reduced to 20\%. Results are from scenario 2 in the situation with continuous covariates. The upper-left plot shows the curve-wise estimates of TVEs for covariate $X_{1}$. The upper-right table shows the percentage of simulations in which the null hypotheses of proportional hazards for $X_{1}$ and $X_{2}$ were rejected using joint Wald tests. The lower plot shows the bias in the estimated TVE curve at three time points for $X_{1}$ (in black) and $X_{2}$ (grey)}\label{fig:sim.10pcmiss}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.65]{scen2_conts_X1_10pcmiss.jpeg}
\end{minipage}
\begin{minipage}[b]{0.35\textwidth}
\centering
\begin{tabular}{rrr}
\hline
& X1 & X2 \\
\hline
Complete data & 100 & 5 \\
Complete case & 100 & 3 \\
MI-Approx & 99 & 0 \\
MI-SMC & 100 & 1 \\
MI-TVE-Approx & 99 & 2 \\
MI-TVE-SMC & 100 & 5 \\
\hline
\end{tabular}
\vspace{1.2cm}
\end{minipage}
\includegraphics[scale=0.8]{biasplot_scen2_contsX_10pcmiss_relcompdata.jpeg}
\end{figure}
\begin{table} \label{table:rotterdam}
\caption{Results from the Rotterdam Breast Cancer Study.}\label{table:rotterdam}
\begin{subtable}[t]{1\linewidth}
\caption{p-values from joint Wald tests of the null hypothesis of no TVEs for each covariate, based on the model which gave the smallest p-value, and the form of that model (`p (form*)'). $^\star$`lin' denotes a TVE of linear form $f_{X}(t;\beta_{X})=\beta_{0}+\beta_{1}t$. `k3', `k4', `k5' denote restricted cubic spline forms for the TVE with 3, 4 and 5 knots.}
{\footnotesize
\begin{center}
\begin{tabular}{lrrrrrr}
\hline
Covariate & Complete-data & Complete-case& MI-Approx & MI-SMC & MI-TVE-Approx & MI-TVE-SMC \\
\hline
Age & 0.011 (k4) & $<0.001$ (k4) & $<0.001$ (k4)& $<0.001$ (k4) & $<0.001$ (k4) & $<0.001$ (k4) \\
Size 1 & $<0.001$ (lin) & $<0.001$ (lin) & $<0.001$ (k3)& $<0.001$ (k3) & $<0.001$ (k3) & $<0.001$ (k3) \\
Size 2 & 0.003 (lin) & 0.002 (lin) & 0.003 (lin)& 0.003 (lin) & 0.004 (lin) & 0.003 (lin) \\
Grade & 0.054 (lin) & 0.195 (lin) & $<0.001$ (k4)& $<0.001$ (k4) & $<0.001$ (k4) & $<0.001$ (k4) \\
enodes & $<0.001$ (lin) & $<0.001$ (lin) & $<0.001$ (k4)& $<0.001$ (k4) & $<0.001$ (k4) & $<0.001$ (k4) \\
Hormone therapy & 0.122 (k5) & 0.272 (k5) & 0.622 (lin)& 0.522 (lin) & 0.581 (lin) & 0.586 (lin) \\
Chemotherapy & 0.007 (k4) & 0.006 (k3) & 0.004 (k3)& 0.004 (k3) & 0.004 (k3) & 0.004 (k3) \\
log(pgr+1) & $<0.001$ (k3) & $<0.001$ (k3) & $<0.001$ (k4)& $<0.001$ (k4) & $<0.001$ (k4) & $<0.001$ (k3)\\
\hline
\end{tabular}
\end{center}}
\end{subtable}
\begin{subtable}[t]{1\linewidth}
\caption{Estimated log hazard ratios and standard errors (`Est(SE)') from the final model selected using the MI-MTVE algorithm, for covariates with no TVE. Log hazard ratios for covariates with TVEs are shown graphically in Figure \ref{fig:rotterdam}, and the corresponding covariates are labelled 'TVE' in the table.}
{\footnotesize
\begin{center}
\begin{tabular}{lllllll}
\hline
Covariate & Complete-data & Complete-case& MI-Approx & MI-SMC & MI-TVE-Approx & MI-TVE-SMC \\
\hline
Age &-0.013 (0.002)&TVE (k4)&-0.013 (0.002)&-0.013 (0.002)&-0.013 (0.002) &-0.013 (0.002)\\
Size 1 &TVE (lin)&0.250 (0.066)&TVE (lin)&TVE (lin)&TVE (lin)&TVE (lin) \\
Size 2 &0.151 (0.081)&0.160 (0.089)&0.132 (0.081) &0.133 (0.081)&0.137 (0.082)&0.129 (0.082) \\
Grade &0.375 (0.065)&0.367 (0.072)&0.364 (0.066)&0.367 (0.067)&0.356 (0.068) & 0.371 (0.067)\\
enodes &-1.697 (0.084)&TVE (lin)&TVE (k4)&TVE (k4)&TVE (k4)&TVE (k4) \\
Hormone therapy &-0.413 (0.085)&-0.334 (0.099)&-0.441 (0.090)&-0.430 (0.088)&-0.432 (0.091)&-0.442 (0.089) \\
Chemotherapy &-0.447 (0.073)&-0.423 (0.077)&-0.428 (0.074)&-0.451 (0.074)&-0.434 (0.074) &-0.446 (0.074)\\
log(pgr+1) &TVE (k3)&TVE (k3)&TVE (k4)&TVE (k4)&TVE (k4) &TVE (k3)\\
\hline
\end{tabular}
\end{center}}
\end{subtable}
\end{table}
\begin{figure}\caption{Results from the Rotterdam Breast Cancer Study. Plots showing estimated log hazard ratios as a function of time, for variables found to have a TVE using one or more methods. Thick lines indicate the estimates and the thin lines indicate corresponding 95\% confidence bounds.}\label{fig:rotterdam}
\begin{center}
\includegraphics[scale=0.8]{rotterdam_example.jpg}
\end{center}
\end{figure}
\clearpage
\bibliographystyle{wileyj} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,738 |
Q: Why playwright miss pattern url? I need to handle request with certain url and im trying to do it like this:
await page.route("**/api/common/v1/play?**", handle_data_route)
But it also handles a url like this: api/common/v1/play_random?
A: Try using a regular expression instead
await page.route(/^.*\/api\/common\/v1\/play\?.*$/, handle_data_route)
| {
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# Elogios para
# Casi una mujer
de _Esmeralda Santiago_
"Una historia universal... convertida en una historia especial por la sencillez y la honestidad de Santiago al decirla."
_—The Baltimore Sun_
"Santiago captura la fuerza, los contornos y la dinámica de la familia latina."
_—Latina_
"Cautivante... Santiago mantiene con tal equilibrio la voz de una mujer joven a punto de madurar, que sentimos que estamos ahí mismo con ella."
_—Ft. Lauderdale Sun-Sentinel_
"Santiago no protege los sentimientos de nadie sino que expone la verdad como la ve, y transporta a los lectores, en apremiantes marejadas, a otros mundos."
_—The Dallas Morning News_
"Santiago escribe con una elegancia tal para los detalles, el humor y las emociones complejas, que arrastra a los lectores hacia un encantador, aunque a veces desgarrador, viaje personal."
_—The Orlando Sentinel_
Obras por la misma autora
_Cuando era puertorriqueña_
_El sueño de América_
#
# _Esmeralda Santiago_
Casi una mujer
Esmeralda Santiago llegó a los Estados Unidos desde Puerto Rico a los trece años de edad, hizo su escuela intermedia en Brooklyn y su escuela superior en la Performing Arts High School en la ciudad de Nueva York. Después de los extraordinarios años descritos en este libro, se graduó de Harvard University y obtuvo una Maestría del Sarah Lawrence College. Santiago es la autora de _Cuando era puertorriqueña_ y _El sueño de América_ , y es co-editora, con Joie Davidow, de _Las Christmas_. Santiago vive en Westchester County, Nueva York, con su esposo Frank, su hijo Lucas, y su hija Ila.
#
_Una nota sobre la traductora_
Nina Torres-Vidal es puertorriqueña. Profesora de lenguas y literatura de la Universidad del Sagrado Corazón en San Juan, Puerto Rico, sus intereses investigativos son la literatura comparada, la literatura autobiográfica, los estudios del género y la teología feminista.
Primera Edición en Español de Vintage, Septiembre de 1999
© 1999 por Alfred A. Knopf, Inc.
Todos los derechos reservados bajo las Convenciones Panamericanas e Internacionales sobre Derechos de Autor. Publicado en los Estados Unidos por Vintage Books, una división de Random House, Inc., New York, y simultáneamente en Canada por Random House of Canada Limited, en Toronto. Originalmente publicado en carpeta dura en los Estados Unidos, por Perseus Books, miembro de Perseus Book Group, Reading, Massachusetts, en 1998. Copyright © 1998 by Esmeralda Santiago
Vintage es una marca registrada y Vintage Español y colofón son marcas de Random House, Inc.
Biblioteca del Congreso Catalogando-en-Datos para Publicación Santiago, Esmeralda.
[Almost a woman. Spanish]
Casi una mujer / Esmeralda Santiago : traducción de Nina Torres-Vidal.
p. cm.
ISBN 0-375-70526-0
eBook ISBN: 978-0-8041-5339-3
1. Santiago, Esmeralda—Childhood and youth. 2. Puerto Rican women—New York (State)—New York Biography. 3. Puerto Ricans—New York (State)—New York Biography. 4. New York (N.Y.) Biography. 5. Brooklyn (New York, N.Y.) Biography. I. Title.
974.7′1004967295′0092 — dc21
[B] 99-33347
Fotografía de la autora © Frank Cantor
www.vintagebooks.com
v3.1
# Índice
_Cubierta_
_Otros libros de este autor_
_Sobre el autor_
_Una nota sobre la traductora_
_Página del título_
_Derechos de autor_
_"Martes, ni te cases ni te embarques ni de tu familia te apartes."_
_"Te puede pasar algo."_
_"A mí no me importa lo que hagan esas americanas."_
_"Negi, ¿tú vas a ser famosa?"_
_"Pero siguen siendo ilegítimos..."_
_"¿Qué es un traje de Cleopatra?"_
_"¿Tú no quieres sonar puertorriqueña?"_
_"A mí no me importa que vaya el mundo entero."_
_"Ella no es exactamente Método."_
_"Deja de pensar y baila."_
_"Tiene que ser pecado faltarle el respeto así a la Virgen."_
_"¿Quién tú te crees que eres?"_
_"Las perlas traen lágrimas."_
_"Tenía la música por dentro..."_
_"¿Qué tamaño de brasier tú usas?"_
_"No se vería bien."_
_"Tu cara ya no es inocente."_
_"¿Dónde tú estabas anoche?"_
_"Para ese aire de niña-esclava."_
_"Así tiene que ser."_
_Reconocimientos_
# "Martes, ni te cases ni te embarques ni de tu familia te apartes."
#
En los veintiún años que viví con mi mamá, nos mudamos por lo menos veinte veces. Atacuñábamos las cosas en maletas descascaradas, en cajas de cartón con anuncios en letras llamativas a los lados, en fundas, en sacos de arroz vacíos, en latas de galletas que olían a levadura y harina. Lo que no podíamos cargar, lo dejábamos: gaveteros a los que les faltaban gavetas, sofás llenos de chichones, los quince cuadros que pinté un verano. Aprendimos a no apegarnos demasiado a nuestras pertenencias porque eran tan temporeras como las paredes que nos cobijaban por unos meses; como los vecinos que vivían un poco más abajo en la misma calle, o como el muchacho de ojos tristes que me amó cuando yo tenía trece años.
Nos mudamos del campo a la ciudad, al campo, a un pueblito, a una gran ciudad, a la ciudad más grande de todas. Ya en Nueva York, nos mudamos de apartamento en apartamento, en busca de calefacción, de menos cucarachas, de más cuartos, de vecindarios más tranquilos, de mayor privacidad, de mejor acceso al _subway_ y a la casa de nuestros parientes.
Nos movíamos en círculos alrededor de los vecindarios que queríamos evitar: aquéllos donde no había puertorriqueños o donde el graffiti nos advertía que andábamos por territorios de pandillas, aquéllos donde la gente vestía mejor que nosotros, donde a los caseros no les caían bien los puertorriqueños o no aceptaban el _welfare_ o meneaban la cabeza cuando veían a nuestra familia de tres adultos y once niños.
Evitábamos los vecindarios con muy pocas tiendas, con demasiadas tiendas, con las tiendas que no eran tiendas nada o con ninguna tienda. Le dimos vueltas a nuestro primer apartamento como le dan vueltas los animales al lugar donde van a dormir y después de diez años de dar vueltas, Mami regresó a lugar donde comenzó nuestro peregrinaje: a Macún, el barrio puertorriqueño donde todo el mundo se conocía y conocía la vida y milagros de los demás, y donde los cachibaches que dejamos atrás fueron bien aprovechados por gente que se mudaba menos que nosotros.
Para cuando Mami regresó a Macún, yo también me había mudado. Cuatro días después de cumplir los veintiún años, me fui de casa, olvidando el refrán que canturreaba de niña: "Martes, ni te cases ni te embarques ni de tu familia te apartes". Un martes brumoso no me casé, pero sí me embarqué y sí me aparté de mi familia. En el buzón, le dejé una carta a Mami en la que le decía adiós porque no tuve el valor de despedirme en persona.
Me fui a la Florida a dar mis propias vueltas de una ciudad a otra. Cada vez que empacaba mis cosas dejaba un pedacito de mí en los cuartos que me albergaban —nunca mi hogar— siempre, los sitios donde vivía. Me felicitaba por lo fácil que se me hacía dejarlos, por lo bien que empaquetaba todas mis pertenencias en un par de cajas y una maleta.
Años después, cuando visité Macún, fui al lugar donde empezó y terminó mi niñez. Parada en lo que quedaba de nuestro piso de losetas azules, contemplé el verdor agreste que me rodeaba, lo que había sido el patio de nuestros juegos, el rincón donde la mata de berenjena se convertía en árbol de Navidad, el sitio aquél donde me corté el pie y donde la tierra se chupó mi sangre. Ya no me parecía familiar, ni hermoso y no había ni una pista que me sugiriera quién había sido yo allí, o en quién me convertiría dondequiera que fuese después. Los morivivís y el culantro sofocaban el batey, las enredaderas habían arropado el piso de cemento, los cohitres se habían trepado por lo que quedaba de las paredes y las habían convertido en montoncitos verde-tierno que albergaban lagartijos de un color olivo pardusco o verde brillante, coquíes y picaflores. No había un solo indicio de que alguna vez habíamos estado allí, excepto el montecillo de losetas azules donde estaba parada. Relucía bajo el sol de la tarde, de un color tan intenso que me pregunté si no estaría parada sobre un piso ajeno porque yo no recordaba que nuestro piso hubiese sido nunca tan azul.
# "Te puede pasar algo."
#
Llegamos a Brooklyn en el 1961 en busca de atención médica para Raymond, mi hermano menor, a quien una cadena de bicicleta por poco le cercena los dedos del pie cuando tenía cuatro años. En Puerto Rico los médicos querían amputarle el pie, que con frecuencia se le hinchaba y se le enrojecía, porque no acababa de sanar. En Nueva York, era la esperanza de Mami, los médicos podrían salvárselo.
El día que llegamos, la tarde caliente y húmeda se había astillado en truenos y relámpagos y los últimos rayos de sol se sumergían en el resto de los Estados Unidos. Tenía trece años y era ya lo suficientemente supersticiosa como para creer que los rayos y los truenos guardaban un significado más allá del meteorológico. Conservé en mi memoria las imágenes y los sonidos de aquella noche triste como si, algún día, con un chispazo de luz, su sentido último me fuera a ser revelado para transformar mi vida para siempre. Cuando esa luz llegó, nada cambió, porque lo importante no era el clima de Brooklyn, sino el hecho de que yo estaba allí para notarlo.
Con Mami apretándome una mano y Edna, de seis años, la otra, nos fuimos abriendo paso entre el gentío de pasajeros. Raymond, de cinco años, se aferraba a la otra mano de Mami; su vaivén al andar provocaba sonrisas de pena en la gente que se echaba a un lado para dejarnos pasar.
Al final del túnel, nos esperaba Tata, la mamá de Mami, vestida de encaje negro y tacos altos, un broche de _rhinestones_ puntiagudo en el hombro izquierdo. Cuando me abrazó, el prendedor me pinchó el cachete dejándome una sutil hendidura en forma de flor que yo me iba sobando rítmicamente, mientras el taxi volaba por las calles empapadas, flanqueadas por altos edificios angulosos.
Nueva York resultó más oscuro de lo que esperaba y, a pesar de la lluvia purificadora, también más sucio. Acostumbrados a las curvas sensuales de los campos de Puerto Rico, mis ojos tuvieron que ajustarse a la bidimensionalidad agresiva y uniforme de Brooklyn. Gruesas gotas de lluvia golpeaban las calles duras, capturaban el empañado resplandor plateado de las luces de la calle, rebotaban contra la acera en destellos de luz y entonces, como diminutas joyas efímeras, desaparecían en la oscuridad. Mami y Tata bromeaban con que yo me había desilusionado porque las calles no estaban pavimentadas en oro. Pero en realidad, no era esa la imagen que yo tenía de Nueva York. Más bien estaba desilusionada por la oscuridad y cifré mis esperanzas en la promesa de luz escondida en las resplandecientes gotitas de lluvia.
Dos días más tarde, recostada contra la pared de nuestro edificio de apartamentos en la Calle McKibbin, me preguntaba dónde terminaría Nueva York y dónde empezaría el resto del mundo. Era difícil calcularlo. No había horizonte en Nueva York. Dondequiera que miraba, mis ojos tropezaban con un laberinto vertical de rectos edificios marrones y grises con esquinas cortantes y sombras profundas. Cada dos o tres bloques había un parquecito infantil, de cemento, cercado con una verja de alambre. Entremedio había lotes atestados de matojos, basura y carros corroídos de moho.
Del edificio de al lado salió una nena con una cuica en la mano. Me miró de arriba a abajo tímidamente. Yo me hice la que no la veía. Ella pisó la cuica, estiró las puntas por encima de su cabeza como para medirle el largo y empezó a dar brinquitos, despacio, dejando escapar un jum del fondo de su garganta cada vez que tocaba la acera. Tchis, tcha, jum, tchis me dió la espalda; tchis, tcha, jum, tchis, se volvió hacia mí y me sonrió. Yo le devolví la sonrisa y ella se acercó saltando.
"¿Tú eres hispana?" preguntó, mientras dibujaba en el aire amplios arcos con la cuica.
"No, yo soy puertorriqueña."
"Es lo mismo. Puertorriqueña, hispana. Eso es lo que somos aquí." Dio un brinco cerrado, se paró de pronto y con un gesto rápido, me ofreció la cuica.
"¿Quieres?"
"Claro." Salté en una pierna, luego en la otra. "¿Así es que si uno es puertorriqueño le dicen hispano?"
"Ujúm. Cualquiera que hable español."
Di una vuelta como la que había dado ella, pero más rapidito. "¿Tú quieres decir que si uno habla español uno es hispano?"
"Bueno... ajá. No... Es que tus papás tienen que ser puertorriqueños o cubanos o algo así."
Torcí la cuica hacia la derecha, después hacia la izquierda, como los boxeadores. "Okay, suponiendo que tus papás son cubanos y tú naciste aquí pero no hablas español, ¿tú eres hispana?"
Se mordió el labio inferior. "Supongo," dijo finalmente. "Tiene que ver con que uno sea de un país hispánico. Tú sabes, es como que tú y tus papás, este, aunque ustedes no hablen español, son hispanos, ¿tú sabes?"
Me miró dudosa. Yo asentí y le devolví la cuica.
Pero no, no sabía. Yo siempre había sido puertorriqueña y no se me había ocurrido nunca que en Brooklyn me convertiría en otra cosa.
Más tarde, pregunté, "¿Nosotros somos hispanos, Mami?"
"Sí, porque hablamos español."
"Pero una nena me dijo que uno no tiene que hablar el idioma pa' ser hispano."
Entrecerró los ojos. "¿Qué nena? ¿Dónde tú conociste una nena?"
"Allá afuera. Vive en el edificio de al lado."
"Y a ti, ¿quién te dio permiso pa' salir pa' la acera? Esto no es Puerto Rico. Te puede pasar algo."
"Te puede pasar algo" era una variedad de peligros que acechaban fuera de las puertas cerradas de nuestro apartamento. Me podían asaltar. Yendo y viniendo de la escuela me podían arrastrar hasta cualquiera de los edificios oscuros y abandonados y me podían violar y matar. Podían acercárseme los miembros de alguna ganga si me perdía y caía en su territorio. Podía ser seducida por esos hombres que acosan a las muchachitas que andan solas y están dispuestas a hablar con extraños. Yo oía el sermón de Mami con la mirada baja y las debidas muestras de respeto y humildad. Pero por dentro, estaba que trinaba. Dos días en Nueva York y ya me había convertido en otra persona. No fue difícil imaginar que peligros aun mayores me esperaban.
Nuestro apartamento en la Calle McKibbin era más sólido que cualquiera de nuestras casas en Puerto Rico. Las escaleras de mármol, las paredes de yeso, y los pisos de losa estaban pegados a la tierra, muy diferentes a los cuartos de madera y zinc montados en zocos donde yo me había criado. Unos angelitos gordos con las nalguitas al aire danzaban en torno a unas guirnaldas de yeso en el techo. En la cocina había una bañera con agua corriente, caliente y fría, y un inodoro dentro de un closet con un lavamanos y un botiquín. El callejón entre la ventana de nuestra habitación y la pared del edificio contiguo era tan estrecho que yo me estiré un poco para tocar los ladrillos y dejé mi huella en el hollín grasiento que los cubría. Arriba, un pedacito de cielo empujaba una tenue luz amarilla hacia el suelo cubierto de cajas de detergente vacías, trapos viejos, zapatos sueltos, botellas, vidrios rotos.
Mami tenía que salir a buscar trabajo, así es que Edna, Raymond y yo bajamos a quedarnos en el apartamento de Tata. Estaba levantándose todavía cuando le tocamos a la puerta. Me senté en la mesita cerca del _counter_ de la cocina a leer los periódicos que había traído la noche anterior Don Julio, el novio de Tata. Parados en el medio del cuarto, Edna y Raymond clavaron sus ojos en un televisor pequeño puesto sobre una mesa bajita. Tata lo prendió, trasteó un rato los botones y la antena hasta que desaparecieron unas franjas horizontales y salieron unos muñequitos en blanco y negro persiguiéndose unos a otros sobre un paisaje plano. Los nenes cayeron sentados, las piernas cruzadas, los ojos fijos en la pantalla. Contra la pared, debajo de la ventana, dormía de espaldas a nosotros Tío Chico, el hermano de Tata. A cada rato, un ronquido lo despertaba, pero él se masticaba la baba, murmuraba algo y volvía a dormirse.
En lo que Tata fue a lavarse al baño del pasillo, yo me sintonicé al televisor. Un punto brincaba sobre las palabras de una canción que era interpretada por un tren que iba bailando sobre los rieles con perros, gatos, vacas y caballos que se salían por las ventanas de los vagones. Quedé hipnotizada por ese puntito que saltaba sobre unas palabras que no se parecían en nada a como sonaban. "Chilbii cominraun demauntin uenchicoms, tuut-tuut," cantaba la locomotora y la bolita bajaba y subía _"She'll be coming 'round the mountain when she comes,"_ sin tuut tuut. Los animales, vestidos con sombreros de vaquero, mamelucos de mahón y pañuelos en el cuello agitaban picos y palas al aire. El tuut-tuut se iba sustituyendo por un guau-guau o un miau-au o un muu-muu. Era una cancioncita alegre y tonta que hacía reir a Edna y a Raymond. Pero a mí se me hacía difícil disfrutarla porque estaba enfocada en las palabras que pasaban volando, en la bolita que brincaba rítmicamente de sílaba en sílaba sin tiempo apenas para conectar las letras con el sonido, y además tenía la distracción adicional de un rebuzno, un ladrido o la risa de uno de los nenes.
Cuando regresó del baño, Tata preparó café en la estufa de dos hornillas. Un humo fragante pronto invadió el cuartito y según ella filtró la harina por el colador de bayeta desgastado, se levantó Tío Chico como si el aroma fuera una alarma más fuerte e insistente que los animales cantores en la pantalla del televisor, el chocar de las ollas contra la estufa o el chirrido de las patas de la silla cuando me acomodé para poder ver a Tata y los muñequitos a la vez.
"Adiós, mira a quién tenemos por aquí," dijo Tío Chico mientras se estiraba hasta que sus dedos largos y huesudos tocaron el techo. Vestía la misma ropa del día anterior: unos pantalones oscuros despintados y una camiseta de manga corta, las dos piezas arrugadas y con un agrio olor a sudor. Les pasó por encima a Edna y a Raymond que apenas se movieron para dejarlo pasar y de dos zancadas se escurrió hacia el baño. Según cerró la puerta, pareció que las paredes se juntaron, como si el cuerpo flacucho de Tío Chico aumentara las dimensiones del cuartito estrecho.
Tata tarareaba la canción de los muñequitos. Sus manos grandes agarraron una cacerola, le echaron leche y la batieron ligerito hasta que hirvió e hizo espuma. Yo estaba embobada con su gracia, con su porte, con los rizos cenizos y despeinados que enmarcaban sus pómulos altos. Levantó sus pícaros ojos caramelo y, sin perder el ritmo, sonrió.
Tío Chico regresó duchado y afeitado y vistiendo una camisa y unos pantalones limpios tan arrugados como los que se había quitado. Dejó caer la ropa sucia en un rincón cerca de la cama de Tata y arregló su catre. Tata me pasó una taza de café con leche endulzado y ladeando la cabeza, me indicó que le dejara mi silla a Tío Chico.
"No, no, está bien," dijo, "yo me siento acá."
Se acomodó en la orilla del catre, los codos en las rodillas y los dedos rodeando el tazón que le dio Tata. De entre sus manos, ascendía el humo en un espiral transparente. Tata les sirvió a Edna y a Raymond y después se sentó con su café en una mano y el cigarrillo en la otra a hablar bajito con Tío Chico, que también fumaba. Yo acerqué la cara al aromático vapor del café para evitar el olor del humo mentolado que circulaba de su lado del cuarto al nuestro hasta posarse como una suave manta gris que se nos derretía en la ropa y el pelo.
Yo no hablaba inglés, así es que el orientador escolar me ubicó en una clase para estudiantes que habían obtenido puntuaciones bajas en los exámenes de inteligencia, que tenían problemas de disciplina o que estaban matando el tiempo en lo que cumplían dieciséis años y podían salirse de la escuela. La maestra, una linda mujer negra un par de años mayor que sus estudiantes, me señaló un asiento en el medio del salón. No me atreví a mirar a nadie a los ojos. Unos gruñidos y murmullos me seguían y aunque yo no tenía idea de lo que significaban, no me sonaron nada amistosos.
La mesita del pupitre estaba tallada con esmero. Tenía muchos nombres seguidos de un apóstrofo y un año. Algunas obscenidades cuidadosamente labradas no me dijeron nada pero podía reconocer la maestría con que estaban hechas las letras sombreadas y las orillas alrededor de la _f_ y de la _k_. Supuse que una niña había escrito el mensaje en cursivo porque las íes en vez de puntos tenían corazones y margaritas. Debajo, unas líneas escritas en una tímida letra finita, como arañazos de pollo, alternaban con unas en agresivas letras de bloque.
Me apreté las manos debajo de la mesa para controlar el temblor y me puse a examinar las líneas rectas y las curvas serradas esculpidas en el pupitre por aquéllos que lo habían ocupado antes que yo. Con los ojos fijos en la superficie guayada, me concentré en la voz de la maestra, en las ondas de sonidos extraños que pululaban sobre mi cabeza. Hubiera querido salir flotando de ese salón, alejarme de ese ambiente hostil que permeaba cada rincón, cada grieta. Pero mientras más trataba de desaparecer más presente me sentía hasta que, exhausta, me dejé ir, y floté con las palabras, convencida de que si no lo hacía, me ahogaría en ellas.
Para la clase de Educación Física, las muchachas teníamos que usar un mameluco en algodón verde yerba, de manga corta y patas abombachadas, abotonado al frente hasta la cintura, donde cerraba con una banda tan corta que no daba más que para amarrársela en un abultado nudo. El verde yerba no le quedaba bien a nadie, pero mucho menos a las muchachas adolescentes con las caras llenas de barritos rojos. El uniforme tenía un elástico en la parte de abajo para que no se nos vieran los panties cuando nos cayéramos o nos sentáramos. El elástico nos quedaba flojo a las que teníamos las piernas flacas y los bombachos nos colgaban hasta las rodillas y aleteaban cuando corríamos. Como el uniforme era de una sola pieza, era casi imposible ir al baño en los tres minutos que teníamos entre clases. En vez de tenerlo puesto todo el día, podíamos traerlo a la escuela y cambiarnos antes de la clase, pero nadie lo hacía porque periódicamente los muchachos invadían el cuarto de los _lockers_ para vernos en ropa interior. Con el uniforme puesto era difícil mantener una buena higiene cuando estábamos "malas", para lo que hubiéramos necesitado por lo menos tres manos, así es que nuestras mamás nos preparaban excusas. Lo malo era que si una no usaba el uniforme durante los días de Educación Física, todo el mundo se enteraba de que tenía la menstruación.
Una muchacha compró dos uniformes de educación física. Le cortó la bombacha a uno, le hizo un repulgo en la cintura y usaba la parte de arriba debajo de la blusa. Así, nadie sabía si estaba o no con el período. Le pedí a Mami que me hiciera lo mismo, pero me contestó que nosotros no teníamos dinero para botar en esa tontería.
Los viernes por la mañana teníamos Asamblea. Lo primero que hacíamos era ponernos la mano derecha sobre el pecho y cantar "The Star-Spangled Banner". Se nos estimulaba a cantar tan fuerte como pudiéramos y en un par de semanas ya me había aprendido la canción de memoria.
Ojo sé. Can Juice ¿Y?
Bye de don surly lie.
Whassoprowow we hell
Add debt why lie lass gleam in.
Whosebrods tripen sand bye ¿Star?
True de perro los ¡Hay!
Order am parts we wash,
Wha soga lang tree streem in.
No tenía la más mínima idea de lo que decía o significaba la canción y a nadie se le ocurrió explicármelo. Era una de esas cosas que se suponía que supiera y que, al igual que el juramento diario a la bandera, tenía que hacerse con entusiasmo o las maestras nos daban deméritos. El juramento a la bandera, escrito en una letra muy ornamentada, estaba en un cartel que colgaba debajo de la bandera de cada salón. En cambio, durante años "The Star-Spangled Banner" continuó siendo un misterio para mí, su letra disparatada, la única canción en inglés que podía cantar de principio a fin.
Una tarde fría de octubre, Mami, Don Julio y yo fuimos al aeropuerto a recoger al resto de mis hermanas y hermanos que se habían quedado en Puerto Rico con nuestro padre en lo que Mami reunía el dinero para sus pasajes. Delsa, Norma, Héctor y Alicia eran más chiquitos, más oscuros de lo que los recordaba, más foráneos. Se acurrucaban unos contra otros cogidos de la mano. Sus miradas, como dardos, volaban de esquina a esquina del enorme terminal, a las cientos de personas que decían adiós, que se abrazaban y se besaban, al equipaje que tropezaba con ellos. Como pajaritos, alzaban las cabezas, boquiabiertos, hacia las voces incorpóreas y amplificadas que berreaban órdenes desde el techo. Me pregunté si me habría visto así de asustada y vulnerable dos meses atrás.
Nos habíamos mudado a un apartamento más nuevo y más grande en la Calle Varet. Tata y Tío Chico habían estado cocinando toda la mañana y al entrar al apartamento la fragancia del achiote, el ajo y el orégano y la risa de la familia reunida y charlando, nos dio la impresión de que era Navidad.
Teníamos muchos parientes en Brooklyn. Paco, el hijo de Tío Chico, era bajito y musculoso. Siempre tenía la cara y los brazos magullados, los ojos hinchados e inyectados de sangre y la nariz vendada como resultado de su trabajo como luchador. Su nombre profesional era El Santo. En el cuadrilátero usaba un leotardo blanco, una correa de piel blanca, una máscara blanca, una capa de satín de un blanco leche con un cuello alto salpicado con _rhinestones_. Era de los buenos, y aunque generalmente ganaba las peleas, cogía siempre una pela de los tipos vestidos de negro.
Jalisco, el hermano de Paco, trabajaba en una fábrica. Era alto y delgado como el papá. Se acicalaba el bigote en forma de una pelusa negra y recta sobre los labios, a lo Jorge Negrete, el cantante y actor mexicano. Cuantas veces venía Jalisco, yo lo rodeaba como mariposa febril, ofreciéndole algo de tomar o de comer o recordándole que me había prometido que cantaría "Cielito Lindo" después de la comida. Mami nunca me dejaba sola con él.
Las dos hermanas de Tata vivían a unos pocos bloques de nuestro apartamento. Tía Chía y sus hijas —Margot, Gury y La Muda— eran muy cercanas a mi mamá. Llegaban arrastrando bolsas llenas de ropa y zapatos que ya no usaban. Gury, la menor, era esbelta y de hablar suave. Su ropa me servía, aunque Mami dijo que las faldas estrechas, las blusas transparentes y los tacones altos que prefería Gury no eran apropiados para las nenas de mi edad.
Su hermana La Muda era sorda y muda. Según Mami, La Muda había nacido con una audición perfecta, pero de chiquita se enfermó y cuando se recuperó estaba sorda.
"¿Entonces, por qué no le dicen mejor La Sorda...?" empecé, pero Mami me advirtió que estaba siendo irrespetuosa.
La Muda leía los labios. Si no hablábamos con la cara hacia ella, nos sacudía por los hombros y nos hacía repetirle lo que habíamos dicho mientras sus ojos se enfocaban en nuestras bocas. Muy pronto aprendimos a interpretar su lenguaje, una danza de gestos realizados con murmullos, gorjeos y gruñidos que no parecían venir de su garganta, sino de una fuente más profunda dentro de su vientre. Sus manos eran grandes, bien cuidadas, adornadas con numerosas sortijas de oro y piedras que relucían según sus manos volaban aquí y allá.
A La Muda le gustaba que le leyéramos el periódico, mejor dicho, Mami o Don Julio lo leían en voz alta mientras la muchachería le actuaba las noticias. Los ojos de La Muda volaban de los labios de Mami a nuestra representación de los crímenes del día, los accidentes de carro, los resultados del hipódromo, dramatizados carrera a carrera, alrededor de la mesa de la cocina. Su risa, frecuente y contagiosa, era profunda pero desentonada, como si por no poder oírse a sí misma no lograra coger el tono.
Su novio era alguien que habíamos conocido en Puerto Rico. Era un hombre de pelo oscuro, flaco, lacónico, que usaba siempre un traje crema. Cuando lo conocimos, mis seis hermanos y hermanas y yo le cogimos miedo, pero él se sacó unas barajas del bolsillo, nos hizo unos trucos y desde ese día le llamamos Luigi, que nos sonaba como el nombre perfecto para un mago.
Titi Ana, la otra hermana de Tata, tenía dos hijas que estaban más cerca de mi edad que La Muda, Margot o Gury. Alma era un año mayor que yo y Corazón, uno menor. Entre ellas hablaban inglés y, cuando hablaban con nosotras o con su mamá, su español era vacilante y tenía acento. Mami decía que estaban _americanizadas_. El modo en que pronunciaba _americanizadas_ hacía sonar la palabra como algo terrible que tenía que evitarse a toda costa; otro "algo" para añadirse a la lista de "algos" que acechaban detrás de la puerta de la calle.
Cuando entraron al apartamento, mis hermanas y hermanos se sometieron a los besos y abrazos de unas personas que eran extrañas para ellos, pero que se presentaron como el primo tal o la tía mascuál. Delsa estaba al borde de las lágrimas. Norma agarraba a Alicia, asustada de que se fueran a perder en el revolú. Héctor se paseaba entre los hombres, seguido por Raymond que chachareaba sobre las hazañas de Paco en el cuadrilátero, o sobre lo generoso que era Don Julio con el menudo que le sobraba.
Con su rostro, habitualmente sombrío, iluminado por un asomo de sonrisa, Luigi nos hacía trucos nuevos y los nenes se tranquilizaron un poco como si este recuerdo de nuestra vida en Puerto Rico hubiera bastado para disolverles el miedo. Margot había traído su tocadiscos portátil y unos discos que se oían a todo volumen en la cocina, mientras que en el cuarto de al frente, el televisor estaba prendido en la película de terror que daban por las tardes. Los nenes vagaban de cuarto en cuarto, aturdidos, con una sobredosis de Twinkies, Yodels y papitas fritas que Don Julio nos había traído.
La fiesta de bienvenida duró hasta entrada la noche. Don Julio y Jalisco fueron a la bodega varias veces a buscar más cervezas. Tío Chico encontró un _liquor store_ y regresó con una pipota de vino Gallo. Mami corría de un lado a otro, de los adultos a la muchachería, recordándoles a los hombres que había menores en la casa y que debían dejar de beber.
Uno a uno los parientes se fueron yendo y una vez más los nenes se entregaron a los besos y abrazos. Los bolsillos nos sonaban con los chavitos que las tías, los tíos y los primos nos habían repartido como para agradecernos la fiesta. Luigi acompañó a La Muda de regreso desde el apartamento. Sus dedos pálidos le apretaban la cintura, su traje, demasiado grande, le bailaba alrededor de su desgarbado cuerpo de espantapájaros. Cuando salieron, los adultos intercambiaron sonrisas misteriosas.
Tío Chico y sus hijos fueron los últimos en marcharse. Tata y Don Julio se metieron al cuarto de ella y corrieron la cortina que separaba su lado del apartamento del nuestro. "Hora de dormir," nos recordó Mami. Nos preparamos; Delsa y yo en la litera de arriba, Norma y Alicia en la de abajo, Héctor en el sofá, Raymond en dos butacones que se juntaron, Edna y Mami en la cama grande. Mami apagó la luz y los suaves crujidos de mis hermanos y hermanas, acomodándose para pasar su primera noche en Brooklyn, me llenaron de un gozo secreto que nunca admití, pero que me reconfortaron y me serenaron como nada lo había hecho desde que nos fuimos de Puerto Rico.
# "A mí no me importa lo que hagan esas americanas."
#
Como todas las demás mamás puertorriqueñas que yo conocía, Mami era estricta. Su razón para traerme a Nueva York con los nenes más chiquitos fue que yo era "casi señorita" y ella no quería dejarme sola en Puerto Rico durante lo que ella llamaba "una etapa crítica en mi vida." Mami le decía a su amiga Minga que las muchachitas de mi edad tenían que ser vigiladas por sus madres y protegidas de los hombres que siempre buscaban aprovecharse de una niña en cuerpo de mujer. Si bien mi cuerpo no era exactamente el de una mujer, yo entendía lo que Mami quería decir. Los años que me pasé escuchando disimuladamente sus conversaciones me habían enseñado que en los hombres no se podía confiar. Todos engañaban con sus pocavergüenzas, esos actos vergonzosos que incluían beber, jugar y botar dinero en mujeres —que no eran sus esposas— mientras sus hijos pasaban hambre. Para tapar las pocavergüenzas, los hombres decían embustes. Un hombre podía estarle diciendo "mi amor" a su esposa mientras con el rabo del ojo estaba mirando a cualquier otra mujer que le pasara por el lado.
"Una muchacha hace bien en dudar de cualquier hombre que le hable bonito," declaraba Minga. "Para ella, esas palabras son la cosa más bella que ha oído, ni se imagina que él las ha dicho ya mil veces antes... y las va a seguir diciendo mientras haya alguna pendeja que le haga caso."
Según Mami y sus amigas, las mujeres hacían pocavergüenzas también. Coqueteaban con hombres que ya estaban cogidos por mujeres más dignas que ellas y sonsacaban a esos hombres inútiles e irresponsables.
Después de oír innumerables historias de hombres embusteros y mujeres listas, decidí que no me convertiría nunca en una de esas putas calculadoras, pero tampoco iba a ser una pendeja que creyera todo lo que un hombre me dijera o que me haría de la vista larga mientras me estuviera engañando. Había un punto medio entre puta y pendeja que yo estaba tratando de descifrar, un espacio seguro donde las mujeres decentes vivían, progresaban y criaban a sus familias. Mami pertenecía a ese grupo, así como sus amigas y las mujeres de su familia. Sus sermones y las conversaciones que tenían, con toda la intención de que yo las escuchara, eran para ayudarme a distinguir entre una puta y una pendeja. Pero había siempre una advertencia. Un paso en falso y corría el riesgo de convertirme en una o de ser percibida como la otra.
En la escuela, me hice amiga de Yolanda, una nena que hablaba bien el inglés pero que conmigo hablaba español. Yolanda era la única puertorriqueña que yo conocía que fuera hija única. A ella le daba curiosidad saber cómo era eso de tener seis hermanos y hermanas y yo a ella le preguntaba qué hacía todo el día sin tener a nadie con quién jugar o pelear.
"Ahí, tú sabes, ver televisión, leer y también tengo mis álbumes."
Ella coleccionaba retratos en libretas de tres argollas, y los organizaba por temas. "Estas son flores," me dijo, mientras bajaba una libreta gruesa de una tablilla que había encima de su cama. La abrió en una página llena de flores de las que venían en las etiquetas de los potes de leche evaporada Carnation. "Y éstos son labios." Páginas y páginas de labios de hombre y de mujer, algunos tenían bigotes, otros eran sólo las sonrisas incorpóreas de artistas de cine. "Este otro es de letras." Organizadas en orden alfabético había cientos de letras pegadas en las páginas, las mayúsculas esparcidas a la izquierda, las minúsculas a la derecha. Otros álbumes tenían etiquetas de productos de lata, de cajas de toallas sanitarias, de marcas de ropa. Uno tenía anuncios de productos para el pelo y de belleza, sacados de periódicos y revistas. El álbum más grande tenía retratos de diferentes medios de transportación: carros, trenes, barcos cruceros, lanchas, bicicletas de dos sillines. Llegué a la conclusión de que, definitivamente, Yolanda pasaba demasiado tiempo sola.
"¿Te gustaría venir a mi casa?" la invité un día. Tenía que pedirle permiso a su mamá, pero ella estaba segura de que no habría problemas. Al día siguiente me dijo que su mamá no le había dado permiso. "Le rogué," me explicaba Yolanda con los ojos llorosos, "pero es que ella es tan estricta conmigo." Me dio pena, pero entendí, porque Mami también era estricta. Pero cuando le conté a Mami que la mamá de Yolanda no la dejaba venir a casa, se puso furiosa.
"¿Y qué es lo de esa mujer? Tú puedes ir pa' llá pero la princesa de ella no puede entrar aquí?" Después de eso, ya no me dejaron ir más al apartamento de Yolanda.
Un día Yolanda me pidió que la acompañara a la biblioteca. Le dije que no podía porque Mami nos tenía prohibido que nos quedáramos en ningún sitio, sin permiso, de regreso a casa. "Pídele permiso y vamos mañana. Si traes un papel que diga dónde vives, te pueden dar una tarjeta," me sugirió Yolanda, "y puedes sacar libros prestados. Gratis," añadió cuando titubeé.
Yo había pasado por la Biblioteca Pública de Bushwick muchas veces y me habían llamado la atención sus pesadas puertas de entrada enmarcadas por columnas y las anchas ventanas que miraban desde lo alto al vecindario. Alejada de la calle, detrás de un cantito de grama seca, la estructura de ladrillos rojos parecía estar fuera de lugar en una calle de edificios de apartamentos en ruinas, y enormes e intimidantes proyectos de viviendas. Adentro, los techos eran altos con aditamentos y lámparas colgantes sobre largas mesas marrón, colocadas en el centro del salón y cerca de las ventanas. Los estantes alrededor del área estaban repletos de libros cubiertos de plástico. Cogí uno, de una de las tablillas de arriba, lo hojeé y lo devolví a su sitio. Caminé todos los pasillos de arriba a abajo. Todos los libros eran en inglés. Frustrada, busqué a Yolanda, me despedí en voz baja y me dirigí a la salida. Cuando iba saliendo, pasé por el Salón de los Niños, en donde una bibliotecaria estaba leyéndole a un grupo de niños y niñas. Leía despacio y con expresividad, y después de leer cada página viraba el libro hacia nosotros para que pudiéramos verla. Cada página tenía sólo unas pocas palabras y una ilustración que clarificaba su sentido. Si los americanitos podían aprender inglés con esos libros, yo también podría.
Después de la sesión de lectura, busqué en los anaqueles los libros ilustrados que contenían las palabras que necesitaría para mi nueva vida en Brooklyn. Escogí libros del alfabeto, de páginas coloridas donde encontré: _car, dog, house, mailman_. No podía admitirle a la bibliotecaria que esos libros tan elementales eran para mí. _"For leettle seesters,"_ le dije, y ella asintió, me sonrió y estampó la fecha de entrega en la parte de atrás del libro.
Paraba en la biblioteca todos los días después de clase y en casa me memorizaba las palabras que iban con las ilustraciones en las enormes páginas. Algunos conceptos eran difíciles. La nieve era representada como inmensos copos multifacéticos. Hasta que vi la nieve de verdad, me la imaginaba como una cortina elaborada, tiesa y plana que podría capturar con la punta de los dedos.
Mis hermanas y hermanos también estudiaban los libros y nos leíamos en voz alta las palabras tratando de adivinar la pronunciación.
_"Ehr-rahs-ser,"_ decíamos en lugar de _"eraser." "Keh-neef-eh,"_ por _"knife." "Dees,"_ por _"this"_ y _"dem"_ por _"them"_ y _"dunt"_ por _"don't."_
En la escuela, escuchaba con cuidado y trataba de reconocer aquellas palabras que sonaban como las que habíamos leído la noche anterior. Pero el inglés hablado, a diferencia del español, no se pronuncia como se escribe. _"Water"_ se convertía en _"waddah", "work"_ en _"woik"_ y las palabraschocabanunasconotras en un torrente de sonidos confusos que no guardaban ninguna relación con las letras cuidadosamente organizadas en las páginas de los libros. En clase, casi nunca levantaba la mano porque mi acento provocaba burlas en el salón cada vez que abría la boca.
Delsa, que tenía el mismo problema, sugirió que habláramos inglés en casa. Al principio nos destornillábamos de la risa cada vez que nos hablábamos en inglés. Las caras se nos contorsionaban en muecas, nuestras voces cambiaban y las lenguas se nos trababan al tratar de reproducir torpemente los sonidos. Pero, según los demás se nos fueron uniendo y practicábamos entre nosotros, se nos fue haciendo más fácil y ya no nos reíamos tanto. Si no sabíamos la traducción para lo que estábamos tratando de decir, nos inventábamos la palabra, hasta que formábamos nuestro propio idioma, ni español ni inglés, sino ambos en la misma oración, y a veces, en la misma palabra.
"Pasa mí esa sabaneichon," le decía Héctor a Edna para pedirle que le pasara una sábana.
"No molestándomi," le soplaba Edna a Norma cuando ésta la molestaba.
Veíamos la televisión con el volumen bien alto aunque Tata se quejaba de que oír tanto inglés le daba dolor de cabeza. Poco a poco, según aumentaba nuestro vocabulario, se fue convirtiendo en un vínculo entre nosotras, uno que nos separaba de Tata y de Mami que nos observaba perpleja, mientras su expresión pasaba del orgullo, a la envidia, a la preocupación.
Una mañana, Mami me dijo que no podía ir a la escuela ese día porque tenía que acompañarla a hacer una diligencia. "No empieces con tus preguntas," me advirtió tan pronto abrí la boca.
Cogimos dos guaguas y caminamos dos cuadras hasta que llegamos a un cansado edificio de bloques con tela metálica en las ventanas. Adentro, el área de espera estaba repleta de mujeres, sentadas en unas sillas plásticas anaranjadas, todas con un montón de papeles en la mano. Un mostrador dividía el salón y en la parte de atrás había tres filas de escritorios de metal gris, llenos de estibas de cartapacios, folletos, formularios y otros papeles.
SOLICITUD PARA ASISTENCIA PÚBLICA, aparecía en la parte superior del formulario, DEPARTAMENTO DE BIENESTAR PÚBLICO: AYUDA A FAMILIAS CON HIJOS DEPENDIENTES (AFHD). "Toma," Mami me entregó una pluma, "llénalos con tu mejor letra."
"¿Pero, pa' qué es esto?"
"Pa' poder conseguir ayuda hasta que yo encuentre otro trabajo." Hablaba bajito y miraba de un lado a otro, pendiente de que alguien la fuera a oir.
Llené los formularios lo mejor que pude, dejando en blanco los espacios donde no entendía las preguntas.
Según fue pasando la mañana, entraron más mujeres, unas solas, otras cargando muchachos. Era fácil distinguir a las que ya conocían la oficina del mantengo. Ésas escudriñaban el salón para tener una idea de cuántas habían llegado antes, iban donde la recepcionista, tomaban los formularios, los llenaban en un momentito, como si tuvieran ya memorizadas las preguntas y las respuestas. Las primerizas en el _welfare_ titubeaban en la puerta, miraban de derecha a izquierda hasta que localizaban el escritorio de la recepcionista y entraban como si las estuvieran pinchando. Miraban suplicantes a la recepcionista, trataban de contarle sus historias. Con un gesto displicente, ella las interrumpía, les pasaba los formularios, les daba las instrucciones de cómo llenarlos, "siéntate, espera", siempre las mismas palabras como si no quisiera molestarse siquiera en pensar formas nuevas para decir la misma cosa.
Yo no me había traído un libro, así es que me entretuve mirándolo todo. Mami me dio un codazo y me dijo que dejara de estar mirando a la gente. Bajé la mirada enseguida. Cuando estaba a punto de quejarme de que tenía hambre, unos hombres y unas mujeres entraron desperdigados por la puerta de atrás y se sentaron en los escritorios detrás del mostrador.
Cuando nos llegó el turno, el trabajador social nos llevó hasta una esquina del salón. Era un hombre corpulento, de pelo negro que tenía cara de ser sido pintado o de ser una peluca. Tomó los formularios que yo había llenado, hizo unas marcas de cotejo en algunos de los cuadritos y con el bolígrafo, dio unos golpecitos en los espacios en blanco. Le habló a Mami, que enseguida se volvió hacia mí como si yo hubiera entendido lo que él había dicho. Repitió la pregunta; esta vez a mí. Me concentré en el movimiento de sus labios, en su expresión, en el tono de su voz, pero no tenía ni idea de lo que me estaba preguntando.
"Yo no sé," le dije a Mami.
Mami chasqueó la lengua.
"Plis, no spik inglis," le dijo con una linda sonrisa al trabajador social. Él volvió a hacer la pregunta y nos señaló los espacios en blanco.
"Me parece que quiere el nombre y las fechas de nacimiento de los nenes," le interpreté. Mami sacó los certificados de nacimiento de su cartera y se los estiró con la mano, uno por uno, mientras el hombre anotaba la información.
"Dile," me dijo Mami, "que me dieron _leyof_."
" _My mother_ leyof," le traduje.
"Dile que la fábrica cerró. Que se mudaron a otro Estado. Que no tengo dinero ni para la renta ni para comida." Se ruborizaba, hablaba rápido, bajito. "Yo quiero trabajar. Dile eso," me dijo en una voz más fuerte. "Cerraron la fábrica," repetía.
_"Fabric no,"_ le dije. _"She work wants."_ Los ojos del hombre se fruncían y los carrillos le temblaban según me animaba a seguir con el movimiento de su cabeza. Pero yo no tenía más palabras para él. Escribió algo en los papeles, miró a Mami. Ella me miró a mí.
"Dile que no quiero que mis hijos sufran. Dile que necesito ayuda hasta que la fábrica abra de nuevo o yo encuentre otro trabajo. ¿Ya le dijiste que yo quiero trabajar?"
Le dije que sí, pero en realidad no estaba segura de que el trabajador social me hubiese entendido. _"My mother, she work want. Fabric close,"_ le explicaba al trabajador social, moviendo las manos como La Muda. _"She no can work fabric no. Babies suffer. She little help, she no lay off no more."_ Estaba exhausta, me sudaban las manos y me dolía la cabeza según seguía hurgándomela para encontrar palabras. Tenía la quijada tensa por el esfuerzo de pronunciarlas. Frenética, buscaba la combinación de palabras adecuadas, las que dijeran lo que Mami quería decir para convencer a ese hombre de que ella no estaba pidiendo ayuda porque fuera una vaga, sino porque las circunstancias la obligaban. Mami era una mujer orgullosa y yo sabía lo difícil que era para ella pedirle ayuda a alguien, especialmente a un extraño. Yo le quería hacer saber que ella tenía que estar desesperada para haber venido a un sitio como éste.
Batallé durante el resto de la entrevista, llevando hasta los límites mi escaso vocabulario en inglés. Cuando terminó, el trabajador social se levantó, le dio la mano a Mami, me la dio a mí y dijo, lo que a mí me sonó como: yo me comunico con ustedes.
Salimos de la oficina en silencio, la espalda de Mami, tan tiesa y estirada que parecía que le habían puesto una faja. Yo por mi parte, iba como una bola de tensión, en pánico de pensar que había fracasado como intérprete y que por mi culpa no conseguiríamos ayuda y no tendríamos ni dónde vivir ni qué comer.
"Lo hiciste muy bien," me aseguró Mami esa noche frente a Tata y a Don Julio, "ya sabes bastante inglés."
"Es más fácil pa' los muchachos," murmuró Don Julio entre sorbos de cerveza. "Cogen el idioma, así." Y sonó los dedos.
Me sentía agradecida de la fe que Mami tenía en mí, pero no pude estar tranquila hasta que nos contestaron del _welfare_. Unos días más tarde nos aprobaron la solicitud. Para entonces ya había decidido que aun cuando me pareciera que mi cabeza no podría contener tantas palabras nuevas, tenía que aprender el inglés suficiente para nunca más volver a quedar atrapada entre dos idiomas.
Desperté a mitad de noche con algo caminándome por el cuello. Lo espanté, pero se me enredó en el pelo cerca del lóbulo de la oreja. Desperté en la oscuridad buscando frenética lo que se me había enredado en el pelo. Para cuando llegué al interruptor de la luz al lado de la puerta, había pinchado entre el pulgar y el índice una cucaracha tostada, crujiente, de muchas patas, antes de que se me metiera en el oído.
"Apaga esa luz," protestó Delsa desde su lado de la cama. Tiré la cucaracha al piso y la espacharré con un zapato antes de que se me escapara.
"¿Qué haces?" Mami se sentó en la cama.
"Una cucaracha por poco se me mete en el cerebro." Me sentía sucia y me picaban los dedos, como si todavía la tuviera entre ellos.
"Mañana fumigo," dijo con un gesto agobiado y se acomodó de nuevo en la cama. Sacudí la sábana para asegurarme que no había ninguna otra cucaracha escondida entre los pliegues.
"Deja eso," Delsa agarró molesta su lado de la frisa. Norma y Alicia gimieron en el sueño y se viraron.
Yo sabía que donde había una cucaracha, había cientos. Imaginaba hordas de cucarachas marrón oscuro, colocadas en las grietas de los zócalos de los pisos, esperando que yo apagara la luz para empezar a marchar alrededor del cuarto. Las había visto escabulléndose para esconderse, cuando entraba a la cocina de noche a tomar agua. Las cucarachas andaregueaban por el _counter_ dentro de las tazas y los vasos, alrededor de los cuchillos de pelar frutas, en el espacio que quedaba entre la azucarera y la tapa. Mami traía un veneno cada vez más poderoso para fumigar por las esquinas del apartamento. El veneno nos hacía toser y nos irritaba los ojos. Después que fumigaba, la ropa se quedaba con la peste del Black Flag o del Flit durante un montón de días. Pero las cucarachas no se morían. Se iban un tiempito en lo que se disipaba el gas acre y venenoso y volvían otra vez más envalentonadas y en mayor cantidad.
Antes de tomar agua, lavábamos el vaso ya lavado. Antes de cocinar, enjuagábamos las ollas y los utensilios que ya habían sido fregados. Antes de servir, pasábamos por agua cada plato, envase, tasa o cuchara y lo secábamos con una toallita de cocina limpia. Guardábamos la comida en envases bien sellados, metíamos en la nevera lo que no cabía en los gabinetes, barríamos y mapeábamos el piso todas las noches antes de acostarnos. Pero, no importaba cuánto laváramos, estregáramos, limpiáramos y enjuagáramos, las cucarachas volvían siempre a desfilar por el piso, el _counter_ , los gaveteros y los marcos de las ventanas.
Acostada en la cama, me imaginaba un ejército de cucarachas marchando en filas muy ordenadas hacia la cama que compartía con Delsa halaba las sábanas tratando de taparme las orejas, pero según yo halaba para mi lado, Delsa halaba para el suyo. Trataba de cubrirme la cabeza con la almohada, pero la guata se balanceaba sobre mi frente y no se amoldaba a la forma que yo trataba de imponerle alrededor del cráneo, por el lado de la cabeza, pasando por el lóbulo de la oreja. No me atrevía a quedarme dormida porque me imaginaba que las cucarachas estaban a punto de metérseme por dentro. Tenía miedo de salir de la cama. ¿Y si la legión de cucarachas andaba marchando por el piso? Antes de que pudiera llegar a encender la luz, las pisaría con los pies descalzos. Me retorcía tratando de borrar esa imagen de mi mente. Con la punta de la sábana, me estrujaba el sitio donde me había tocado la cucaracha, pero no importaba cuánto me frotara, la seguía sintiendo. De hecho, sentía montones de insectos caminándome por encima, pero según yo los espantaba por un lado, se movían a otro. Me viraba hacia la derecha, luego hacia la izquierda, pensando que si no me quedaba en una sola posición mucho rato, las cucarachas no tendrían tiempo de metérseme por los diferentes orificios que, yo suponía, eran su meta.
Cuando sonó el despertador, me escurrí de la cama exhausta. Caminé en puntillas para pisar las menos posibles, si era que todavía quedaban cucarachas en el piso. El linóleo estaba desnudo, brillante, limpio, excepto por la porquería amarillenta de la cucaracha que estaba cerca del zapato que había usado para espacharrarla a mitad de noche. No se veía ninguna cucaracha viva, pero eso no era ningún consuelo. Yo sabía que estaban escondidas en las fisuras de los zócalos, dentro de las grietas del marco de la puerta, debajo de la cama.
Según el otoño fue haciéndose invierno y los días refrescaron, descubrimos que nuestro apartamento no tenía calefacción. Mami fue hasta la bodega a llamar al casero. A veces se escuchaban unos golpes secos y metálicos y los radiadores se ponían tibios, pero nunca lo suficiente para calentar todos los rincones del apartamento. Tata prendía la estufa y nos pasábamos la mayor parte del tiempo sentados alrededor de la mesa de formica frente al horno abierto. Inevitablemente, uno de nosotros cayó enfermo con catarro y, en lo que el diablo se pela un ojo, todos nos contagiamos y nos pasamos la mitad de la noche tosiendo y con un pitito asmático en el pecho.
Mami y Tata corrían de un lado a otro con una palangana llena de agua caliente donde habían derretido una cucharadita de Vick's Vaporub. Mientras Mami nos sostenía la palangana debajo de la nariz, Tata nos hacía una casita de campaña con una toalla que nos puso por encima de la cabeza. Después que cada uno había inhalado todo el vapor que pudo, Tata nos emplastó el pecho y la espalda con más Vick's Vaporub y unas hojitas medicinales y nos hizo poner el abrigo más calientito que teníamos. Al día siguiente, Mami preparó una pócima con jarabe Breacol de base, ligado con unos ingredientes de su propia fórmula, cuyos sabores no desaparecían ni aun con las porciones tan generosas de miel que le añadía al pote. El sirop, negro y amargo, olía a alcanfor y clavos quemados. Nos obligó a tomarlo y a las pocas horas ya se nos había pasado el moquillo y se nos había quitado la tos. A partir de entonces, tan pronto uno de nosotros estornudaba o se ponía moquilloso, Mami sacaba la botella pegajosa y eso bastaba para que nos curáramos en el acto. Nosotros le decíamos Tutumá, un misterioso nombre para esa extraña y poderosa medicina que no nos teníamos que tomar para que nos curara.
Tata sostenía que el primer invierno en Nueva York era siempre el más difícil porque, como veníamos del trópico, nuestra sangre no estaba lo suficientemente espesa. Para espesarse la suya, tomaba cerveza o vino diariamente, lo que también contribuía a aliviarle los dolores de hueso que ella juraba no le cedían ante nada más. Para espesarnos la nuestra, Tata nos cocinaba sopones, asopaos y guisos sustanciosos con ñames, yautía y otras viandas puertorriqueñas.
"Pero, si comemos la misma comida que comíamos en Puerto Rico, no se nos va a espesar la sangre," argumenté yo un día. "Si no se nos espesó cuando vivíamos allá."
"Tiene un punto," rió Don Julio.
"Vamos a seguir teniendo la misma sangre aguá de siempre," insistí.
"Lo que esos muchachos necesitan," sugirió Don Julio, "es comida americana."
Tata no se dejó convencer. "La comida americana no alimenta."
"Pero mira lo grandes y saludables que son los muchachos americanos," concedió Mami. "Su comida debe de estarles haciendo algo."
"Parecen papas sancochás," afirmó Tata.
"Pero tienen la sangre espesa," argumentó Delsa, "y nunca se enferman."
A pesar de la desconfianza de Tata en la comida americana, Mami estaba dispuesta a probar cualquier cosa con tal de espesarnos la sangre. Ante nuestra insistencia, compró un par de latas de los productos que habíamos visto anunciados por televisión: espagueti Franco-American, ravioli Chef Boyardee, sopa de pollo con fideos Campbell.
"Fo, qué baboso," Tata miraba con recelo la olla de ravioli de lata que Mami nos estaba calentando. "Yo no sé cómo se pueden comer eso," decía, haciendo muecas, mientras nosotros limpiábamos el último chispito de salsa de tomate del plato.
Mami nos dio comida americana de lata durante una semana, pero los catarros no se nos quitaban nada más que con Tutumá. Así es que perdió la fe en la comida americana y sólo nos la daba como algún antojo especial, pero nunca para sustituir la nutritiva comida puertorriqueña que ella y Tata seguían preparando. Cuando Tata le preguntaba por qué nos dejaba comerla, Mami le explicaba: "Tienen que aprender a comer como americanos por si alguna vez los invitan a una casa americana, no se vayan a portar como jíbaros a la hora de comer."
Eso calló a Tata y me dio a mi una idea. "Mami, todas las nenas se maquillan para ir a la escuela."
"A mí no me importa lo que hagan esas americanas. Tú eres puertorriqueña y muy nena pa' pintorretearte."
Era bueno ser saludable, grande y fuerte como Dick, Jane y Sally. Era bueno aprender inglés y saber cómo comportarse entre los americanos, pero no era bueno actuar como ellos. Mami nos hacía ver claro que a pesar de que vivíamos en los Estados Unidos, nosotros seguiríamos siendo cien por ciento puertorriqueños. El problema era que se hacía difícil saber dónde terminaba lo puertorriqueño y empezaba lo americanizado. ¿Estaba yo americanizada si me gustaba más la pizza que el pastelillo? ¿Sería más puertorriqueña sí la falda me cubría las rodillas? Si yo recortaba un retrato de Paul Anka de una de las revistas y la pegaba en la pared, ¿era menos puertorriqueña que cuando recortaba los retratos de Gilberto Monroig? ¿Quién podría decírmelo?
Alma y Corazón, las primas de Mami, nacieron en Puerto Rico, pero su mamá, Titi Ana, se las trajo a Brooklyn, cuando eran chiquitas. Vivían en la esquina de la Calle Varet y la Avenida Bushwick, en el último piso de un edifico de seis, que tenía unas amplias ventanas al frente. Los pasillos y las áreas comunes tenían los pisos en mosaicos negros y blancos. Por una ventana enorme entraba la luz hacia una escalera cuyos peldaños y el pasamanos eran de un mármol fresquito, fresquito, gastado en el centro de tantos años de sube y baja. Había cuatro apartamentos en cada piso; dos que daban a la Calle Varet y dos al fondo. Siempre que subía al sexto piso, paraba a coger aliento en cada descanso, a escuchar los sonidos detrás de las puertas o a disfrutar de los sabrosos olores de lo que se estaba cocinando. Detrás de una puerta, alguien estaba viendo una novela; se oían unas voces apagadas, como en sordina, acentuadas por una música de órgano. En el apartamento de al frente me olió a café cola'o y un poco más arriba, alguien cocinaba bacalao con berenjena. En el próximo nivel, el sofrito se doraba en aceite caliente y en el apartamento de al lado, las habichuelas se estaban ahumando. Detrás de otra puerta, se oía un merengue a todo volumen, pero frente a ese apartamento, en los apartamentos del fondo, no se oía nada ni salían olores por debajo de la puerta. Cuando llegué arriba y toqué a la puerta de Titi Ana, tenía hambre y los oídos me chillaban.
Corazón abrió los tres cerrojos y la cadena de la puerta para dejarme entrar. Tenía una botella de Coca-Cola en la mano. "Sírvete," me dijo y me señaló la nevera. "Alma está por ahí," me señalo la puerta al lado de la cocina y se metió en su cuarto. En la nevera de Titi Ana había siempre un paquete de Coca-Cola, helado en el congelador y bizcochos Hostess en el gabinete encima del fregadero. Agarré un refresco y le toqué en la puerta a Alma.
Estaba sentada en su cama leyendo un pesado libro de hombres con bigotes grandes. "Tengo examen mañana," me dijo mirándome, "de historia." El cuarto de Alma me era familiar, no sólo por el tiempo que había pasado allí desde que llegué a Brooklyn, sino por lo mucho que se parecía a los cuartos de todas las muchachas que había conocido, cuyos papás tenían chavos para gastar en algo más que en las necesidades básicas. Su cama era blanca y estaba cubierta con una colcha de vuelos, con florecitas, en combinación con las cortinas y con la falda de la coqueta. El piso de linóleo tenía también un motivo floreado, lo que producía el efecto de que Alma vivía en una esplendorosa, llana y eterna primavera. La ventana daba a los techos de los edificios de dos o tres pisos.
"Ahí está el Archie nuevo." Me indicó la tablilla donde guardaba los paquines con los más recientes puestos encima del paquete.
Archie, Verónica, Betty, Reggie, Jughead eran los únicos adolescentes americanos que conocía. En el vecindario puertorriqueño donde vivíamos no había americanos y los pocos que iban a la misma escuela que yo, se apartaban en un impenetrable grupito de muchachas habladoras con rabos de caballo y suéteres cardigan y de muchachos de piernas largas y caras llenas de espinillas. Al igual que Archie y sus amigos, no eran italianos, ni judíos ni negros ni puertorriqueños. Tenían nombres cortos, fáciles de recordar como Sue, Matt, Fred, Lynn. Eran los presidentes de los clubes, los organizadores de los bailes, los editores del periódico de la escuela y del anuario. Parecían actores de televisión: blancos, vestidos con ropa que nunca se arrugaba ni se ensuciaba, con cada pelo en su sitio y un aire de superioridad que los hacía clase aparte.
Mis vecinos, en su mayoría de tez oscura o identificados por su país de origen, vivían en edificios maltrechos y casi en ruinas. A través de Archie había conocido otro Estados Unidos —los suburbios horizontales y bien acicalados de los americanos blancos. A través de él descubrí que la vida de los jóvenes americanos era muy diferente a la mía, sus preocupaciones, tan extrañas para mí como serían las mías para ellos.
Archie y sus amigos vivían en un mundo sin adultos, tomaban sus propias decisiones sobre dónde ir y cómo llegar sin consultar con nadie, excepto con ellos mismos. Mi mundo estaba dominado por adultos, sus reglas estaban escritas en piedra, en español, en Puerto Rico. En mi mundo no había concesiones por el hecho de que estuviéramos viviendo en los Estados Unidos, de que el inglés, poco a poco, se iba convirtiendo en nuestra lengua, de que éramos extranjeros inmersos en la cultura americana.
Archie nunca comía en su casa. Él y sus amigos comían en la fuente de soda de Pop y su dieta consistía en sándwiches, hamburgers, papas fritas, batidas —alimentos que se podían comer sin cubiertos. En nuestro apartamento, Mami y Tata se pasaban en la cocina preparando asopaos espesos, arroz y habichuelas, fricasé de pollo, comidas abundantes que exigían tiempo para saborearse y una conexión estrecha con la cocinera que no se alejaba, pendiente de que comiéramos lo suficiente y preguntándonos si estaba buena.
A Betty y a Verónica les preocupaba mucho salir con muchachos. A mí no me dejaban salir a ninguna parte con un muchacho que no fuera mi hermano. No teníamos teléfono, así es que, a diferencia de Betty y Verónica, no podía sentarme con unas piernas bien formadas colgando sobre el brazo de una butaca tapizada, charlando con amigas invisibles sobre muchachos. No teníamos una butaca tapizada. Yo no tenía amigas.
Archie y sus amigos a veces cargaban libros, pero nunca se les veía en clase, tomando exámenes o estudiando. Su existencia giraba en torno a su vida social, mientras que la mía estaba definida por mis obligaciones como estudiante y como hermana mayor. Ni a Betty ni a Verónica se les pedía que sirvieran de ejemplo para sus hermanas o hermanos menores. Existían sólo para sí mismas, sus únicas responsabilidades eran verse bonitas y mantener contentos a los novios.
Desde la cocina de Titi Ana, me zambullía en el mundo brillante y sin sombras de Archie, celosa de esa vida tan simple, de diversiones y problemas triviales, tan distante de la mía. Nadie nacía ni moría nunca en el mundo de Archie, nadie compartía la cama con una hermana o se bañaba en la cocina o sufría por un papá ausente. Yo quería vivir en esos espacios y paisajes horizontales, pintados en colores primarios, donde "algo" nunca pasaba, donde los adolescentes como yo vivían dichosos, ignorantes de la violencia y la suciedad, donde las cucarachas no acechaban en la noche, donde nadie tenía siete hermanos y hermanas, donde las abuelas no bebían cervezas hasta quedarse dormidas y donde las mamás no necesitaban que una les sirviera de intérprete en la oficina del _welfare_.
Un día, Mami se apareció de sorpresa en la escuela. Empecé a temblar cuando vi el gesto que hizo al ver mi falda que estaba a media pierna cuando salí de casa por la mañana y ahora andaba por encima de la rodilla. Me escudriñó las líneas borrosas alrededor de los ojos y los residuos de colorete en los cachetes. Todas las mañanas, camino a la escuela, Yolanda y yo nos metíamos en la entrada de un edificio de apartamentos en la Avenida Bushwick y nos enrollábamos las faldas hasta el largo donde las usaban las otras muchachas. Nos hacíamos la línea alrededor de los ojos con un lápiz de cejas que Yolanda le había cogido a su mamá. En la escuela, las muchachas que se compadecían de las que teníamos mamás anticuadas, con frecuencia compartían con nosotras el lápiz de labios y el colorete y nos ayudaban a hacernos "tisin" y a arreglarnos el pelo en grandes moños, tiesos por el "esprei". De regreso a casa, nos desenrollábamos la falda a su largo normal, nos quitábamos con saliva lo que nos quedaba del maquillaje, nos cepillábamos el pelo y nos hacíamos de nuevo el rabo de caballo liso y decente.
Tan pronto como Yolanda divisó a mi mamá, bajó la cabeza para que Mami no le viera la cara. Mami me agarró del brazo, cruzó la calle arrastrándome hasta que finalmente me pude zafar de su apretón. Yo evité mirar a los muchachos que se reían y chocaban cinco y levantaban el pulgar para agitar a Mami mientras le gritaban _"Go mamma,"_ según íbamos pasando. Mami los fulminaba con una mirada que los callaba y les borraba de los labios la sonrisita descarada. "Títeres," murmuraba, "no respetan a nadie."
"¿Por qué tú siempre me estás velando?" gritaba, según íbamos subiendo la escalera del edificio. Sabía que me esperaba una paliza, y hubiera podido atenuarla si hubiese mostrado una dosis mayor de humildad. Pero a mi no me importaba que Mami me matara cuando llegáramos a casa. Me había humillado delante de todo el mundo en la escuela, y yo jamás volvería allí.
"Yo no te estaba velando. Fui pa' llevarte de compras," dijo en voz baja, consciente de que los vecinos podían estar mirándonos por debajo de las cadenas de sus puertas entreabiertas, pendientes de la gritería.
"¿Por qué no esperaste que llegara a casa?" chillaba, dándole a la puerta de la cual no tenía llave. Héctor la abrió y la mantuvo abierta hasta que entramos al cuarto lleno de gente y yo tiré mis libros al piso.
Mami me agarró por el pelo. "¿Quién te crees tú que eres," gritaba, "pa' contestarme a mi de esa manera?" Levanté los brazos tratando de soltarme y me aguanté el pelo contra el cráneo, mientras ella me lo halaba hacia afuera. "No te creas que porque estamos aquí te vas a portar como esas americanas frescas," gritaba Mami con la cara roja, los ojos achinados, los labios tirantes. Me tiró en la litera de abajo donde estaban Norma y Alicia con los ojos inmensos del susto. Tata salió de la parte de atrás del apartamento y se interpuso entre nosotras, pero Mami ya había acabado. Me acosté boca abajo en la cama, asfixiada de rabia, ahogándome en los sollozos que venían después de sus pelas. Me sobé el cuero cabelludo que me ardía, gemí sin llorar, me di con la frente contra el colchón, hasta que Norma me empujó con los pies. "Muévete," me dijo, "nos estás aplastando las muñequitas de papel." Levanté la cabeza y me topé con la mirada vacua de unas muchachitas rubias de ojos azules y labios rojos, con vestidos cortos, ajustados y reveladores. De un manotazo, las barrí de la cama y las pisé cuando me levanté para subirme a la litera de arriba. Los gritos de Norma y de Alicia apagaron los míos.
Esa noche, acostada al lado de Delsa, me alejé de ella y de mí misma y del apartamento en la Calle Varet, de Brooklyn, de Nueva York. Volé hacia la tibia brisa de una tarde puertorriqueña, hacia el aire oloroso a jazmines, hacia el coquí cantando en la hierba. Me acerqué al lado de mi papá que estaba mezclando cemento, moviendo la pala rapidito en el fango gris, raspando las orillas, juntándolo con la mezcla en el centro. Mientras trabajaba, cantaba un chachachá de Bobby Capó. La carretilla llena de cemento chirriaba según Papi la empujaba y la acercaba al muro que estaba construyendo. Las venas de sus brazos morenos se le marcaban por el esfuerzo, los músculos de la espalda se le abultaban hasta la cintura. Me quedé dormida contándole de mi día, del trayecto a la escuela por las anchas aceras, de los salones repletos, de las gangas en las que se metían los muchachos para protegerse de otras gangas, de cómo en los Estados Unidos no éramos puertorriqueños, sino hispanos. Le dije que Mami estaba desencantada conmigo y me acusaba de haberme americanizado cuando yo lo único que quería era ser como las otras muchachas de mi edad. Hablé con él como lo hacía cuando vivíamos juntos y Mami y él se contentaban después de alguna pelea. Le pedí que viniera a buscarnos y nos sacara de Brooklyn, como nos rescataba de los sitios donde nos llevaba Mami cada vez que se peleaban. Uno de estos días se iba a aparecer en el umbral de la puerta como hacía en Puerto Rico, y convencería a Mami de que había cambiado, de que todavía la quería. Le escribiría largos poemas románticos sobre hogares felices y el amor que siente un hombre por la madre de sus hijos. La ablandaría con regalos, una flor en un vaso de papel, un helado de coco medio derretido. Le había resultado antes; le resultaría otra vez. Mami cedería y aceptaría volver con él y todos regresaríamos a Puerto Rico, donde nunca tendríamos frío, donde reiniciaríamos nuestras vidas en nuestra lengua, en nuestra patria, donde volveríamos a ser una familia.
Papi escribió para anunciarnos que se había casado con una mujer a quien ninguno de nosotros había oído mencionar y que se había mudado a un pueblo que nunca habíamos visitado.
Sentada en la orilla de la cama de Mami, leí la carta una y otra vez, miré su letra clara y pareja, los márgenes amplios, tan familiar todo y tan doloroso.
La nueva esposa de Papi me cayó mal enseguida y juré que nunca la aceptaría, ni los visitaría. Mis cartas para él, hasta entonces, llenas de noticias, de temores y confusión, se volvieron saludos breves, lista de notas e informes médicos sobre el progreso de los tratamientos de Raymond, a quien, con mucho éxito, le estaban salvando el pie.
Por las mañanas, camino a la JHS 49, añoraba mi vida en Macún. Extrañaba el aire húmedo de rocío, el crujir del cascajo en la carretera de tierra, el canto del gallo, el zumbido de las abejas, el brillante sol amarillo del amanecer puertorriqueño. Me resistía a la cuadrada uniformidad de las calles de Brooklyn, los edificios imponentes de esquinas angulosas, las aceras manchadas con costras de flema y gomas de mascar pegajosa. Cada día que pasábamos en Brooklyn, era como un telón que caía entre mi otra vida y yo, la vida en la que yo sabía quién era, en la que yo no sabía qué era ser pobre, ni sabía que mi papá y mi mamá no se querían, ni sabía lo que era perder a un padre.
Con el matrimonio de Papi, se deshicieron nuestros lazos con Puerto Rico. Él era nuestro vínculo más fuerte con la Isla, puesto que la mayor parte de la familia de Mami estaba en Brooklyn y los hermanos y las hermanas de Papi no habían sido nunca una presencia significativa en nuestro hogar.
Cuando traté de averiguar si Mami se sentía tan desencantada como yo, no me hizo caso y contestó que Papi tenía derecho a hacer su vida y que nunca debíamos culparlo, ni faltarle el respeto. Pero yo no podía quitarme de encima la sensación de que vagaba a la deriva. Al no tomarnos en cuenta a la hora de casarse, Papi nos había excluido a todos del resto de su vida.
# "Negi, ¿tú vas a ser famosa?"
#
Nos dimos cuenta de que Mami estaba enamorada porque tarareaba y cantaba boleros mientras limpiaba o planchaba. Estaba enamorada, porque tan pronto encontró otro empleo se compró ropa nueva, algo que no había hecho desde que nos mudamos a Brooklyn. Definitivamente, estaba enamorada; lo sabíamos porque sus ojos castaños brillaban, tenía la sonrisa a flor de labios y parecía que iba a reventar de orgullo cada vez que nos miraba como si hubiéramos sido los hijos más perfectos que madre alguna pudiera desear. Estábamos seguros de que Mami estaba enamorada porque Tata discutía con ella por cualquier tontería y se paraba en la ventana cuando Mami salía, a velar para dónde cogía. Dos o tres veces por semana, Mami cruzaba al frente después del trabajo, se quedaba como una hora y regresaba de lo más contenta. No se quedaba nunca hasta más tarde de las nueve, pero el que oía a Tata, pensaba que Mami se pasaba por la calle hasta las tantas de la madrugada.
Cuando finalmente conocimos a Francisco, quien vivía con sus padres frente a casa, confirmamos que Mami estaba enamorada porque lucía tranquila cerca de él y la mirada inquieta había desaparecido de su cara. Tenía treinta años y Francisco, veintiocho; pero los dos años de diferencia a nosotros no nos molestaban tanto como parecían molestarle a Tata.
Un brillante día, de finales de invierno, húmedo por la nieve que aún se estaba derritiendo, salimos en fila india de nuestro edificio, cada uno cargando una caja o una maleta. La gente que pasaba nos miraba perpleja. Teníamos miedo de irritar a Tata, así es que salíamos y entrábamos del apartamento en puntillas, cargando nuestras pertenencias hasta que mudamos todo, menos las cosas más pesadas, a un apartamento que quedaba un poco más abajo del nuestro. Paco y Jalisco llegaron por la tardecita para ayudarnos a mudar los muebles y antes del anochecer estábamos acomodados en un apartamento de dos cuartos.
Dos o tres días después de la mudanza, Francicso vino a comer. Después, él y Mami se quedaron en la cocina hablando, mientras nosotros veíamos _Candid Camera_ en el cuarto de al frente. Se fue temprano, pero volvió al día siguiente y así, todos los días durante una semana, cada vez se requedaba hasta un poquito más tarde, hasta que una mañana amaneció allí.
"¿Como le decimos?" le pregunté a Mami, cuando ya era obvio que se había mudado con nosotros. "No le podemos decir Papi..."
Frunció el ceño, como hacía siempre que me ponía irrespetuosa. "No, no es tu papá," dijo por fin, mientras casaba unas medias.
"Y es demasiado joven para ser Don Francisco."
"Si, lo es." Cogió unos panties y los estiró con la mano. Me daba cuenta de que estaba abochornada, y de que debía suspender las preguntas y dejarla en paz.
"¿Entonces, cómo le decimos?"
"Franky, así le dice su familia," dijo seca, pasándome los panties y un par de camisas dobladas. "Pon esto en la gaveta de Edna." Las cejas se le juntaron sobre los ojos, lo que significaba que no estaba dispuesta a contestar ni una pregunta más.
Guardé la ropa pero no podía dejar de pensar en el asunto. Franky no sonaba lo suficientemente oficial para quien era nuestro padrastro. Bueno, no exactamente. Puesto que no estaba casado con Mami, técnicamente no era su esposo. Pero ella no había estado casada con Papi y él había sido su esposo. ¿O no? Oficialmente no. Papi era nuestro padre, así lo decían nuestros certificados de nacimiento. ¿Pero qué quedaba él de ella? Y ahora que Francisco era parte de nuestra familia, ¿qué quedaba de nosotros?
No le podía preguntar a Mami. Era una falta de respeto meterme en su vida personal. Pero yo sabía que las mujeres casadas miraban por encima del hombro a las que no lo eran. "Ah, ésa vive con él," decían con un gesto despectivo de la mano y una expresión de repugnancia.
También sabía que el matrimonio con traje blanco, velo y corona, con un desfile por la nave central de la iglesia, con cura, damas en trajes vistosos y ujieres en etiqueta era el sueño de Mami para mí y para mis hermanas.
"¡Qué felicidad!" decía anhelante, "ver a una hija camino al altar, de blanco, con velo y corona."
Mami no se había casado por la iglesia, pero se suponía que nosotras sí lo hiciéramos. Nunca íbamos a la iglesia, pero algún día nos pararíamos frente a un sacerdote y haríamos los votos que ella nunca hizo.
"Yo me sacrifico por cada una de ustedes," nos decía una y otra vez. Una boda bonita, por la iglesia, era una de las recompensas que esperaba de cada una por su sacrificio.
Poco tiempo después de que la barriga de Mami empezara a crecer con el bebé de Francisco, lo llevaron a él a la sala de emergencia con dolor de estómago. Cuando Mami regresó del hospital, nos dijo que Francisco tenía cáncer.
"Pero no se preocupen," nos dijo, "se va a mejorar pronto."
Por su rostro tenso, los labios apretados y la mirada asustada, nos dimos cuenta de que sólo estaba tratando de tranquilizarnos.
Nos mudamos a un apartamento que quedaba un poco más abajo en la misma calle, para que Tata pudiera vivir con nosotros. Don Julio trajo el catre y el gaveterito de Tata, su radio, su ropa, algunas fotos de cuando era joven, su altar. Ahora que Francisco estaba enfermo, Tata no se quejaba tanto de que fuera tan joven o de que Mami nos estuviera dando un mal ejemplo al vivir con él. Ahora, en vez de quejarse, cocinaba y nos atendía para que Mami, al salir del trabajo, pudiera irse directamente al hospital a estar un rato con Francisco.
Unas semanas más tarde, el casero nos pidió que nos fuéramos porque había demasiada gente viviendo en el apartamento de tres cuartos que él le había alquilado a una mujer con dos niños. Nos mudamos por quinta vez en un año. En el apartamento nuevo de la Calle Ellery, la bañera estaba otra vez en la cocina, cubierta por una plancha de metal esmaltado que servía de mesa durante el día y que se removía de noche para podernos bañar. Cuando bajaba la temperatura, los radiadores permanecían fríos y el viento silbaba a través de las grietas de los marcos de las ventanas.
Tuvimos que transferirnos de escuela. La Escuela Intermedia 33, donde estudié mi noveno grado, ocupaba la mayor parte de un bloque de la ciudad. El patio de cemento y la cancha de balonmano estaban cercados por una verja. En la parte de adentro, las paredes estaban forradas con los mismos bloques color ámbar que cubrían la parte exterior. Los pisos eran de un _vinyl_ lustroso que chirreaba cuando me ponía los tenis que sólo estaban permitidos los días de Educación Física.
Obtuve puntuaciones altas en una batería de exámenes que me administró Míster Barone, el consejero académico. Yo no tenía idea de para qué eran los exámenes, ni por qué tenía que tomarlos, pero Míster Barone me explicó que mostraban aptitud y potencial y que en lugar de pedir ingreso a una escuela vocacional, debería solicitar a una escuela que me preparara para seguir estudios universitarios. A pesar de que se me hacía cada vez más fácil entender el inglés escrito, el inglés oral todavía me confundía, por lo tanto, accedí a aspirar a una educación académica sin entender bien de qué se trataba y demasiado avergonzada para preguntar. Fue idea de Míster Barone que solicitara a Performing Arts High School en Manhattan.
"¿Por qué tan lejos?' preguntó Mami, "¿no hay escuelas en Brooklyn?"
"Es una escuela especial."
Frunció el ceño. "¿Especial?"
"Tengo que solicitar..."
"Escuela privada. No tenemos chavos."
Le expliqué que era una escuela pública para muchachos y muchachas que quisieran ser actores, bailarines, músicos.
Se me quedó mirando. "¿Desde cuándo tú quieres ser artista?"
"Yo no sé. Es sólo una escuela."
"Vas a salir bien allí," interrumpió Tata, "con lo dramática que tú eres."
"No hay actores puertorriqueños en televisión," nos recordó Delsa.
"¿Ricky Ricardo?" preguntó Raymond.
"¡Babalú!" Edna le dio a un tambor imaginario que tenía al lado y Alicia y Héctor se le unieron formando una línea de conga y cantando "¡Babalú, Babalú Oyé!"
"¡Estecen quietos!" dijo Mami, "que la gente de abajo se va a creer que hay salvajes acá arriba."
"Ricky Ricardo es cubano y es cantante, no actor," continuó Delsa tan pronto los nenes se tranquilizaron, "y todos sabemos que Negi no sabe cantar."
"Y aunque supieras," advirtió Norma, "Mami nunca te va a dejar poner esos vestidos que usan las _vedettes_ , que se les ve to'. ¿Verdad, Mami?"
"Déjensen de boberías," contestó Mami, volviendo a la ropa que estaba remendando.
"Te lo dije," rió Norma.
Mami sonrió, pero no dijo nada más.
A mí nunca se me había ocurrido escoger la actuación como futura profesión, pero cuando Míster Barone me sugirió Performing Arts y yo accedí, me hizo tanta fiesta que me gustó. No le dije que a lo mejor Mami no me dejaría ir aunque me aceptaran. Me ayudó a prepararme para la audición que era requisito, seleccionó un monólogo, reclutó a Míster Gatti, el maestro de inglés para que me practicara la pronunciación de las palabras que yo me memorizaba fonéticamente sin entender su significado. Misis Johnson, la maestra de Economía Doméstica, me enseñó cómo entrar a un salón como una dama y cómo sentarme con las piernas juntas.
No perdía oportunidad de dejarle saber a Mami que me estaba preparando para mi audición. Me paraba frente al espejo de su tocador a practicar mi monólogo, tratando de vencer mi eterno hábito de hablar con las manos, algo que según Misis Johnson, distraía muchísimo. Me sentía como una muñeca de papel, tiesa y plana con una sonrisa pegada al rostro.
"Usted pertenece a un tipo muy común en este país, Sra. Phelps," empecé. Mis hermanas y hermanos se reían ante mis intentos de ser dramática y repetían algunos pasajes de mi monólogo, trincando la cara por el esfuerzo de mantenerse serios.
_"Stop molestationing me,"_ gritaba y Mami o Tata los espantaban para el otro cuarto, donde yo los oía riéndose.
Durante semanas, mis hermanas y hermanos me relajaron con mi falta de talento, mientras en la escuela, Míster Barone, Míster Gatti y Misis Johnson me ayudaban a prepararme. Nadie de la JHS 33 había pasado nunca a Performing Arts High School, y Míster Barone hizo lo que estaba a su alcance para que todo el mundo se enterara de que yo había solicitado. Ahora, además de mi familia, todo el noveno grado cuestionaba mis habilidades artísticas.
"Mira, spic!" me dijo con sarcasmo Lulú, un día cuando entraba yo al baño de las muchachas. "¿Qué es lo que tú te crees? ¿Que eres mejor que nosotras? Pues tú no eres más que una spic, y que no se te olvide." Me empujó dentro de uno de los cubículos y por un momento pensé que me iba a dar un puño en la cara, pero se contentó con escupirme, reírse y dejarme sentada en el inodoro, tan asustada que por poco me meo encima.
Me limpié la cara con papel de inodoro, me bajé los panties y oriné aguantándome las lágrimas. No me iba a ver llorar. Tampoco me vería pelear porque yo no podría ganarle. Lulú y sus amigas eran fuertes, una ganga de muchachas que se sentaban en la parte de atrás del salón a pasarse papelitos unas a las otras, fumaban en la escalera y buscaban bulla con todo el que no les cayera bien. Ellas sabían que yo les tenía miedo y hacían todo lo posible para que yo siguiera asustada. Me hacían tropezar en la clase de Educación Física; me empujaban en las escaleras; me cogían comida de la bandeja del almuerzo. Por culpa de Lulú y sus amigas, yo iba al baño en la escuela solamente cuando no podía aguantar más. Por culpa de ellas, regresaba a casa por el camino más largo para evitar las esquinas por donde ellas se pasaban, las mañanas y las tardes, fumando, riéndose y acosando a todo el que pasaba.
Durante meses, Lulú y su ganga me habían pasado por alto. Yo era una más de las muchachas con las que ellas tropezaban en los cambios de clases. Pero, tan pronto oyeron que yo estaba solicitando a Performing Arts, Lulú y sus amigas empezaron una campaña para ponerme de nuevo en mi sitio.
"Ahí va la actriz," decía con sorna Luz Mari cuando le pasaba por el lado en el pasillo.
"Se cree blanca," murmuraba Violeta cuando me excusaban de la clase de Estudios Sociales para que practicara mi monólogo con Míster Gatti.
"¿Qué es, mi'jita?" me retaba Denise mientras esperaba mi turno para subir la soga en la clase de Educación Física, "Eli Whitney no es lo suficientemente buena para ti?" Casi todos los estudiantes de la _JHS 33_ iban a parar a la escuela vocacional más cercana que adiestraba secretarias, enfermeras, mecánicos de automóvil y técnicos de refrigeración.
"Es sólo una escuela," me defendía, pero de nada valió. Lulú y su ganga, para quienes yo había sido invisible, me consideraban una traidora por haber aceptado la sugerencia de mis maestros.
"Es que están celosas," sugirió mi amiga Natalia de regreso a casa un día. "Ellas van a estar preñás y cogiendo _welfare_ antes de que terminemos la _High_."
Natalia vivía con su mamá y sus hermanas en un edificio que quedaba a unas puertas del nuestro. Había nacido en Nueva York, su inglés era perfecto y hablaba español lo suficientemente bien como para que yo pudiera hablarle una mezcla de los dos sin confundirla demasiado. La mamá de Natalia, como la mía, trabajaba en una fábrica de ropa en Manhattan, aunque la mía cosía brasieres y fajas y la de ella hacía ropa deportiva.
Los sábados nos saludábamos a la distancia mientras ayudábamos a nuestras mamás a subir por la escalera, los carritos llenos de compra. Durante la semana, Natalia les preparaba el desayuno a sus hermanas y las acompañaba a la escuela antes de irse ella para la nuestra. Su mamá recogía a las nenas por la tarde, así es que Natalia y yo nos regresábamos juntas casi todos los días. Al principio, cuando la conocí, pensé que era cristiana porque nunca usaba maquillaje, ni faldas cortas ni colores brillantes. Entonces descubrí que se veía así porque su mamá, como la mía, era bien anticuada.
Cuando nuestras mamás se dieron cuenta de que las dos eran igual de estrictas con nosotras, Natalia y yo pudimos ser amigas sin problemas porque ninguna de las dos podría convertirse en una mala influencia para la otra. Las dos éramos nenas "buenas" de quienes se esperaba que hiciéramos lo que se nos mandaba, que le sirviéramos de ejemplo a nuestras hermanas y hermanos, y que asumiéramos esa responsabilidad con seriedad. Sin embargo, Natalia era mejor que yo en eso de servir de modelo. Era buena por naturaleza. En cambio, a mi me irritaba la idea de que todo lo que hiciera fuese observado por seis hermanas y hermanos que, a su vez, pudieran hacer lo mismo. Me preocupaba que si yo tropezaba, Delsa, Norma, Héctor Alicia, Edna y Raymond, caerían detrás de mí, como una fila de dóminos, para no pararse jamás.
Natalia y yo hablábamos mucho de nuestro futuro. Ella solicitó a la Bronx High School of Science porque soñaba con convertirse en doctora de un hospital importante, como el Mount Sinai.
"Voy a tener un apartamento en Park Avenue, con portero y ascensor," fantaseaba apretando las manos contra su pecho como para contener toda la felicidad que sentiría.
"Cuando yo sea una actriz famosa, voy a regresar a Puerto Rico a una finca en el campo," le dije, "y voy a tener pollitos y un gallo y a lo mejor, un perro."
"Pero, ¿por qué tú quieres hacer eso?"
"Porque..." ¿Me atrevería a decirle que añoraba regresar a Macún? ¿Que extrañaba el ritmo pausado del campo puertorriqueño, los apacibles y silvestres montes verdes, la variedad en la textura de los caminos de tierra, de polvo a gravilla, a arena, a fango? Yo me reía y le decía que las riquezas que soñábamos alcanzar estaban destinadas a regresarme a mi lugar de origen, mientras que ella soñaba con algo completamente diferente a lo que había conocido antes. Ella reía por compromiso y yo sufría pensando que la había ofendido al implicar que mi niñez había sido más feliz que la suya.
"Negi, ¿tú vas a ser famosa?" me preguntó Raymond, dos o tres días antes de la audición.
_"Leaf me alone,"_ le dije molesta y preocupada de que quizás me había metido en camisa de once varas. Había memorizado el monólogo que había escogido Míster Barone y había practicado cómo entrar a un salón como una dama, cómo sentarme en vez de tirarme en la silla, cómo mantener las manos en la falda en vez de usarlas para darle énfasis a mis palabras. Me parecía que ya estaba actuando y todavía ni había visto la escuela.
"Mami, la audición es la semana que viene, ¿me puedes llevar?" Le enseñé el papelito donde Míster Barone había apuntado la dirección de la escuela: 120 West 46th Street. Lo estudió como si hubiese tenido escrito mucho más que dos números y dos palabritas.
"¿Cuándo tienes que ir?" preguntó después de un rato y yo me derretí del alivio. Le dí los detalles y de paso le mencioné que Misis Johnson me había dicho que no tenía que vestirme formalmente, pero que sería bueno que fuera bien arregladita. "Yo vi un traje que te quedaría bien lindo," me ofreció Mami, y yo no le dije que si me iba a comprar algo, prefería escogerlo yo misma.
Unos días después, Mami me trajo un _jumper_ en tela escocesa roja y un par de zapatos nuevos. "Esta es una fajita para medias," me dijo, desempacando una pieza de ropa interior blanca adornada con encaje que tenía unas ligas que terminaban en unos botones de goma que encajaban en unos aritos de metal. "Esto es lo que estamos haciendo ahora en la fábrica. Ésta, la hice yo misma."
Había visto a Mami ponerse las medias, alisárselas con los dedos y abrochárselas. La había visto de espaldas al espejo, asegurándose de que las costuras estuvieran derechas y ajustándoselas con cuidado. Hasta ahora no me había dejado usar medias, por eso yo sabía que la fajita y el paquete plano que contenía unas medias sin costura era una concesión especial de parte de Mami, un reconocimiento de que ya no era una niña, aunque ninguna de las dos estaba lista para considerarme una mujer.
"Gracias, Mami," la abracé emocionada.
"Para ocasiones especiales," me dijo y me besó en la cabeza. "Se te van a ver bien con tu traje y tus zapatos nuevos."
Durante el resto de la semana, Tata me servía grandes porciones de comida como para engordarme para lo que venía. Conscientes de toda la atención que estaba recibiendo, mis hermanas y hermanos me seguían con enormes ojos de incredulidad, en busca de aquello que todo el mundo parecía ver, menos ellos.
Yo me sentía más o menos igual. Tanta gente grande mimándome por un lado, mientras que por el otro, Lulú y su rebaño intensificaban las amenazas y los insultos como para evitar que se me subieran los humos a la cabeza. Percibía que ser aceptada en Performing Arts sería importante, no solo para mí, sino para Míster Barone, que se pavoneaba por la escuela diciéndole a todo el que quisiera escuchar que yo iba para esa escuela. Esto, a pesar de que faltaban muchos días para la audición y yo podría no impresionar a la gente de la escuela con mis talentos dramáticos. Y era importante para Mami, que se las echaba muchísimo diciéndole a los parientes que yo iba a ser artista, lo que traía a mi mente las mismas imágenes que le traía a Norma: mujeres curvilíneas en escasos y provocativos vestuarios y plumas en el pelo.
El día de la audición Mami me llevó a Manhattan. Era la primera vez que había salido de Brooklyn desde que llegamos a Nueva York. El tren elevado viajaba al mismo nivel que las ventanas de arriba de los almacenes y de los edificios de apartamentos que quedaban a escasos pies de los rieles. Traté de asomarme para ver qué había detrás de esas ventanas, dentro de los apartamentos que parecían poder tocarse con la mano. Pero, el tren se movía con demasiada rapidez y yo sólo alcanzaba a ver unas imágenes borrosas que podían o no ser de gente en habitaciones sombrías.
La escuela quedaba a un bloque del brillo y la conmoción de Broadway. Era un día frío y ventoso y Mami y yo íbamos encogidas dentro de los abrigos, con los ojos lagrimeando por los vientos helados. Las dos o tres cuadras entre la estación de Times Square y la escuela estaban repletas de gente que parecían no notar el frío, y que admiraban las enormes pizarras eléctricas en los lados de los edificios, o se embelesaban con unos carteles que cubrían los frentes de muchas tiendas, de mujeres con las partes privadas cubiertas por unas franjas negras, pero que aun así revelaban que estaban desnudas.
En la esquina de la 46th Street y Broadway había un Howard Johnson's y entramos un momento a calentarnos. Las mesas que bordeaban las ventanas estaban ocupadas por gente que parecía llevar años allí. Mami y yo nos sentamos en el counter donde nos atendió una mujer de pelo plateado espumoso, con sombra turquesa, pestañas postizas, lápiz de labio rosa brillante y una cara más arrugada que una pasa. Nos decía _"honey"_ o _"darling"_ y después que nos sirvió el dulce y el café, volvió varias veces para preguntarnos si queríamos algo más y para servirnos más café.
Yo estaba nerviosa, pero eso no me impidió comerme mi _danish_ de piña y la mitad del de Mami y tomarme dos tazas de café cargado, con crema y mucha azúcar.
"Para ser tan flaca, _come_ ," le dijo la mesera a Mami, que asintió y sonrió como si hubiese entendido.
Caminamos la media cuadra que nos faltaba hasta llegar a la escuela y tan pronto me llamaron al salón me arrepentí de tener tanta comida en el estómago. Las entrañas me daban vueltas y más vueltas, y si la entrevista no acababa pronto, iba a vomitar frente a las tres señoras que tenían en sus manos mi futuro como "artista." Pero, logré decir el monólogo, hacer una pantomima y salir por las pesadas puertas rojas de la escuela antes de vomitar, entre dos carros estacionados, mientras Mami me aguantaba el pelo hacia atrás y me preguntaba: "¿Estás bien? ¿Te sientes mejor?"
De regreso a casa me preguntó qué había pasado en la audición. "Na'," le dije, "contesté unas preguntas y dije mi monólogo."
No le podía decir que estaba tan nerviosa que se me había olvidado todo lo que aprendí con Míster Barone, Míster Gatti y Misis Johnson. Dije el monólogo volando, tumbé una silla, contesté preguntas sin entender exactamente qué me estaban preguntando. No le podía decir a Mami lo mal que lo había hecho después que ella había gastado un dinero que necesitábamos tanto en comprarme un ajuar nuevo con todo y zapatos. Me daba vergüenza regresar a JHS 33 y decirle a Míster Barone que había echado a perder la audición. Todo el mundo se reiría de mí por creída, por creerme que podía entrar a Performing Arts y por fracasar, a pesar de toda la ayuda que recibí. Me ví en la escuela con Lulú y Violeta, con Luz Mari y Denise, quienes no me dejarían olvidar nunca que yo me creía mejor que ellas. Por las mañanas, mientras yo viajara en la guagua hacia Eli Whitney, Natalia estaría en el tren camino al Bronx High School of Science. Ya no tendríamos nada de qué hablar porque ella estaría ocupada, preparándose para la Universidad, mientras yo estaría cosiendo ropa interior, en una fábrica, al lado de Mami.
El tren salía como un bólido de los túneles, traqueteando sobre el puente de Williamsburg camino a Brooklyn. El horizonte de Manhattan retrocedía como una enorme pared entre nosotras y el resto de los Estados Unidos. Viré la cara para que Mami no me viera, y lloré. Al principio las lágrimas eran de vergüenza por lo que yo creía había sido una audición espantosa. Pero según nos acercábamos a nuestra parada en Brooklyn, lloraba porque las semanas de intensa preparación me habían dejado añorando una vida que, ahora estaba segura, no tendría nunca.
# "Pero siguen siendo ilegítimos..."
#
Según le iba creciendo la barriga a Mami, más le costaba moverse porque le dolían la espalda y las piernas. Dejó el trabajo y yo la acompañé de nuevo a la oficina del _welfare_.
"Necesito ayuda hasta que nazca el bebé y el papá salga del hospital," me hizo traducir.
"¿Y cuánto tiempo llevan de casados usted y el señor Cortez?" preguntó la trabajadora social.
"No estamos casados," contestó Mami. "Vivimos juntos hace diez meses."
La trabajadora social frunció la boca. "¿Su primer esposo le pasa pensión a los hijos?"
"No."
"¿Cuánto tiempo llevan divorciados?"
"Dile," me pidió Mami, "que tu papá y yo no estuvimos casados."
La trabajadora social agarró la pluma y su letra zurda y sesgada se deslizó lentamente por el papel rayado, como filas de alambre de púa.
"Entonces los siete hijos mayores también son ilegítimos," dijo y Mami se puso colorada, aunque todavía yo no había traducido.
"Su padre los reconoció a todos," me hizo traducir, sacando los certificados de su cartera.
"Pero siguen siendo ilegítimos," insistió la trabajadora social, sin hacerle ningún caso a los documentos.
"¿Y eso qué tiene que ver?" preguntó Mami en español y yo traduje, encendida también de la vergüenza porque había subido la voz y yo sabía que estaba a punto de formar un revolú.
La trabajadora social no contestó y siguió escribiendo en su libreta. "Eso es todo," dijo finalmente. "Le dejaremos saber."
Al regresar a casa, la busqué. "Ilegítimo" significaba nacer de padres que no estaban casados. Por la manera en que los labios de la trabajadora social se fruncieron, ilegítimo me había sonado a algo peor. Tenía un sinónimo, "bastardo," que yo había escuchado usar como insulto. Sin yo percatarme, la trabajadora social nos había ofendido a Mami y a mí. Ojalá me hubiera dado cuenta, para poderle contestar algo. ¿Pero qué le iba a decir? Tenía razón. Éramos ilegítimos. Ahora me preocupaba que Mami no fuera a conseguir la ayuda que necesitábamos del _welfare_ porque ella y Papi no se hubieran casado pero, unos días más tarde, nos aprobaron el caso.
La palabra, sin embargo, permaneció en mi conciencia mucho tiempo.
Dos meses después de que nació su hijo, murió Francisco. La mirada de Mami, generalmente viva y curiosa, se volvió opaca, se volcó hacia adentro, donde no podíamos alcanzarla con nuestros besos y abrazos. Sobre su tocador había velas encendidas de día y de noche, su calor rondaba el aire como si hubiera sido el espíritu de Francisco el que anduviera rondando, vigilante, para ver si, cómo y por cuánto tiempo, le guardaríamos luto.
No podia llorar la desilusión de que nuestra familia se hubiera roto de nuevo. Papi se había negado a seguir a Mami a Nueva York, renuente a ayudarnos a enfrentar una ciudad tan injusta y fría. Francisco nos había dejado tan rápido como llegó, llevándose consigo la promesa que le había hecho a Mami de amarla para siempre, de ser el hombre de la casa, de formar con nosotros una familia completa, con una mamá, un papá y unos hijos. Cada vez que pasaba por el altar, me detenía a ver las llamas anaranjadas que flotaban sobre la cera derretida. Ponía mi mano sobre ellas y sentía su calor, una tibieza fuerte, como un abrazo, una promesa.
Traté de imaginarme la vida de Papi. Se había mudado y yo me preguntaba, cómo sería su nueva casa. ¿Sería en el campo o en un pueblo? Su nueva esposa, ¿sería más bonita que Mami? ¿Sería tan buena cocinera? ¿Se sentarían las hijas de ella cerca de Papi mientras él leía algún poema que acababa de escribir, como hacía yo? Le escribía cartas sosas, sin atreverme a preguntarle nada sobre su vida, temerosa de que fuera a contarme lo feliz que era.
Si Papi se hubiera venido con nosotros, Mami nunca se hubiera enamorado de Francisco, él nunca se hubiera muerto y nosotros no estaríamos cogiendo _welfare_ otra vez. Mami y Papi peleaban, sí, pero siempre se contentaban. Igual que yo cuando me peleaba con mis hermanas y hermanos; eventualmente, seguíamos como si nada. Si nosotros podíamos hacerlo, ¿por qué ellos no?
Resentía a los hombres que se pasaban parados en las esquinas o sentados en las entradas de los edificios, los codos en las rodillas, la mano sosteniendo una lata de cerveza o acunando un cigarrillo humeante entre las piernas. Podrían haber sido el papá de alguien, pero no encontraban nada mejor que hacer que estar ligando a las muchachas y a las mujeres que pasaban para mascullarles promesas entre dientes.
Una mañana, al llegar a la escuela, Míster Barone se me acercó corriendo. "¿No es maravilloso? ¡Felicitaciones!" Mi cara debe haberle dicho que no tenía idea de lo que estaba hablando, así es que, se detuvo, cogió aire y me habló despacio. "Llegó una carta. Te aceptaron en Performing Arts."
"¡Dios mío!" Míster Barone me llevó a la oficina donde la secretaria, las demás consejeras académicas y el principal me dieron la mano. "No lo puedo creer," repetía una y otra vez. "No puede ser verdad."
"Trabajaste duro," me dijo Míster Barone, "te lo mereces."
Camino a mi salón hogar, me encontré con Natalia. "Adivina qué. ¡Me aceptaron!"
Natalia pegó un grito, dejó caer los libros, me abrazó. "¡Ay, Dios mío! ¡Estoy tan orgullosa de ti!" Se recogió enseguida, un poco cortada, por su despliegue de entusiasmo. Yo me bajé para ayudarle a recoger sus libros.
"Estoy loca por decírselo a Mami," dije. "Ella necesita buenas noticias."
"Me encantaría verle la cara cuando lo sepa," Natalia metió unos papeles sueltos dentro de una libreta. Parecía estar a punto de abrazarme otra vez, pero se apretó los libros contra el pecho. "Me alegro tanto por ti," dijo y se fue de prisa por el pasillo.
No me di cuenta de que estaba sonriendo hasta que Lulú me pasó por el lado frente a los laboratorios de Ciencia, me agarró por el brazo y me dijo, "¿De qué te ríes?"
"De nada," le contesté poniéndome seria, "de nada me río." Lulú tenía unos ojos hermosos, redondos, verdes, de pestañas espesas. Parpadeó, estuvo a punto de decirme algo, pero se detuvo cuando se asomó una maestra.
"Chicas, muévanse, sonó el timbre," nos advirtió.
Lulú chasqueó la lengua, me empujó con la fuerza suficiente para dejarme saber que me podía hacer daño. "Quítate esa sonrisa de comemierda de la cara," dijo amenazante y se fue por el otro lado.
Cuando llegué por fin a mi salón hogar, Míster Gatti estaba escribiendo en la pizarra, unas preguntas para un _quiz_ sin avisar. Me sonrió y me guiñó el ojo cuando me senté. El chirrido delator del altoparlante que había en la parte de al frente del salón nos avisó que venía un anuncio. Metimos la cabeza en los libros con la intención de no hacerle caso al anuncio y de aprovechar esos minutos para repasar.
"Ahem," empezó la voz. "Jóvenes, damas y caballeros," la voz áspera de Míster Barone competía con la estridencia de la estática que acompañaba siempre los mensajes. "Ahem. Es para mí un placer anunciarles que una de nuestras estudiantes de noveno grado, Esmeralda Santiago, ha sido aceptada al Performing Arts High School."
Cortada y complacida a la vez, no escuché el resto de lo que dijo. Míster Gatti me dio la mano. Andrea, la muchacha que se sentaba al lado mío me dio una palmada en el hombro. Alguien aplaudió y los demás compañeros se le unieron excepto, naturalmente, los muy _cool_. El resto del período lo pasé en las nubes, consciente de que algo muy bueno me había pasado por fin, temerosa de que fuera demasiado bueno y desapareciera antes de que terminara el día.
Llegué volando a casa, irrumpí en nuestro apartamento sombrío y encontré a Mami organizando unos papeles en su cama.
"Me aceptaron, Mami. Entré a Performing Arts." Me miró sorprendida. "La escuela especial, ¿te acuerdas? En Manhattan."
Se le agrandaron los ojos. "¡Ay, mi'ja, qué bueno!" dijo, halándome para abrazarme. Yo me le pegué. Los abrazos de Mami andaban escasos en esos días y quería quedarme en sus brazos, oler el perfume floral de su jabón, tan tenue que tuve que arrebujar la cara en su cuello para encontrado.
"¿Qué hizo Negi?" Aparenció Alicia y con ella Edna y Raymond. Como siempre, cuando alguno de nosotros recibía la atención de Mami, los demás se le arrimaban para ver cómo podían recibir su parte también.
Mami me sentó al otro lado de sus papeles. "A tu hermana la aceptaron en la escuela para artistas en Manhattan," les dijo y me sentí orgullosa porque sentí el orgullo en su voz.
"¿Tú, una artista?" preguntó Héctor desde el otro cuarto.
"Va a aprender a ser artista para llegar a ser rica y famosa algún día," sonrío Mami.
Entré en pánico. ¿Eso era? "Es sólo una escuela superior, Mami, para poder ir a la Universidad."
"¿Tú no dijiste que era para estudiar arte dramático y baile?" dijo molesta.
"Pues, sí..."
"¿Y vas a salir por televisión con Ricky Ricardo?" preguntó Raymond.
"No sé..."
"Esa es demasia'o fea pa' salir por televisión," desde su esquina, Héctor metió la cuchara.
Todos se rieron. Mami me abrazó y me dio un beso en la cabeza. "Voy a preparar la comida," dijo. En ese apartamento, nunca más se volvió a mencionar Performing Arts.
Una semana más tarde, Natalia faltó a la escuela. Como estuvo ausente varios días, fui a saber de ella. A pesar de que vivíamos a sólo dos o tres puertas de por medio, no nos visitábamos nunca y me sentía rara en aquel pasillo extraño, tocando a una puerta que no estaba siquiera segura que fuera la de ella. Nadie contestó, volví a tocar, acerqué el oído para ver si oía un televisor o un radio que impidiera que me escucharan. Todo estaba en silencio, pero la puerta del apartamento de al frente se entreabrió.
"¿Quién es?" preguntó en español una voz frágil y cuando me viré, un ojo y media cara arrugada se asomaron por debajo de la cadena de seguridad.
"Estoy buscando a Natalia Pons. Creo que vive aquí."
"Se mudaron."
"Imposible, si yo la vi hace poco y no me dijo na'."
"Se fueron, eso es to' lo que sé. Todavía no se ha muda'o nadie al apartamento, pero ya vendrá alguien." Cerró la puerta, puso varios seguros y se encuevó de nuevo en su apartamento.
No le creí. Natalia no me había dicho que se mudaba. Cuando le pregunté a Míster Barone por qué Natalia no estaba viniendo a la escuela, me dijo que su familia se había regresado a Puerto Rico.
_"But she never bean there,"_ le dije.
Alzó los hombros. "Su mamá está enferma."
Mami averiguó que la mamá de Natalia había tenido un accidente de trabajo y que el tío de Natalia había venido a buscarlas para llevárselas a Puerto Rico. No tenía ningún sentido, pero así se hacían las cosas en nuestro vecindario. La gente iba y venía sin avisar, sin despedidas. Mi propia familia se mudó cinco veces en un año y nunca hubo un adiós, ni siquiera una mirada atrás. Se suponía que cada mudanza fuera para mejorar y yo quería pensar que para Natalia, el regreso a Puerto Rico sería bueno. Pero yo sabía que el español de Natalia era en realidad espanglés, una mezcla de español e inglés que servía sus propósitos, pero que sólo entendían las personas que hablaban los dos idiomas. ¿Cómo le iría en Puerto Rico? ¿Podría estudiar medicina todavía? Si la aceptaban en el Bronx High School of Science, ¿iría?
Me dio pena con ella y conmigo. Mi mayor deseo, regresar a Puerto Rico, se hizo realidad para ella. Pero su sueño era el opuesto al mío. Quería quedarse en Nueva York, tener éxito al estilo americano, rodeada de todas esas cosas que pensábamos que nos traerían la felicidad: el apartamento en Park Avenue, el carro de lujo, la ropa, las cenas elegantes, las noches de teatro. Me encogí como hacía Mami, asustada de soñar; no, más bien, temerosa de decir mi sueño en voz alta porque, mira lo que le pasó al de Natalia.
La tienda de dulces que quedaba frente a la JHS 33 era de una pareja de viejitos. Vivían en la parte de atrás de la tienda, en un cuarto que quedaba detrás de una puerta holandesa que se dividía en dos, para que la esposa del dueño pudiera hablarle desde la mesa redonda donde se sentaba frente a un montón de retazos de tela que iba convirtiendo en alegres edredones. Las manos del señor estaban manchadas e hinchadas; sus dedos, redondos y lisos como salchichones. La muchachería decía que lo del viejito era contagioso, así es que nunca lo tocábamos cuando nos daba el cambio. Echaba las monedas en un envase de plástico que había encima del mostrador y yo las cogía, las dejaba caer en el bolsillo y me limpiaba las manos en la falda para quitarme los gérmenes.
En la acera, frente a la tiendita, había una especie de atril de metal para los periódicos. El viejito cobraba el dinero del periódico a través de una ventanilla al frente de la tienda. Por las mañanas se sentaba junto a esa ventana para velar a los muchachos que entraban a la escuela, y para estar pendiente de los títeres que eran locos con agarrar los periódicos y salir corriendo.
Cuando las gangas estaban revueltas, con frecuencia me metía en la tiendita, me entretenía hojeando una revista o me tomaba mi tiempo en comprar algún dulce, mientras velaba desde el mostrador para ver si los muchachos se habían ido. El hombre detrás del mostrador sabía que su tienda era un refugio para los que no éramos lo suficientemente fuertes ni valientes para enfrentarnos a los bravucones. Si alguno de nosotros se requedaba mucho en la tienda sin comprar nada, el viejito se asomaba por la ventana, por encima del atril de periódicos y miraba la acera de arriba a abajo. Con un áspero "¿qué estás esperando?" nos hacía una seña, nos ladraba el precio del artículo y se enfurecía cuando lo volvíamos a poner en su sitio porque no teníamos los chavos. "Fuera de aquí," nos gruñía, pero nosotros sabíamos que en realidad nos estaba avisando que no había moros en la costa.
Los insultos y las amenazas de Lulú se volvieron más frecuentes después de que Míster Barone anunció que yo había sido aceptada en Performing Arts. Ahora que Natalia no estaba, yo iba y venía sola, por eso salía de la escuela tan pronto sonaba el timbre, porque sabía que Lulú y su ganga eran demasiado _cool_ para salir corriendo como si alguien las estuviera persiguiendo. Una tarde cuando ya había cruzado la calle, tranquila porque una vez más había logrado evitarlas, salió Lulú de la puerta de uno de los edificios abandonados que quedaba en la cuadra cerca de la tiendita de dulces. Detrás de ella estaban Luz Mari y Denise. Me rodearon y me empujaron dentro de un frío y oscuro pasillo que apestaba a orines y a madera podrida. Me patearon, me dieron puños; con voces estridentes me lanzaron un coro de obscenidades, mientras los puños agudos y certeros me golpeaban el pecho, la barriga, la parte baja de la espalda. Yo me defendía con patadas, arañazos y puños como los que usaba cuando peleaba con mis hermanas y hermanos, sólo que más duros. Las muchachas me enterraban las uñas en los brazos, la cara, la nuca. Yo lanzaba golpes contra los seis puños que me golpeaban las costillas, las seis piernas que me pateaban las espinillas y la genitalia, las tres bocas dientúas que aullaban, chillaban y escupían, los seis ojos que relucían en la húmeda oscuridad con un feroz odio verde. Me defendí, pero estaba en desventaja y perdí. Quedé con la ropa destrozada y sucia, los brazos raspados, arañados, las piernas llenas de golpes, el pecho y la espalda que me latían del dolor. Mientras peleábamos, ellas me habían gritado en inglés y yo les contestaba en español, las malas palabras que no me permitían decir en casa, pero que ahora me fluían de la boca como el ácido.
Me dejaron tirada contra un montón de cartones húmedos, me gritaron lo que me imagino fueron más insultos, aunque no estaba muy segura. No lograba entender qué era lo que querían de mí, ni qué podía hacer para que dejaran de hacerme caso, como antes. No me requemé en el pasillo apestoso. Podía oír las alimañas corriendo por el fondo del edificio abandonado. Me limpié y busqué mis cosas. Cuando salí a la calle, el viejito de la tienda de dulces estaba en la acera. Me llamó para que entrara y me dio un Yoo-Hoo helado. De la parte de atrás apareció su esposa con un paño húmedo, murmurando en un idioma que no era ni inglés ni español y me limpió el sucio y las lágrimas de la cara, sus ojos fríos buscando alguna herida abierta en mis brazos o en mis cachetes.
"Esas muchachas," dijo el viejito y golpeó el mostrador con su mano hinchada. No me miró mientras su esposa me pasaba alcohol en los golpes, lo que hacía que los arañazos y los verdugones me ardieran y me dolieran más. Miraba por la ventana hacia la calle frente a la escuela, alicaído y con una expresión triste en el rostro.
"Vete a casa, cuéntale a mamá," dijo la señora, acompañándome hasta la salida de la tienda. Les di las gracias, traté de mirarlos a los ojos, pero no me miraron. Me hicieron un gesto con la mano, renuentes a aceptar mi agradecimiento. Me arrastré hasta casa, sentía cada paso como agujas en las costillas y las caderas. Mami estaba en el baño cuando llegué, así que me escurrí al cuarto de al frente, me cambié y me puse una ropa que me cubriera los golpes en los brazos y en las piernas, y me pasé el resto de la noche inclinada sobre un libro para que Mami no fuera a ver los arañazos en los cachetes y el labio hinchado. Después de comer, me di un largo baño caliente y ahogué mis sollozos con el chapaleteo del agua y cantando a todo pulmón, corridos mexicanos sobre amores traicioneros y la revolución. Si Mami notó algo, no dijo nada, ni tampoco mis hermanos o hermanas, cuyas propias luchas con los bravucones habían tenido resultados similares.
Durante el resto del año, evité volver a la tiendita de dulces, avergonzada sin saber por qué; la amabilidad de la dueña sin nombre, todavía un peso que no lo aliviaba ni siquiera el hecho de que Lulú no había vuelto a molestarme.
Un día llegué de la escuela y encontré a Mami en rolos. "¿Tienes muchas asignaciones?" me preguntó y me puso una tasa de café al frente.
"Tengo que estudiar para los exámenes finales."
"Hay que comprarte el traje de graduación."
Ya había perdido la esperanza de que alguien se diera cuentra de que en menos de un mes me graduaría de escuela intermedia. Mis quince años habían pasado sin pena ni gloria durante los tiempos tristes y, como pintaba la cosa, parecía que el último día de clases sería igual.
"¿Podemos ir?" preguntó Edna.
"No, ustedes se quedan con Tata. No nos vamos a tardar mucho." Antes de la muerte de Francisco, Edna y Raymond hubieran llorado, discutido y hubieran prometido ser los mejores nenes del mundo si Mami aceptaba llevarlos. Pero ahora, sólo pusieron carita triste.
"Me voy a cambiar." Corrí hasta el cuarto de al frente donde estaban las camas literas, la cuna de Franky y el catre de Tata, colocados en fila. Las ventanas que daban a la calle estaban abiertas y Delsa, Norma y Alicia estaban en la acera brincando cuica.
Tata estaba recostada en su cama arrullando a Franky y cuando entré, me miró y sonrió. Agarré un vestido de uno de los ganchos que Mami había atornillado en la pared porque el apartamento no tenía closets. Con dos toallas pilladas con la litera de arriba, creé un espacio privado en el que me cambié el uniforme y me puse el vestido de algodón.
Mami estaba en su cuarto que servía también de pasadizo entre el cuarto de al frente y la cocina. Su cama estaba pegada a la pared, contra un rincón, debajo de una ventana que daba a un oscuro respiradero. Cuatro gaveteros que no hacían juego con nada y tenían una gaveta para cada uno de nosotros y dos para Mami, estaban alineados contra la pared. Mami estaba de pie frente al que tenía un espejo, peinándose los rizos.
"Volvemos en un par de horas," le dijo a Tata cuando salimos. Edna y Raymond se quedaron medio tristones.
"Tráenos dulces," pidió Raymond, cuando Mami fue a cerrar la puerta.
Delsa, Norma y Alicia soltaron la cuica cuando nos vieron salir. Antes de que pudieran preguntar para dónde íbamos, Mami examinó la calle.
"¿Dónde está tu hermano?"
"Fue a la esquina," contestó Alicia.
"¿A qué esquina? ¿Quién le dijo a él que se podía ir a andareguear así?"
"Héctor siempre hace eso, Mami. Se va cuando le da la gana..." Norma le dio un codazo a Delsa para que no siguiera. "Él vuelve enseguida," continuo Delsa en voz baja. "No es pa' tanto."
"No se queden afuera mucho rato," les advirtió Mami y tomó el camino hacia Broadway.
"¿Adónde van?" preguntó Alicia.
"A comprar mi traje de graduación," les grité, feliz de ver las caras de envidia de mis hermanas. Corrí detrás de Mami, que con paso decidido, había llegado ya hasta la esquina.
Estábamos a principios de mes, que era cuando llegaban por correo los cheques del _welfare_ y del seguro social. Broadway estaba lleno de gente comprando, saliendo y entrando de las tiendas o esperando en las paradas de guaguas, cargada de bolsas repletas de cosas. Por encima, el tren elevado pasaba traqueteando a cada rato y frenaba con un chirrido en la estación de Flushing Avenue. Las vigas que sostenían las vías del tren dividían la calle en cuatro carriles: los dos del centro, en los que el tránsito se movía en ambas direcciones y los dos de los extremos que se usaban para el tráfico local y que siempre estaban congestionados con carros que se parqueaban en doble fila, con guaguas lentas y camiones de entrega.
Seguí a Mami hasta la oficina donde cambiaban cheques, una tienda con un enorme letrero sobre la puerta y un montón de hombres vegetando al frente. Para esta época del mes estaban siempre allí, esperando a sus mujeres para que les entregaran el dinero del cheque que habían cambiado. Uno besó y abrazó a la mujer cuando ella le entregó el dinero. Otro, ni la miró; se metió los chavos en el bolsillo y no le dio ni las gracias. Un tercero, empezó a pelear con la mujer tan pronto ella salió. Ella le decía que necesitaba el dinero para alimentar a los muchachos y pagar la renta y la electricidad. Pero, él se lo arrebató, lo contó y siguió andando y la dejó allí, llorando y maldiciendo en voz baja, mientras la gente que pasaba, abría un amplio círculo alrededor de ella para no acercársele demasiado.
Adentro había dos filas largas frente a dos hombres parados detrás de un cristal grueso. Los cajeros vestían camisas blancas, pantalones negros, tirantes y un casquete. Tenían un tirabuzon a cada lado de la cara, como los vendedores de la marketa y las tiendas de muebles usados de la Avenida Graham.
Nos paramos en fila detrás de una señora flaquita que tenía una lucha con una nena. La nena gritaba y pataleteaba y arañaba la mano que la tenía fuertemente agarrada por la muñeca. La gente que estaba esperando las miraba y se retiraba un poco de ellas, pero sin dejar su sitio en la fila. La mujer le gritaba a la nena que basta, basta, basta y la halaba por la mano y la abofeteaba, lo que la hacía llorar aún más y luchar con más fuerza. La mujer levantó la vista, como retándonos a decirle algo y todos miramos para otro lado. En su jaula, los cajeros eran los únicos que se atrevían a mirarla, con un desprecio, dirigido a ella, a la niña, a todos nosotros.
Cuando nos llegó el turno, Mami sacó un bolígrafo de su cartera y firmó el cheque del _welfare_ frente al cajero. No miró al hombre, aislado detrás de la lámina de cristal, ni él la miró a ella. La transacción se hizo en silencio; la atmósfera estaba cargada con la vergüenza de ella y con el desdén de él hacia gente como nosotras: mujeres de piel oscura en _welfare_.
Antes de salir, Mami metió el dinero en su monedero y lo guardó con cuidado en la cartera que llevaba bien pillada con el brazo. Los hombres nos miraron a la expectativa, y nos dieron la espalda molestos cuando se dieron cuenta de que no éramos las mujeres que ellos esperaban.
"¿A cuál tienda vamos?" le pregunté a Mami que me iba mostrando el camino.
"A aquélla." Miró hacia Dolores's Ladies Shoppé, que estaba al cruzar la avenida y donde, de regreso de la escuela unos días antes, había visto en la vitrina algo perfecto para mí; un traje amarillo, sin mangas, de falda amplia y una banda ancha en la cintura.
"¿Mi traje tiene que ser negro o puede ser de cualquier color?"
Me miró extrañada al cruzar la calle y no me contestó hasta que llegamos al otro lado. "Puedes usar el color que tú quieras."
Mi suspiro de alivio la hizo sonreír y me puso una mano en el hombro cuando entramos a Dolores's Ladies Shoppé, donde esperaba mi vestido, amarillo como la cáscara de limón; el corpiño y la falda de encaje, la banda de organza formando un lazo en la espalda.
"Te hace ver amarillenta," me dijo Mami cuando me lo probé.
Me miré en el espejo de cuerpo entero, vi el reflejo dorado en mis brazos y en mis piernas morenas y la luz que el traje reflejaba en mi cara. "Yo creo que se me ve bien."
"A lo mejor le gustaría más este azul celeste," buscó Dolores entre las bolsas de plástico transparente que cubrían cada pieza de ropa que colgaba en las paredes de su tienda.
"A ella no le gusta ese color," dijo Mami, mientras buscaba con Dolores entre las bolsas plásticas.
Entrecerré los ojos para lograr una perspectiva diferente en el espejo, traté de verme como me vería un extraño y vi a una jovencita, con el pelo castaño peinado en _flip_ , los ojos oscuros, maquillados con sombra azul y bordeados con una línea negra que terminaba con un rabito en la esquina. En la boca, un beso rosado tan pálido que los labios se me veían casi blancos. En los pies, altos tacones puntiagudos. Me parecía a una de las Chiffons, las cantantes de "He's So Fine." Cuando volví a abrir los ojos, me vi como era de verdad, con mi melena recogida en un rabo de caballo suelto, sin maquillaje, con mocasines y medias a la rodilla.
"Aquí hay uno," dijo Mami. "Éste es más tu color." Me mostró un vestido azul marino, con cuello cuadrado, mangas tres cuarto y talle a la cadera. Era como los vestidos que siempre me compraba, sencillo, modesto, no como los trajes atrevidos que usaban las americanas.
Me miré en el espejo de nuevo. "Éste es el que me gusta." Sentí que nos estábamos preparando para una discusión. "Es mi graduación, tengo que ponerme algo bien lindo." Le di la espalda.
Mami se puso tensa pero no iba a formar un lío delante de Dolores, que se había quedado cerca de nosotras sosteniendo en su mano dos vestidos tan aburridos y conservadores como el que me había enseñado Mami. El traje amarillo era luminoso, me hacía sentir linda y especial.
"Tú me dijiste que podía usar cualquier color," le recordé a Mami, cuyo traje negro y suelto le colgaba de sus hombros sin adorno, apenas tocando su busto y sus caderas sin acentuar su redondez. La ropa negra, el vientre todavía hinchado después del parto y las piernas marcadas de venas varicosas, la hacían lucir sólida, pesada, pegada a la tierra.
El escote de mi vestido amarillo me llegaba un poco más arriba de mis coquetitas recién estrenadas. La banda apretaba una cintura pequeña y la falda ancha, más amplia aún por la enagua can-can parecía elevarme del piso, de la alfombra sucia y áspera frente al espejo estrecho de Dolores's Ladies Shoppé. Parada una al lado de la otra, Mami y yo parecíamos la noche más oscura junto a la mañana más luminosa, cada una resuelta a hacer su voluntad, sabiendo que una tendría que ceder frente a la otra, esperando hasta el último momento posible de incertidumbre antes de rendirse.
"Está bien, llévate el vestido amarillo," suspiró con voz quebrada, exhausta, triste.
"Yo no sé qué te ha pasa'o ti," murmuró Mami de regreso a la Calle Ellery, "has cambiado."
Abracé el plástico con mi vestido amarillo. "Me estoy poniendo grande, Mami." Me reí para que no me fuera a acusar de malcriada.
"Grande, sí," continuó implacable, "y terca e irrespetuosa." Me miró con el rabo del ojo. "No te creas que porque vas pa' esa escuela de blanquitos te voy a estar aguantando pocavergüenzas." Dobló en la esquina y yo seguí detrás, indolente, atrapada entre mis pensamientos.
Cuando Mami y yo íbamos a la oficina del _welfare_ o del Desempleo, uno de los blancos de los formularios que teníamos que llenar pedía que indicáramos nuestra raza: Blanca, Negra, Otra. Técnicamente, Mami era blanca. Su piel era de un blanco cremoso, sin los cálidos tonos marrón que sus hijos con Papi habíamos heredado. Mi recuerdo de mis abuelos paternos era que eran blancos, pero Papi y algunos de sus hermanos y hermanas eran de un marrón oscuro que evocaba un ancestro africano no tan lejano. Franky, el hijo de Mami y Francisco, era de tez más clara que el resto de los siete hermanos y hermanas. Tenía piel blanquita y el pelo y los ojos oscuros del padre.
Cuando me tocaba indicar mi raza, siempre marcaba "otra" porque ni negra ni blanca eran apropiadas. Pretender ser blanca cuando claramente no lo era hubiera estado mal. Si hubiera podido "pasar", que no era el caso, me hubiera asaltado la pregunta que hacían los puertorriqueños cuando alguien se las echaba demasiado por su piel blanca: "¿Y tu abuela dónde está?" Preguntar por la abuela implicaba que en Puerto Rico, nadie tenía nunca su cuadro racial completo y los reclamos de pureza de raza eran medio sospechosos.
Yo no estaba ajena al asunto racial en Puerto Rico. Había notado que la gente que no tenía la piel blanca, la evidiaba y los que la tenían, despreciaban a quienes no la tenían. Los bebés de tez blanca eran más mimados por la familia que los más oscuritos. El pelo "bueno" era el lacio, no el "kinky," y era más deseable que el pelo "malo" que en su forma más extrema se llamaba pasitas, pasurín o sereta. Los ojos azules o verdes proclamaban blancura, aunque estuvieran rodeados de piel oscura.
Yo no era ni negra ni blanca; era trigueña, color del trigo. Tenía el pelo bueno y mis facciones no eran ni africanas ni europeas, sino una combinación de ambas. Estando en la escuela en Puerto Rico no había sobresalido por el color de mi piel, ni por mis facciones. Nunca fui ni la más blanca ni la más oscura de mi salón. Pero, cuando vivíamos en la ciudad me relajaban por ser una jíbara del campo. En el campo, mi experiencia en la ciudad, levantaba sospechas.
En las Escuelas Intermedias 49 y 33 de Brooklyn yo era una puertorriqueña recién llegada a una escuela en que la mayoría de los estudiantes eran puertorriqueños, italianos y negros. Me distinguía, como los demás recién llegados, por mi afán de hablar inglés. Los pocos americanos que había en nuestra escuela, todos blancos, vivían y se movían en sus propios vecindarios y grupos, cerrados al resto de nosotros.
Cuando Mami me acusó de querer ir a una escuela para "blanquitos", intuyó que la mayoría de la gente en Performing Arts sería blanca y por lo tanto, más rica que nosotros. En Puerto Rico, al igual que en los Estados Unidos, ser blanco significaba privilegio económico y cuando Mami hablaba de "los blanquitos," se refería a la gente de un alto nivel social, más que a un color de piel. La implicación de que al asistir a Performing Arts estaba aspirando a más de lo que me correspondía me dolió, pero yo no estaba en las de defenderme de Mami. Cualquier respuesta al juicio que tenía formado sobre mí y sobre lo que quería hacer con mi vida le hubiera confirmado sus sospechas de que, desde que llegamos a los Estados Unidos, yo había cambiado. Me había vuelto, demasiado independiente, insistía; demasiado aferrada a mis propias ideas, demasiado exigente. Toda la atención que había recibido cuando solicité a Performing Arts, se me había subido a la cabeza. Me había vuelto ambiciosa, difícil de complacer, siempre queriendo más de lo que tenía o me merecía.
Tenía razón, había cambiado. Algunas noches, acostada al lado de mi hermana, me preguntaba si ella también estaría cambiando. Si la Delsa de Brooklyn sería diferente a la Delsa de Puerto Rico. Más allá de la fluidez que iba logrando en inglés, Delsa seguía siendo la misma muchacha nerviosa, responsable y trabajadora de siempre. Ella no iba a solicitar a Performing Arts en Manhattan. Ella iba para Eli Whitney a estudiar enfermería, una profesión de verdad, que le daría un buen salario y un trabajo permanente. Pensándolo bien, ninguna de mis hermanas ni hermanos parecía sentir la insatisfacción con la vida que sentía yo.
Yo quería un vida diferente a la que tenía. Quería mi propia cama, en mi propio cuarto. Quería poder bañarme sin tener que botar de la cocina a la familia entera. Quería libros sin fechas de devolución. Quería ropa bonita, escogida por mí. Quería maquillarme, arreglarme el pelo, caminar en tacos altos. Quería mi propio radio para poder escuchar a La Lupe en la estación hispana, o los 40 Éxitos de Cousin Brucie en la americana. Quería poder comprarme una Pepsi o un Baby Ruth siempre que se me antojara. En Puerto Rico no había deseado ninguna de esas cosas. En Puerto Rico no sabía siquiera que estaban a mi alcance. Pero en Brooklyn, cada día estaba lleno de deseos, a pesar de que Mami estaba siempre pendiente de que tuviéramos todo lo que necesitábamos. Sí, había cambiado. Y no para mejorar. Cada vez que Mami decía que yo había cambiado, era porque había hecho algo malo. La había desafiado, o le había faltado el respeto, o ya no me gustaba lo que me gustaba antes. Cuando ella me decía que había cambiado, lo que quería decir era que me estaba americanizado, que me creía que me merecía más, que era mejor que nadie, que era mejor que ella. Me miraba con resentimiento, como si la hubiera traicionado, como si yo hubiera podido evitar convertirme en quien me estaba convirtiendo, como si yo hubiera sabido.
# "¿Qué es un traje de Cleopatra?"
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En el verano de 1963, nos mudamos de nuevo, esta vez a un apartamento que quedaba en los altos de una farmacia, en el tercer piso de un edificio de la congestionada Pitkin Avenue. Delsa y yo compartíamos un cuarto que daba hacía la calle y hacia un Woolworth's y una tienda Thom McAnn que quedaban al cruzar.
A diferencia de otros lugares donde habíamos vivido en Brooklyn, en Pitkin Avenue no había niños que salieran a jugar después de clases. Era un bloque comercial atiborrado de tiendas, unas encima de las otras, que tenían las vitrinas emplastadas con anuncios de ventas especiales y adornadas con las mismas decoraciones alusivas a diferentes festividades y que, año tras año, eran exhibidas por los dueños, que observaban con desconfianza y resentimiento a sus clientes puertorriqueños y negros. Una vez cerraban las tiendas, la calle se dormía, el tránsito se hacía más liviano, y las guaguas que subían y bajaban por la Pitkin y Rockaway Boulevard, transitaban calmosas como para conservar su energía para los días frenéticos.
El trabajador social del _welfare_ que atendía nuestro caso le dijo a Mami que ella cualificaba para recibir beneficios como sobreviviente. Debido a que Mami y Francisco no habían estado casados legalmente, hubo que arreglar un montón de papelería que me tocó a mí interpretar y llenar. Ya había mejorado bastante en contar la historia de Mami, en transmitir su frustración por estar en _leyof_ cuando ella lo que quería en realidad era trabajar; pero todavía era un reto mantenerme tranquila para que el inglés no me abandonara justo en el momento en que tuviera que hablar. Tras muchas visitas y entrevistas, nos aprobaron la reclamación. Sin embargo, una vez fue confirmada, el _welfare_ le redujo a Mami su pago de AFND, así es que el seguro social no nos ayudó demasiado.
Después de semanas de estar buscando, Mami encontró trabajo en Manhattan. La tristeza no se le quitó cuando empezó a trabajar. Su dolor era como una caja transparente que le permitía coser brasieres en la fábrica, hablar con nosotros, cocinar, ir de compras; pero que la mantenía contenida en sí misma, intocable. Por las mañanas me despertaban sus movimientos silenciosos por el apartamento, mientras se preparaba para irse a trabajar. Se levantaba temprano, se daba una ducha y se ponía un saquito negro sencillo o una falda y una blusa negra. Se cepillaba el pelo, se lo recogía en un moño tirante, y se empolvaba la nariz y la frente. Nunca desayunaba, ni siquiera una taza de café. Bajaba en puntillas por la escalera de madera que crujía a pesar de su cuidado.
Saqué la cabeza por la ventana. Las aceras estaban vacías, la oscuridad apenas rota por los aros de luz que formaban las lámparas de la calle. Mami miró para la izquierda y para la derecha antes de tirarse. Enderezó la espalda, alzó la barbilla, agarró bien la cartera y caminó hasta la esquina donde dobló a la derecha hacia la estación del tren. Su figura indefinida se abrió paso a través de la oscuridad, sin mirar para atrás ni para el lado; su mirada fija en algún punto frente a ella. Se veía tan triste y sola que me asustó que fuera a desaparecer en la ciudad y no regresara jamás. Al doblar la esquina, sus pisadas se esfumaron entre los ruidos de Brooklyn. Traté de tranquilizar el miedo que me hacía latir la cabeza con mil pensamientos aterradores. Ella nos recordaba constantemente todas las cosas que nos podían pasar. Pero, ¿y si "algo" le pasara a ella? ¿Temía ella tanto por sí misma como por nosotros?
Sobre un horizonte serrado el sol aguijoneaba unos tenues jirones de nubes que se tornaban rosados para después derretirse en amarillo. Un rugido suave acompañaba al alba, un gruñido bajo que se volvía más fuerte según se despertaba la ciudad. En pocos minutos, la gente andaba apurada, subiendo y bajando por la calle, cruzando la avenida, entrando y saliendo de las tiendas, sus pasos en _staccato_ atenuados por los primeros bocinazos, las sirenas distantes y el ruido amortiguado de los radios.
Las clases no empezarían hasta dentro de algunas semanas. Los días se estiraban largos y húmedos; el siguiente, igual que el anterior, excepto los fines de semana cuando Mami estaba en casa y hacíamos diligencias o visitábamos a la familia.
La actividad cumbre de la semana era hacer la compra del sábado. Cuando estábamos cogiendo _welfare_ , la compra se hacía en menos de una hora y se arrastraba a casa en un carrito lleno de los alimentos básicos de nuestra dieta: grandes fardos de arroz, habichuelas, latas de salsa de tomate, cebolla, ajo, pimientos verdes, orégano fresco y recao para el sofrito. Mami compraba también, un par de latas de Bustelo, el único café al estilo puertorriqueño que se conseguía en Nueva York, aunque no era tan dulzón como el que se conseguía en la Isla, un paquete de azúcar de cinco libras y leche en polvo para cuando no hubiera chavos para comprarla fresca.
Pero, cuando Mami estaba trabajando, mis hermanos y yo nos pelábamos por ayudarle a hacer la compra, porque entonces habría Corn Flakes de Kellogs y leche fresca, espagueti Franco-American, _ravioli_ Chef Boyardee y otras latas de comida americana. Cuando Mami tenía trabajo, había Quick de Nestlé, pasta de guayaba con queso blanco del país, chuletas, salchichón con galletitas Ritz, Cheez Whiz con Export Sodas, carne guisada con pedazos de calabaza y yautía, y a lo mejor, un pernil. Mami estaba orgullosa de que aun en las malas rachas, nunca pasábamos hambre. "En esta casa siempre hay pan y leche," decía, "y hay siempre una taza de arroz y un puñado de habichuelas."
Pero nosotros no queríamos arroz y habichuelas, pan y leche. Queríamos Ring Dings, Yodels, _pizza_ , Coca-Cola, Frosted Flakes, Jell-O, alimentos que nunca tuvimos en Puerto Rico y que sólo conseguíamos en Brooklyn cuando había suficiente dinero o cuando nuestros parientes, cuando venían de visita, nos regalaban algún menudo por habernos portado bien. Cuando estábamos en _welfare_ , hablábamos de lo que compraríamos cuando fuéramos grandes y tuviéramos trabajo, y pudiéramos gastar el dinero como nos diera gusto y gana.
"Yo voy a comprar la fábrica donde hacen los Sno-Balls," decía Alicia y nosotros nos lamíamos los labios, anticipando el sabor dulce del chocolate y el coco rayado de los bizcochitos rellenos de crema que parecían tetitas nevadas envueltas en celofán.
"Yo voy a abrir una dulcería para poder comer Baby Ruths y Almond Joys todas las veces que me dé la gana," replicaba Raymond y nosotros le decíamos que sí, que una tienda con dulces variados era mucho mejor que una fábrica de una sola clase.
Cuando Mami estaba trabajando y nosotros la ayudábamos con la compra, zigzagueábamos de arriba a abajo por los pasillos del colmado, buscando un dulce nuevo y sabroso para convencerla de que nos lo comprara. En casa saboreábamos cada bocado, nos chupábamos los dedos para aprovechar el último chispito de dulce que nos quedara en la punta, escurríamos la botella de refresco, hasta que no le quedaba ni una gota del líquido burbujeante y espumoso hasta que el vidrio liso y duro nos agarraba con fuerza la lengua.
Ahora que Tata estaba viviendo con nosotros otra vez, Tío Chico se buscó un cuarto por la Bowery. Habíamos oído decir que allí era que vivían los _"bones,"_ pero Mami insistía en que Tío Chico no era un _"bon."_ "A veces bebe demasiado," decía, "pero trabaja y se cuida a sí mismo."
No, Tío Chico no olía como los borrachos que nos encontrábamos por las calles laterales que daban a la Avenida Pitkin. Estaba limpio, aunque tuviera la ropa estrujada, deshilachados los cuellos de la camisa y gastadas las suelas de los zapatos. Se afeitaba un día sí y un día no. Cuando no lo hacía, unos tuquitos negros y blancos le crecían por entre los pliegues profundos que le salían de la nariz hasta la comisura de los labios. Tenía los ojos marrón, como los de Tata, y una nariz bien formada, larga pero no grotesca; bien delineada. Tenía también unas manos de dedos largos, hermosas y elegantes.
Una vez me tocó el seno izquierdo con esos dedos largos, me agarró el pezón y me lo pinchó. Había estado observándome mientras me peinaba y cuando Tata lo llamó para que fuera a la cocina, yo no me moví cuando me pasó por el lado y él estiró la mano y me apretó el seno. "No se lo digas a nadie," me murmuró al oído. Cuando regresó, dejó caer un dólar frente a mí.
Pude haberle dicho a Mami lo que había hecho y pude haber usado el dólar como evidencia, pero no lo hice. Lo gasté en un _sundae_ y me dije que era que él estaba borracho. Desde ese momento, lo evitaba siempre que se aparecía por casa; me iba a otro cuarto, me escondía en el baño, o me sentaba lo más lejos posible de él cuando venía de visita. Sus ojos caramelos y rojovenosos me seguían según yo caminaba por el apartamento. Yo evitaba su mirada, consciente de que compartíamos un secreto vergonzoso, tratando de decidir si la culpa mayor era de él, que me tocó, o mía que se lo permití.
Ahora que Mami estaba trabajando de nuevo, mandó a instalar un teléfono. "Contigo yendo para la ciudad todos los días," razonaba, "necesitamos un teléfono por si te pierdes o algo."
Por las noches nos sentábamos alrededor de la mesa de la cocina a discutir los "algos" que podían pasar. Los informes de los crímenes que se cometían en la ciudad aparecían en los periódicos, ilustrados con fotografías, en un blanco y negro veteado, que electrificaban. Nosotros dramatizábamos los eventos más llamativos del día y les añadíamos detalles que no habían sido reportados, pero que nosotros estábamos seguros que habían ocurrido. El día que un sospechoso de traficar drogas fue encontrado colgado en su celda, Héctor se quitó la correa, se la enlazó sueltecita por el cuello, la haló hacia arriba, sacó la lengua, se puso bizco, tosió, carraspeó y tembló con unos espasmo que nos hicieron reír hasta que se nos salieron las lágrimas. Cuando nos divertíamos demasiado a costa de los muertos, mutilados o victimizados, Mami ponía fin a nuestras parodias. "Ay, bendito, la mamá de ese pobre hombre," suspiraba o "cómo habrá sufrido esa pobre mujer antes de que la mataran." Sus comentarios nos avergonzaban por un ratito, pero no impedían que al día siguiente volviéramos a hacer lo mismo.
Cuando el crimen nos tocó de cerca, cuando asaltaron a Don Julio o cuando a nuestra vecina Minga la empujaron hacia el tráfico de la avenida y le arrebataron la cartera, no nos reímos. Nos acurrucamos alrededor de Mami en un miedo mudo, visualizando los peligros que acechaban detrás de la puerta de la calle, convencidos de que el único lugar seguro en el mundo estaba entre las cuatro paredes que nos cobijaban, tan pequeños y vulnerables, a la sombra de nuestra madre.
Performing Arts High School estaba organizada por departamentos: Baile, Drama y Música. Se podía identificar el área de especialidad de los estudiantes con sólo mirarlos. Los bailarines tenían las pantorrillas musculosas y apenas tocaban el piso al andar; sus pies se abrían desde la cadera, como las manecillas de un reloj marcando las 8:20. Los músicos cargaban con sus estuches negros de formas variadas, tamborileaban con los dedos durante las clases y escuchaban con muchísimo interés cualquier cháchara boba. Los estudiantes de Drama eran los que peor escuchaban, pero los que mejor hablaban. Cuando hablaba con otros estudiantes de Drama, me daba la impresión de que, durante sus breves momentos de silencio, sólo estaban haciendo turno para volverse a escuchar de nuevo.
Nos asignaron salones hogares, cada uno dividido, más o menos equitativamente, entre estudiantes de Música, de Baile y de Drama. Nuestro día se dividía entre las clases de concentración y las materias académicas. Teníamos que mantener un promedio alto en ambas o nos harían transferir. Misis Schein, mi maestra de salón hogar, nos felicitó por haber superado con éxito un proceso que ella catalogó de altamente competitivo. "Ustedes demuestran tener potencial artístico al igual que académico. Al admitirlos a Performing Arts High School estamos afirmando nuestra fe en ustedes como artistas y como intelectuales."
Me complacieron e inspiraron sus palabras, que pude entender gracias a que hablaba en una voz profunda y bien modulada, enunciando claramente cada palabra.
"Tenemos un código de vestimenta," nos informó, "los varones no podrán usar mahones para venir a clases y las muchachas no podrán usar pantalones."
"¿Y si es un día bien frío?" preguntó una muchacha, con el pelo como un nido de ratones y más maquillada que la maestra.
"Pueden usar pantalones debajo de la falda, pero en la escuela se los tienen que quitar y usar un vestido o una falda." Por lo bajo, se oyeron algunas voces de protesta, que se apagaron cuando Misis Schein continuó, "No se puede salir de la escuela en maquillaje de teatro. Es poco profesional."
El profesionalismo era un concepto importante en Performing Arts. La facultad se componía mayormente de actores, bailarines y músicos, activos en sus carreras, que se tomaban muy en serio como artistas, y esperaban que los estudiantes hiciéramos lo mismo. "Ustedes tienen un don," nos dijo cada uno de ellos en su momento, "y nuestro trabajo es ayudarles a desarrollar sus talentos, pero es también nuestra responsabilidad prepararlos para el mundo real."
Ninguno de nosotros, subrayaban, debía esperar convertirse en un éxito de la noche a la mañana. Después de salir de la escuela superior, nos tomaría un promedio de diez años desarrollar al máximo nuestros talentos y ser reconocidos en "el ambiente," antes de poder vivir del arte.
Los diez años de espera me deprimieron. ¿Cómo decirle a Mami que me esperaban tres años de escuela superior y diez años de lucha antes de que pudiera mantenerme yo misma? Había pensado que tan pronto me graduara de Performing Arts, conseguiría un trabajo de actriz y ganaría lo suficiente para ayudarla, pero según la maestra, graduarse de Performing Arts era sólo el principio. "Los únicos que llegan," no nos permitían olvidarlo nunca, "son aquéllos que están comprometidos con su arte, los que están dispuestos a sacrificarse por el privilegio de actuar frente a un público. Vayan haciéndose la idea de que serán artistas muertos de hambre durante un tiempo, antes de ser descubiertos."
Cuando le conté lo que nos habían dicho los maestros, Mami se horrorizó. "Yo no me estoy sacrificando tanto para mandarte a una buena escuela para que después te mueras de hambre," me advirtió. Las dos imaginábamos legiones de actores, músicos y bailarines llenando los formularios del _welfare_ como tantas veces habíamos hecho nosotras.
"A mi no me importan las monerías que te enseñen en esa escuela," aclaraba Mami, "tan pronto te gradúes, te me vas a trabajar."
A los estudiantes de Drama se nos requería estudiar baile, para que desarrolláramos un sentido de cómo se movía nuestro cuerpo en el espacio, y nos preparáramos, por si —a pesar de nuestras aspiraciones dramáticas— conseguíamos trabajo en un musical. A pesar de que la escuela tenía estudios de baile espaciosos, de techos altos, pisos de madera, espejos y buena iluminación, éstos estaban reservados para los estudiantes de concentración. Los actores bailaban en el comedor. Los bancos se acomodaban sobre los topes de las mesas, y las mesas se empujaban hacia una esquina del salón, para dejar libre el piso de losetas. Si bailábamos después de almuerzo, a veces teníamos que barrer las migajas del piso.
Era una suerte que el comedor no tuviera espejos, porque la mayoría de nosotros no estaba acostumbrada a usar la ropa de baile requerida para la clase. Las muchachas usaban unas _tights_ negras sin pie, un leotardo negro de cuello tipo sobre y una falda de baile hasta mitad de muslo. Bailábamos descalzas, como los varones, que vestían _tights_ negros y una camiseta blanca. El primer día entramos al salón tratando de taparnos, los muchachos con las manos cruzadas sobre el bulto aumentado por la requerida correa de baile, y las muchachas encorvadas, abrazándonos los senos.
Miss Lang, nuestra maestra de baile, nos guió por lo que para muchos de nosotros era nuestra primera clase formal de baile. Desgarbados y sin coordinación alguna, nos reíamos mientras ella nos demostraba cómo saltar a través del salón, los pies en punta, la cabeza erecta, la espalda recta. "Pie derecho afuera, brazo derecho arriba," cantaba, mientras marcaba con su tamborcillo de mano el ritmo que la mayoría de nosotros desafiaba con torpes saltos y vueltas. En la primera clase quedó claro que teníamos que desarrollar unos músculos que ni siquiera sabíamos que existían, antes de poder ejecutar los saltos y las piruetas gráciles que nos dejaban espatarrados en el piso. La semana siguiente y durante muchas semanas más, la clase de Miss Lang se dio en el piso, donde ella nos dirigía a través de unos ejercicios de estiramiento rigurosos que nos dejaban sudados y adoloridos. Los otros estudiantes se quejaban de que eran actores y no tenían porqué tomar esa clase tan estúpida, pero a mí me encantaba. Me encantaba el espacio abierto frente a mí en el estudio-comedor. Me encantaba la sensación de ingravidez cuando daba saltos a través del salón, recibía con beneplácito los dolores después de clases, los músculos estirados que después vibraban por horas, el golpe de sangre que me llegaba a la cara, a los brazos y a las piernas. Era cuando sentía este calor único, la única vez durante los inviernos de Brooklyn en que mi cuerpo se movía como yo lo recordaba moviéndose en Puerto Rico —libre, abierto a posibilidades, sin miedo.
La mayor parte de mis compañeros de clase eran nacidos y criados en Nueva York y hablaban con los acentos distintivos de los vecindarios donde habían crecido. Nuestros maestros aseguraban que con sólo oírnos hablar podían reconocer de qué distrito veníamos. En Brooklyn, por ejemplo, _"I am"_ sonaba a _"Oyem," "here"_ sonaba a _"hiah," "bathroom"_ era _"batrum"_ y _"in there"_ era _"inner."_ Yo hablaba inglés de Brooklyn con acento puertorriqueño, una variante en un lugar cuya meta era que habláramos el inglés estándar del este de los Estados Unidos.
Erradicar los acentos, nos decían, era importante para abrir el abanico de posibilidades de los papeles que pudiéramos representar. Un actor tenía que ser lo suficientemente polifacético para poder ajustar su manera de hablar a las necesidades del personaje que estuviera representando. El habla estándar sentaba las bases para otros acentos, incluyendo, si fuera necesario, el que traíamos cuando entramos por primera vez por las puertas de Performing Arts.
Mi maestro de Voz y Dicción era King Wehrle, de Kansas.
"Necesitan un nombre que llame la atención," nos dijo cuando le preguntamos si él había nacido Rey, "yo me cambié el mío cuando vine a Nueva York."
Nos dio una lista de los actores famosos que habían cambiado unos nombres comunes y corrientes por nombres que todo el mundo recordaba: Archibald Leish/Cary Grant; Eunice Quedens/Eve Arden; Betty Joane Perske/Lauren Bacall; Frances Gumm/Judy Garland.
"¿Ustedes creen que a un tipo con un nombre como Marion Morrison le iban a dar un papel de vaquero en el cine?" preguntó Mister Wehrle. "No. ¡Se tuvo que convertir en John Wayne!"
Mister Wehrle nos sugería que cuando fuéramos a hacer el cambio, seleccionáramos nombres con pocas letras, fácil de acomodar en la marquesina del teatro, fácil de recordar y norteamericano, no extranjero. "Ann Bancroft," decía, "no Ana María Italiano, Tony Curtis, no Bernard Schwartz. Kirk Douglas," entonaba en su distinguida voz de locutor, "no Issur Danielovich."
Así es que, además de tener que esperar diez años después de graduarme para empezar a ganarme la vida con mi arte, tenía también que buscarme un nombre nuevo porque Esmeralda Santiago era a todas luces demasiado largo para la marquesina del teatro, difícil de recordar, y definitivamente, extranjero.
Si analizaba a Performing Arts con criterios estrictamente raciales, Mami tenía razón; era una escuela donde casi toda la facultad y el estudiantado eran blancos. En mi clase de décimo grado había 126 estudiantes: catorce negros, tres puertorriqueños y dos asiáticos. Dos de los veinticuatro maestros en los cursos de especialidad, y dos de los veintitrés de los cursos académicos, eran negros.
Pero cuando caminaba por los anchos pasillos de Performing Arts High School, lo que veía no era una escuela para blanquitos. A pesar de que las pieles oscuras estábamos en minoría, las jerarquías basadas en criterios raciales no eran tan marcadas como lo eran cuando estaba en escuela intermedia. En Performing Arts, el estatus lo determinaba el talento. La elite de la escuela consistía de los estudiantes que lograban el papel protagónico de alguna escena o que tocaban un solo instrumental en un concierto de cámara o que bailaban un solo en un virtuoso _pas de deux_. Los demás, cuyos talentos estaban por desarrollarse, mirábamos a las estrellas de la escuela con una mezcla de admiración y envidia. Ellos sí que no tendrían que esperar diez años para "llegar" y "triunfar" en el ambiente.
Reconocía y podía aceptar la jerarquía basada en el talento. A diferencia de la que se establecía por razones raciales, era justa. Pero, entre el estudiantado se daba otra diferencia —más sutil, pero no invisible. Tenía claro, que yo era una estudiante pobre donde muchos eran ricos. En Brooklyn, la mayoría de mis compañeros de clase habían sido de mi mismo vecindario y vivían en condiciones parecidas a las mías; pero Performing Arts se nutría de estudiantes de toda la ciudad. Mi falta de recursos se hacía más evidente cuando hablaba con mis compañeros de clase. Sabía que mi familia era "desventajada"; lo decían las solicitudes del _welfare_. Pero fue en Performing Arts que entendí de primera mano lo que significaba ser "aventajada."
Significaba viajar a Europa durante las vacaciones, tomar clases los fines de semana con maestros de baile y de voz y dicción; significaba hacerse cirugía plástica para reducirse una nariz grande o para afinar una muy chata. Significaba lecciones de tenis, encuentros de natación, prácticas de coro, clubes, tutoría académica, citas con chicos. Significaba tener dinero para almorzar en el Deli frente a la escuela o para regresar a casa en taxi.
Ser "desventajada" significaba que yo sacaba mis leotardos y mis _tights_ de baile de un cajón en la Oficina de Consejería. Significaba tener que lavarlos y ponerlos a secar en los radiadores tibios de nuestro apartamento y ponérmelos húmedos cuando no había dinero para pagar la calefacción. Significaba un boleto para que me dieran un plato de sopa y medio sándwich gratis para el almuerzo. Significaba que si me invitaban a una fiesta a casa de alguna compañera decía que no, porque no teníamos chavos para comprarle regalos a gente rica. Significaba que no invitaba a nadie a casa porque no quería que viera los pañales mojados tendidos en los cordeles que cruzaban de un lado a otro del apartamento o la cantidad de camas que impedía que pudiéramos tener una sala como Dios manda.
Ser "aventajado" significaba poder quejarse de tener tantas cosas que hacer, todas divertidas, y no poder decidir entre quedarse a dormir en casa de Joanie o tomar una clase de baile adicional en el estudio de Madame. Significaba que los trabajos que se le entregaban a la maestra iban pasados a maquinilla en un papel blanco y nítido y no escritos a mano con un bolígrafo barato, en papel de rayas azules de una libreta comprada en Woolworth's. La ventaja no la decidía el talento ni el color de la piel, sino el dinero y los "desventajados" teníamos poco o ninguno.
No era la única estudiante pobre en Performing Arts —ni en mi clase. Éramos muchos. Nos buscábamos unos a otros y revoloteábamos alrededor de los afortunados, cuyos cuentos los lunes sobre sus fines de semana divertidos intensificaban en nosotros el sentido de que nuestro talento nos iba a tener que llevar lejos, bien, bien lejos de donde estábamos.
Aprendimos a actuar haciendo improvisaciones y escenas de obras conocidas. Para las improvisaciones, la maestra nos presentaba una situación y nos permitía que la desarrolláramos frente a la clase o en pequeños grupos; la maestra podía entonces incorporar en medio de nuestra improvisación a otro actor que trajera una motivación diferente o una situación conflictiva. Un ruido estruendoso, también podía interrumpir la escena o la situación podía cambiar con naturalidad, según fuera evolucionando. Además de desarrollar nuestra capacidad de concentración y nuestra rapidez al pensar, la improvisación nos ayudaba a ir descubriendo la composición de una escena, según íbamos comprendiendo las motivaciones del personaje o los subtextos del diálogo.
Montábamos las escenas en pareja. Los maestros nos asignaban obras y escenas apropiadas para nuestros talentos y personalidades, pero evitaban el _"typecasting,"_ para no encasillarnos en el mismo tipo de papel, lo que hubiera sido poco retante para un actor. Para cada escena preparábamos el libreto en lo que se conoce como _"sides"_. Dividíamos cada página por la mitad y a un lado aparecían, en letras mayúsculas, nuestras líneas y nuestros apuntes y al otro, las notas sobre las interpretaciones del subtexto, instrucciones escénicas o motivaciones del personaje.
No teníamos escenografía. Unas cajas de madera con esquinas ásperas y llenas de astillas, lo mismo creaban la ilusión de que estábamos en una cocina sureña que en el Senado Romano, dependiendo de si la escena era de _Member of the Wedding_ o de _Julio César_. Ensayábamos en el sótano, que era de hecho, la planta baja de la escuela, donde estaban los _lockers_ en un extremo del salón y, en el otro, las puertas de entrada y las escaleras que conducían hacia el primer piso. Delimitábamos un área del sótano, acumulábamos cajas para crear nuestra escenografía y trabajábamos independientemente mientras la maestra iba de un grupo a otro, observándonos, cuestionando la motivación, sugiriéndonos otras formas de marcar la escena. Al finalizar la clase la maestra podía pedirnos que representáramos nuestro trabajo-en-progreso frente al grupo. Colocábamos los pupitres en un semicírculo para que todo el mundo quedara en primera fila. A veces se nos pedía que hiciéramos nuestra presentación en jerigonza, para demostrarnos que actuar es mucho más que repetir palabras como el papagayo; que la actuación transmite una experiencia humana que trasciende el lenguaje.
Mi primera asignación fue preparar la primera escena del primer acto del _César_ y _Cleopatra_ de George Bernard Shaw. Mi pareja Harvey, de nariz romana, representaría el Julio César de mi Cleopatra.
¡Estaba fascinada! Durante el verano había leído muchas biografías. Cleopatra era uno de mis personajes históricos preferidos y tenía mucha información sobre quién era ella, cuáles podían haber sido sus motivaciones, cómo era físicamente. Como actores, teníamos que investigar todo lo que pudiéramos sobre los personajes que representábamos, tanto los de ficción, como los históricos, basándonos en la teoría de que mientras más supiéramos sobre ellos, mejor lograríamos darles vida en el escenario.
Me encantaban los preparativos antes de la actuación. Me encantaba leer la obra completa aunque sólo fuera a representar una escena corta. Me encantaba estirar el personaje más allá de lo que había escrito el dramaturgo. Me encantaba diseñar el vestuario y rebuscar en casa hasta encontrar los materiales con qué hacerlo, ya que la escuela no proveía el vestuario excepto para la representación de fin de curso.
Encontré un mantel amarillo que Mami había conseguido en una tienda de baratillos.
"¿Puedo coger esto?"
"¿Para qué?"
"Para hacer un traje de Cleopatra."
"¿Y qué es un traje de Cleopatra?" Frunció los labios en señal de que sospechaba que quería ponerme una moda que ella no me permitía usar.
"Cleopatra era una reina egipcia," le expliqué. "Vivió hace miles de años y usaba trajes apretados."
"¿Y por qué tú te tienes que vestir como ella?" Me cogió el mantel de las manos y lo examinó.
"Es una asignación. Tengo que vestirme como la gente en las obras."
"¿Con un mantel?"
"Ya te lo dije, voy a hacerme un traje con él."
"Tiene una mancha de achiote," me dijo.
"Por eso se me ocurrió que ya tú no lo vas a usar."
"Yo te lo hago," se ofreció, con dudas todavía. Podía imaginarme la idea de Mami de lo que era un traje de Cleopatra; no era nada parecida a la mía.
"Se supone que lo hagamos nosotras mismas," mentí.
"Está bien," aceptó, "pero me dejas verlo antes de que lo termines por si necesitas que te ayude." Quería estar segura de que no fuera muy revelador.
Corté y cosí un vestido tubo tan estrecho que tenía que caminar de lado, como un jeroglífico egipcio.
"Ay, mi Dios," Mami quedó sin aliento cuando lo vio.
"Pareces un guineo," añadió Edna sin que nadie le preguntara.
"Cállate," grité.
"¿Tú te vas a poner eso en frente de la gente?" dijo Mami.
"Es sólo en la escuela, Mami, para una escena en una obra, te voy a enseñar un retrato." A paso de hormiga, caminé hacia el cuarto acompañada por las risas de mis hermanas y hermanos. Para recoger el libro de vestuario que había dejado en el piso, tuve que doblarme con cuidado desde las rodillas. Volví dando pasitos y abrí el libro. "Mira, esta gente es egipcia. ¿Ves qué pegada usaban la ropa?"
Edna, Delsa y Norma miraron el retrato por encima del hombro de Mami y volvieron a mirar mi traje.
"Esa gente no caminaba muy lejos, ¿verdad?" rió Norma. Le mandé una mirada envenenada y ella me sacó la lengua.
Mami estudió la ilustración en la que los trajes se veían transparentes, cosa que el damasco no era. "Se supone que sean lo más parecido posible a los que ellos usaban." Traté de no sonar muy desesperada. "Solamente lo voy a usar en el salón, frente a otros estudiantes y a la maestra."
"¿Y _ellos_ , qué se van a poner?"
No le hice caso a su sarcasmo. "Mi pareja se hizo el traje con una sábana y otra muchacha se hizo el de ella con unas cortinas." Si me concentraba en los materiales, a lo mejor Mami no se fijaba tanto en el entalle. "Nos van a dar nota por el parecido de nuestro traje con el original," volví a mentir.
"Por lo menos no se ve a través," dijo Edna.
"Como si tuviera algo que enseñar," se burló Delsa y chocó cinco con Edna.
"Tienes que soltarle," dijo Mami. "Las costuras te tiran demasiado."
"Okei." No lo tocaría, no había tela de dónde soltarle y, de todos modos, Mami no me iba a ver nunca en él porque yo sólo lo usaría en la escuela como Cleopatra. Me miré en el espejo ovalado encima del tocador que había en el cuarto que compartía con Delsa. Tenía razón, yo no tenía mucho que enseñar. Pero aún así, tampoco parecía un guineo. Aunque mis nalgas fueran planas y no tuviera caderas, los guineos no tenían busto y yo sí. Coloqué las manos en una posición parecida a la que tenían los retratos de jeroglíficos que había visto y me dio la impresión de que me parecía bastante a como debió haber sido Cleopatra. A los quince años, Cleopatra estaba a punto de ser Reina de Egipto, mientras yo tenía que discutir cada cosita con mi mamá. Pensé cómo sería no tener mamá y me dio escalofríos. Tuve que asomarme para asegurarme de que Mami estaba todavía allí, antes de que me volviera el calor al cuerpo.
El Departamento de Drama utilizaba el Método desarrollado por Stanislavsky en su libro _Un actor se prepara_. El actor Método exploraba lo más profundo de su ser para buscar la verdad emocional que le daría base al momento que más tarde tendría que vivir en escena.
Yo me negaba a aventurarme en lo más profundo de mi ser, a revelar mis sentimientos, a examinar mis verdaderas emociones públicamente. Si lo hacía, todo el mundo se enteraría de que era ilegítima, de que compartía una cama con mi hermana y de que estábamos en _welfare_. El resultado era que mis pares me acusaban de estar "indicando," el peor pecado que puede cometer en escena un actor Método. Indicar era fingir el momento a través de ademanes y gestos en lugar de vivirlo.
Era humillante no ser una actriz lo suficientemente buena como para convencer a mis maestros y compañeros, pero sencillamente, no podía entregarme al oficio. No tenía destrezas para actuar dentro de la actuación. Porque el mismo instante que salía del oscuro y abarrotado apartamento donde vivía, entraba en papel, fingía ser alguien que no era. Rechazaba la importancia que le daba el Método a la verdad, puesto que yo la usaba para crear una realidad simulada. Una en la que hablaba un inglés fluido, en la que me sentía como en casa en las duras calles de Nueva York, en la que absorbía la cultura norteamericana sin reparos, mientras lamentaba silenciosamente la disolución de mi otro yo, el de la muchacha puertorriqueña que hablaba español, que se sentía tan a gusto en una polvorienta carretera de tierra tropical. Creé un personaje que evolucionaba según se desarrollaban las múltiples improvisaciones de mi vida. Una protagonista tan alegre y despreocupada como mis amigos de los paquines, Verónica, Archie, Reggie y Jughead.
# "¿Tú no quieres sonar puertorriqueña?"
#
Un día al regreso de la biblioteca, me encontré a mis hermanas y hermanos alrededor de una señora y una muchacha como de mi edad que estaban tomando café y comiendo bizcocho en la mesa de la cocina.
"¿Adivina quién es ésta?" sonrió Mami.
La muchacha se me quedó mirando por entre las pestañas llenas de mascara y la señora, pequeña, prensada en su faja y cuidadosamente maquillada, me miró de arriba a abajo y me encontró poca cosa. No tenía idea de quiénes eran, ni me importaba. "¿Amigas de la fábrica?" tanteé y Mami se rió.
"Esta es tu hermana Margie."
Quedé boquiabierta de la sorpresa, aunque enseguida la cerré porque todos se rieron. Margie; su mamá, Provi; mis hermanas y hermanos que estaban agrupados al lado de la mesa, más cerca de Margie, todos parecían encontrar comiquísimo el que yo no reconociera a alguien a quien no recordaba haber conocido nunca.
"Qué cara tan expresiva tiene," dijo Provi con una risita falsa mientras yo sentía que los cachetes me ardían. Mami entrecerró los ojos e inclinó la cabeza hacia Margie y Provi. Les toqué levemente los hombros con la punta de los dedos, dejando mucho espacio entre nosotras y les besé ligeramente el cachete derecho.
Provi había sido la "esposa" de mi papá, antes de que conociera a mi Mamá. Yo hubiera esperado que Margie se pareciera a él, que tuviera su frente alta, sus pómulos prominentes, su nariz ancha y sus labios llenos. Hubiera esperado que tuviera su color, pero ella era más clara y se parecía más a mi hermana Norma, con el mismo pelo castaño de rizo apretado, los ojos marrón y achinados y su porte distinguido.
Mami me sirvió café y bizcocho. "Provi lo trajo de una repostería cerca de su apartamento en Manhattan." Sonó como una advertencia, pero cuando levanté la vista, Mami se estaba sirviendo café y me estaba dando la espalda.
De espalda a la pared, Margie estaba incómoda en nuestra mesa, mientras mis hermanas y hermanos se empujaban uno al otro para ver quién quedaba más cerca de ella. Héctor sacó su colección completa de tapas de botellas y Edna dibujó flores y pajaritos y se los mostró a Margie en espera de su aprobación. De vez en cuando, Margie me sonreía y yo hubiera querido que nos hubiéramos podido ir a algún sitio a hablar. Pero no había otro sitio, ni sala, ni patio, ni cuarto, que no estuviera lleno de camas o de gente. Me sentía avergonzada y traté de adivinar cómo se sentiría Mami. Pero ella estaba serena, no parecía notar que los ojos de Provi volaban como flechas del fregadero repleto de ollas y cacharros limpios pero estropeados, al cuarto de al lado donde había un cordel enganchado desde la ventana hasta el marco de la puerta. Debajo del cordel, en el piso de linóleo opaco goteaba el agua de los pañales que se estaban secando. A cada rato, Delsa agarraba el mapo, secaba los charcos y se acomodaba otra vez al lado de Margie.
Estaba molesta con la compostura de Mami. Tendría que haber estado tan avergonzada como me sentía yo. Tan pronto me vino esa idea a la mente, la descarté. Mami trabajaba duro para todos nosotros, y si bien yo tenía menos de lo que deseaba, como hija mayor me tocaba más que a mis hermanas y hermanos menores. Cuando ellos se quejaban de que Mami me prefería, yo les porfiaba que no era verdad, pero en el fondo, sabía que sí, lo mismo Tata. Me eché hacia atrás en la silla, hirviendo, alternando la vergüenza con la culpa, envidiosa de la ropa de moda de Margie, el pelo arreglado con rolos, "tisin" y "esprei," su maquillaje impecable, la pulsera de _charms_ que tintineaba en su muñeca derecha y su _Timex_ en la izquierda. A la vez, anhelaba hablar con ella, averiguar si se mantenía en contacto con Papi, si le había dolido que él volviera a casarse, si recordaba a nuestra abuela, a quien, según Provi, yo me parecía.
Mami hablaba con orgullo del mucho inglés que habíamos aprendido en apenas dos años, de la escuela a la cual yo asistía; de lo dulce que era Franky, el bebé; de su trabajo como operaria de la máquina de coser Merrow en una fábrica Maidenform. Las dos hablaban como si hubieran sido amigas de toda la vida que se habían encontrado después de mucho tiempo cuando, en realidad, durante años, Mami se había referido a Provi como "esa mujer" y Provi también debe haberle tenido un par de nombres a Mami, para cuando no estaba sentada en nuestra mesa de comedor tomando café y masticando delicadamente un pedazo del bizcocho demasiado dulce que había traído.
Provi se las echó de su apartamento en Manhattan donde, según señaló, Margie tenía su propio cuarto, de cómo Margie era una de las mejores alumnas de su escuela, de cómo habían vivido tanto tiempo en los Estados Unidos que ya se les estaba olvidando el español cuando todavía estaban aprendiendo inglés.
"Y después, ¿qué haremos?" decía con risa entrecortada. "¡Nos vamos a quedar mudas, sin nada que decir!" Mami y yo intercambiamos una mirada al recordar a La Muda que era todo menos una mujer sin palabras.
Yo interpreté la cordialidad de Provi como una actuación. Acostumbrada a la obsesión de los estudiantes de Drama de encontrar el subtexto en los diálogos, yo oía la cháchara de Provi, pero escuchaba lo que no decía: "No fuiste suficiente mujer para que Pablo se quedara contigo," mientras que lo que Mami no decía, "Yo lo tuve catorce años, cuatro veces más tiempo que tú," caldeaba el ambiente.
Suponía que Provi se alegraba de que Mami hubiera enviudado, que vería la muerte de Francisco como un castigo por el daño que supongo que le habría causado. Mami, más joven y más bonita que Provi, era, me sospechaba, la razón por la cual Papi había dejado a Provi.
Malhumorada en mi esquina de la mesa, escuchaba a nuestras madres chacharear, consciente de que todavía estaban compitiendo por mi padre, que ni siquiera estaba allí, que estaba casado con otra mujer que ninguna de las dos conocía. No oía nada, excepto crítica en los comentarios de Provi, sólo defensa en los de Mami. Me daba pena con Margie, que con los hombros hundidos en la silla, parecía también avergonzada por el comportamiento de su mamá. Me resistí a las sonrisas tensas de Provi y a los frecuentes intentos de Margie de hacer contacto visual conmigo. Cada segundo de su visita era una prueba que teníamos que pasar antes de poder subir al siguiente nivel, pero no estaba segura qué nivel era ése, en qué consistía, o si realmente existía. Margie había llegado demasiado tarde, pero yo no estaba segura de para qué; ni siquiera sabía si había estado esperando por ella.
Cuando cerró la puerta tras ellas, Mami dio un suspiro profundo. Mis hermanas y hermanos se regaron por el apartamento. Tata, que se había quedado en su cuarto durante toda la visita, entró tambaleándose a la cocina y empezó a picar cebollas para la comida.
"Qué linda es Margie, ¿verdá'?" preguntó Mami, sin esperar en realidad una respuesta. Tata refunfuñaba acerca de "esa mujer" y yo estuve tentada a hacer un comentario sarcástico, pero me contuve.
"Tiene un pelo lindo," concedí. "Me gusta como se hace la línea de los ojos con el rabito en la esquina," añadí para decir algo agradable y Mami se me quedó mirando como si hubiera estado viendo lo que antes no era tan obvio.
"Tú tienes mejor pelo," dijo, pasándome los dedos por el pelo. "Es ondulado, no tan rizo como el de ella, puedes hacer más con él." Me tomó la cara y me la inclinó hacia la luz. "En cuanto a su maquillaje, esa línea no te quedaría bien a tí. Tus ojos tienen una forma completamente diferente." Me movió la cara hacia la izquierda, hacia la derecha. "Quizá, el rabito un poquito más corto... ¿Por qué no pruebas a ver?"
Volé hasta el gavetero donde Mami guardaba los cosméticos que no había usado desde la muerte de Francisco. Sin aliento, abrí el _zipper_ de la bolsita. Adentro había un compacto plástico con su espejo y un fino círculo de polvo comprimido alrededor del fondo de metal. La mota de algodón, que una vez fue suave y esponjosa, ahora estaba deshilachada por los bordes. Una cajita de cartón redonda, más pequeña, contenía el colorete en polvo que había regado un fino polvillo rojo sobre dos lápices de labio y el cabo de un lápiz de cejas. Le quité la tapa, afilé la punta con una "yen" y me dibujé una línea en la mano. Cuando traté de hacérmelo en el párpado, la punta dura se rodó y me dejó una pálida línea ceniza que yo me limpié con saliva y papel de inodoro. Cuando finalmente conseguí delinearme la línea oscura en el párpado superior, la extendí hasta formar un ángulo como una sonrisa.
"¿Qué tú crees?" Traté de calmar los latidos que revelaban mi excitación. Mami se apoyó contra el _counter_ y entrecerró los ojos como si estuviera evaluando un objeto costoso.
"Se ve bien," dijo, "pero la próxima vez, hazte los rabitos más cortos."
"Okei." Había dicho la próxima vez. ¡La próxima vez! Corrí hasta el baño, me borré las esquinas de la línea para que no me sobresalieran de los párpados.
"¿Así?"
"Perfecto," sonrió, "se te ve bien."
Tata nos miraba desde su puesto junto a la estufa. "Está creciendo," dijo bajito y yo, con los ojos, le pedí silencio. Se viró con una amplia sonrisa. Mami también sonrió y siguió lavando el arroz.
En mi cuarto, me miré en el espejo, me toqué las gruesas y oscuras líneas que me hacían ver mayor, sofisticada. Delsa estaba acostada, enrollada en una frisa, con los rizos negros asomándose por la parte de arriba.
"Deja," protestó, aunque yo no había hecho ruido. Salí del cuarto y me acurruqué contra la pared en la cama de Norma y Alicia y me puse a ver televisión. Sentía los ojos pesados, como si la línea negra les añadiera peso. Durante un comercial, Alicia se me quedó mirando fijamente y salió disparada para la cocina gritado: "Mami, Negi se pintó."
"Cállate," corrí tras ella y la agarré.
"¿Qué es esa gritería?"
"Negi se pintó," repitió Alicia mientras luchaba por zafarse de mí.
"Deja a tu hermana quieta," gritó Mami y yo no estaba segura si era conmigo o con Alicia.
"La próxima vez que vaya a la farmacia," dijo, rumbo a la cocina de nuevo, "te compro tu propio lápiz."
Yo solté a Alicia, que nos miraba a Mami y a mi con expresión confundida. Tenía nueve años, yo quince y aunque Mami se ponía de mi parte en muchas de las discusiones entre mis hermanas, mis hermanos y yo, las dos sabíamos que algo importante había pasado. Dejé de ser una nena porque Mami no se iba a dejar ganar como mamá, de Provi.
Estaba todavía oscuro cuando salía del apartamento a las cinco y media de la mañana, llevando mis libros y la ropa de baile en el bolso negro de piel que me había dado La Muda. La caminata de quince minutos hasta la estación del elevado era como un túnel de sombras bajo las lámparas fundidas de la calle que alargaban la distancia entre los edificios abandonados y los carros estacionados. Yo caminaba por el medio de la acera con los ojos fijos hacia el frente, pero alerta, anticipando el peligro que podía salir de cualquier sitio, en cualquier momento. Una vez, una rata me pasó corriendo por delante. No sabía qué hacer, tenía miedo de seguir andando y miedo de quedarme quieta. Después de unos segundos, le pasé corriendo por el lado al montón de basura por donde había desaparecido la rata y añadí "mordida de rata rabiosa" a la lista de "algos" que me podían pasar fuera de casa.
Ya a las seis de la mañana los trenes iban repletos, y con frecuencia, tenía que ir de pie todo el camino hasta Manhattan. Aquella mañana tuve suerte. Cuando llegó el tren divisé un espacio en un banquillo de dos asientos frente a la cabina del chofer. Me senté, teniendo cuidado de no molestar a la señora que dormía en el asiento más cercano a la puerta, con sus manos enguantadas apoyadas contra la cartera que llevaba en la falda. Los pasajeros que ya iban en el tren eran negros o puertorriqueños, pero según nos fuimos acercando a East New York, Brownsville, Crown Heights, Prospect Heights y Brooklyn Heights, la gente que estaba esperando en el andén era blanca y mayor que los pasajeros que ya veníamos en el tren. Se abrían paso en el vagón y los demás nos apretujábamos para hacerles espacio.
Un hombre se fue haciendo camino a codazos hasta alcanzar la correa que colgaba sobre el asiento donde yo estaba sentada contra la pared. Colocó el bulto en el piso entre sus piernas, agarró la correa con la mano izquierda y con la mano derecha se desabotonó y se abrió el abrigo. Mantuve los ojos en el libro, apenas consciente del movimiento que había frente a mí, hasta que me di cuenta de que el hombre estaba tan cerca de mí, que no dejaba entrar la luz. Cuando levanté los ojos para pedirle que se moviera, vi que tenía el _zipper_ abierto y el pene colgando fuera del pantalón, a apenas dos pies de mi cara. Bajé la vista enseguida, demasiado avergonzada para decir o hacer algo. De un lado, su abrigo formaba una cortina y del otro, me atrapaba la pared del tren. Yo me hice la que estaba leyendo mientras trataba de decidir qué debía hacer. Podía levantarme y cambiarme de sitio, pero mi bulto estaba debajo de mis pies y si me bajaba a alcanzarlo, quedaría peligrosamente cerca de su arrugado y lánguido pene. Consideré mirar al hombre fijamente a los ojos y decirle que se lo metiera en su sitio, pero no me atreví. Según nos acercábamos a alguna estación y el tren reducía la velocidad, el hombre dejaba caer el brazo de la correa, se tapaba, esperaba que el tren empezara a moverse de nuevo, volvía a levantar el brazo y yo volvía a tener el pene en la cara. Sentía su mirada sobre mí, mientras me debatía tratando de decidir qué hacer. Podría agarrarle el pene y halárselo con fuerza, podría darle un mordisco. Sin tocárselo, podría cerrar de golpe mi libro de biología y pillárselo entre las páginas, pero me quedé sentada, callada, la cara de piedra, fingiendo leer, molesta conmigo misma por ser tan pendeja, y preguntándome qué había hecho para provocarlo.
La clase de Maquillaje para Teatro se daba en un salón frente a la entrada de los camerinos del auditorio. La maestra, Misis Bank, una mujer que no se andaba con cuentos, con una reputación de ser exigente y difícil de complacer, era sin embargo amada por aquellos estudiantes que lograban impresionarla con sus talentos. Yo no estaba entre sus predilectos. Tenía muy poco registro como actriz, para satisfacer sus altas expectativas.
En la primera clase, nos dio una lista de materiales y yo tuve que convencer a Mami de que el gasto era necesario, de que Maquillaje era una clase de verdad en la que me darían nota. Mami protestó por las brochas, lápices, esponjas, motas, cremas y polvos que costaban más que los que usaba ella. Pero, nunca me dijo que no me los compraría.
Misis Bank nos fue llevando rápidamente por los fundamentos del maquillaje teatral. Empezamos por las técnicas para realzar nuestras facciones naturales. Los muchachos, al igual que las muchachas, aprendieron a aplicarse base, delineador de labios, colorete y máscara. Se nos estimulaba a que estudiáramos nuestro rostro; a que conociéramos sus contornos y examináramos las formas que configuraban nuestra apariencia; a que nos viéramos, no tanto como éramos, sino como podríamos ser.
Con este fin, se nos enseñaba a alterar nuestras facciones. A través del uso diestro de la luz y la sombra, aprendimos a afinarnos una nariz ancha, a redondear una demasiado puntiaguda. Los ojos podían verse más grande, los labios más llenos, los pómulos lisos más altos y redondeados, una frente muy ancha, más estrecha. Me encantaba la clase porque podía ponerme todo el maquillaje que quisiera y Mami no podía protestar porque yo le decía que practicar era mi asignación. Me pasaba las horas muertas frente el espejo, maquillándome para verme inocente, sensual, elegante, asiática. Una de las asignaciones fue traer el retrato de un animal y recrear sus facciones en nuestro propio rostro. En casa me maquillaba para parecer un tigre, un camello, un orangután y después corría a mis hermanas y hermanos por el apartamento, haciendo los ruidos propios del animal hasta que Mami o Tata ponían fin a mis rugidos y a sus gritos.
Una de las últimas tareas de ese semestre fue maquillarnos para parecer ancianos.
"Sigan el contorno natural de sus rostros," nos instruyó Misis Bank. "Oscurezcan las líneas que van desde la ventana de la nariz hasta los labios. Aplíquense luz en las orillas para hacerlos parecer más profundos."
La mayoría de nosotros tenía quince o dieciséis años y se nos hacía difícil encontrarnos arrugas, no porque no las tuviéramos, sino porque no las queríamos ver.
"Si fruncen los labios así y después se pintan unas líneas donde quedaron las marcas, lograrán unas líneas muy interesantes."
Seguimos sus intrucciones, riéndonos según iban envejeciendo nuestras caras bajo las motas y las brochas.
"La mayor parte de la gente tiene líneas alrededor de los ojos," señaló. "No olviden el cuello y las manos, también envejecen."
Nos pintamos manchas oscuras en el dorso de la mano. Nos empolvamos el pelo para hacerlo ver canoso. Jay se puso una verruga en la mejilla. Elaine le añadió un temblor a la voz para que le hiciera juego con su frágil carita de vieja.
Al final de la clase, cuando la maestra nos pidió que evaluáramos el trabajo que habíamos hecho, miré con cuidado mis mejillas arrugadas, mis ojos curiosos dentro de dos grandes ojeras y estallé en llanto.
"¿Qué te pasa?" me preguntó alarmada Misis Bank.
"Soy una viejita," gimoteé, en lo que me pareció un modo gracioso de disimular mi vergüenza.
Misis Bank se sonrió. "Ni tanto, todavía no. Tú tienes suerte, que a ti se te va con crema Albolene."
"Qué bueno," reí sin muchas ganas, "soy demasiado joven para ser vieja."
La maestra siguió en lo suyo. Yo me enfrenté al espejo otra vez y vi a Abuela, a quien no veía desde hacía tres años. Si viraba la cara hacia la izquierda, aparecía Tata, la abuela con quien vivía. Me impresionaba verlas observándome desde mi propio rostro. Los ojos tristes de Abuela, la boca sensual de Tata, la nariz pequeña de Abuela, la mirada inteligente de Tata. Pero yo no le iba a admitir eso a Misis Bank ni a los demás estudiantes que se rieron de mis temores a envejecer. Que creyeran lo que quisieran. Nunca sabrían, nunca podrían entender, quién era en realidad.
Tenía una vida secreta, una que no compartía con mi hermana, con quien compartía una cama, ni con mis compañeros de clase, con quienes compartía el sueño de lograr fama y fortuna, ni con mi mamá, cuyos sueños habían quedado en suspenso desde la muerte de Francisco. Mi vida secreta estaba en mi cabeza y la vivía de noche, antes de quedarme dormida, cuando me convertía en otra persona.
En mi vida secreta, yo no era Esmeralda Santiago, ni Negi, ni una muchachita puertorriqueña asustada, sino una mujer segura y poderosa, cuyo nombre cambiaba según me esforzaba por perfeccionarme. Esme, fui una vez. Emmé, otra. Emeraude, mi nombre de la clase de francés. Probé con Shirley, Sheila, Lenore, pero los nombres que no surgían del mío no me sonaban bien. Así es que fui Emma, Ralda o simplemente E.
En esos sueños yo no tenía familia —ni mamá, ni papá, ni hermanas, ni hermanos, ni abuelas, ni primos luchadores, ni tíos borrachos, ni sordomudas. Estaba sola, surgida de una oscuridad innombrable, sin ataduras ni responsabilidades. Era educada, exitosa, profesional. Cualquier cosa que hiciera, la hacía bien, sin pasos en falso, ni errores, ní equivocaciones embarazosas que dieran pie a que otros me juzgaran o se rieran de mí.
Era piloto de mi propio avión y viajaba alrededor del mundo y donde quiera que iba, la gente se alegraba de verme y nadie me preguntaba de dónde era. Era una estrella de cine y mi personaje no moría nunca. Era una científica, rodeada de tubos de ensayo, interruptores, lámparas, mecheros de Bunsen con chispeantes llamas azules, en el momento de recibir el Premio Nobel.
En mi vida secreta yo manejaba un convertible y mi casa, en lo alto de una larga y sinuosa entrada, tenía vista a millas y millas de ondulantes colinas verdes, donde nunca nevaba. Vivía sola en mi casa, en el tope de la loma, rodeada de libros que nunca tenía que devolver a la biblioteca. Y todos los cuartos estaban pulcros y ordenados, aunque yo nunca limpiaba.
En mi vida secreta no era puertorriqueña. No era americana. Hablaba todos los idiomas del mundo, por lo tanto nunca me confundía con lo que la gente decía; y todo el mundo me entendía. Mi piel no tenía un color particular, así es que no sobresalía por negra, blanca o marrón.
Vivía esta vida secreta todas las noches, según me iba quedando dormida, y todas las mañanas me resistía a abrir los ojos en la cama estrecha del estrecho cuarto que compartía con Delsa, el pecho apretado ante la sorpresa y el desencanto de que todo fuera un sueño.
"Eee, eee, eee." Enunciaba las vocales según nos instruía la Dra. Dycke, la directora del Departamento de Drama. "Ay, ay, ay. Eee, eee, eee."
Raymond se asomó por el quicio de la puerta. "¿Qué tú hace'?"
"Practicando. Eee, ay, eee, ay, eee."
"¿Por qué?"
"Para poder aprender a hablar inglés sin acento."
"Ah," siguió andando.
"Eee, eee, ay, ay, eee, eee, ay, ay."
Unos minutos después apareció Edna en la puerta. "¿Qué tú hace'?"
"Practicando. Eee. Eee. Eee."
"¿Practicando qué?"
"¡Pregúntale a Raymond!" Le cerré la puerta en la cara. "Ay. Ay. Ay. Oo. Oo." Se abrió la puerta. "¡Pa' fuera!" grité. "Ah, eres tú."
"Tengo que buscar algo," Delsa señaló al gavetero.
Me eché para atrás para dejarla pasar. "Eeu. Oo. Eeu. Oo. Ay."
"¿Qué _es_ lo que haces?" Sacó una camisa limpia de la gaveta.
"¡Ya!" la empujé hasta el cuarto de al frente. "¡Venga todo el mundo!" grité. "¡Hector! ¡Norma! ¡Mami! ¡Tata!"
"¿Qué quieres?" gritó Norma desde el cuarto de atrás del apartamento. Mami salió de la cocina. "¿Qué es esa gritería?"
"Quiero a todo el mundo aquí para decir esto una sola vez."
"¿Decir qué?"
"¡Norma! ¡Héctor! ¡Alicia! ¡Avancen!"
"Silencio," dijo Mami irritada, "Franky está durmiendo."
"Con calma. Ya no soy tan rápida como antes," llegó Tata arrastrándose hasta el cuarto del frente.
Cuando todo el mundo se había acomodado en las camas, el piso y el sofá, comencé. "Tengo una clase que se llama Voz y Dicción, donde estoy aprendiendo a hablar sin acento."
"¿Por qué? ¿Tú no quieres sonar puertorriqueña?"
"Déjenla hablar," dijo Tata.
"Es parte de mi trabajo de la escuela," dije atravesando a Héctor con la mirada.
"Sonaba como que estabas imitando animales," se burló Edna y todo el mundo se rió.
"Ja, ja, _very funny_." Seria, esperé que se calmaran. "Tengo que practicar y no puedo hacerlo si cada cinco segundos uno de ustedes me interrumpe para venirme a preguntar qué estoy haciendo. Así es que si oyen sonidos extraños saliendo del otro cuarto, es mi asignación. ¿Está bien?"
"¿Por eso era to'a esa gritería?" pregunó Mami.
"Sí. Los muchachos me estaban molestando." Con la mirada fulminé a Edna, Raymond y Delsa. Ellos miraron a Mami, que me dio una mirada seca. Por un momento, me pareció que me iba a regañar o formar tanto lío por tan poca cosa. Pero, se volvió hacia los nenes y les advirtió: "Déjen a su hermana quieta cuando esté estudiando."
"Suena como un zoológico allá adentro," protestó Norma.
"Se me van pa' otro la'o del apartamento cuando ella esté practicando."
Yo me volví al cuarto, seguida de las quejas de Norma. _"But it's not fair."_
No era justo. Desde que había empezado a estudiar en Performing Arts High School, Mami me favorecía. Si yo estaba leyendo y me quejaba de que el volumen del televisor estaba muy alto, ella hacía que los nenes lo bajaran. Si yo me quería acostar temprano, todo el mundo se movía hasta la cocina donde podían formar el barullo sin que yo los oyera. Si yo traía a casa una lista de materiales para la escuela, Mami nunca decía que no. Me daba el dinero para comprarlos o me los compraba sin quejarse de lo que habían costado. Yo sabía lo mucho que trabajaba para mantenernos, así es que no abusaba. Pero, me sentía culpable de que tanto de lo poco que teníamos se gastara en mí y me aterraba el costo que tendría a la larga.
"Vivo para mis hijos," afirmaba Mami. Yo estaba segura de que no importaba cuánto trabajara, nunca podría pagarle por todo lo que ella había tenido que renunciar para que yo pudiera tener todo lo que necesitaba.
Mami se había salido de la escuela elemental y nunca nos permitió olvidar el error que había cometido al no continuar su educación. Aunque nunca se quejaba de que fuéramos una carga, le temblaba la voz cuando nos decía lo difícil que era ser madre y padre de ochos hijos. Aunque nunca los mencionaba, debió haber tenido sueños alguna vez, pero nací yo, y cada año después, con el nacimiento de cada una de mis hermanas y hermanos, esos sueños se fueron esfumando poco a poco, según ella se concentró en asegurarse de que tuviéramos nuestros propios sueños.
"¿Qué tu quieres ser cuando seas grande?" nos preguntaba.
"Doctora," contestaba Delsa. Tenía buenas notas en la escuela. Mejores que las mías, especialmente en Ciencias y Matemáticas. Era más probable que ella llegara a doctora a que yo llegara a ser una buena actriz.
_"Race car driver,"_ anunció Héctor, los ojos brillantes y las manos agarrando un guía invisible. A los once años, Héctor ya trabajaba en la pizzería de al lado. Cada dos o tres días, traía a casa un par de pizzas con mucho chorizo y pepperoni que Gino, el dueño, nos mandaba.
"Su hijo es muy trabajador," le decía Gino a Mami, "usted lo ha criado bien." Mami rebosaba de orgullo ante el elogio y Héctor trabajaba aún más y al final de la semana le daba a Mami la mayor parte de lo que se había ganado.
"¿Y tú, Raymond, qué tu quieres ser cuando seas grande?"
"Policía," respondía Raymond, "y te voy a dar una multa si vas embalao por mi calle," le adviritió a Héctor. El pie de Raymond se había curado después de tres años de tratamiento y ya no cojeaba. Era tan fácil imaginárselo en uniforme, pavoneándose por la calle, buscando bandidos.
"Yo voy a tener mi propio 'biuti'," afirmaba Alicia. A los nueve años, Alicia ya sabía hacerse elegantes peinados con su hermosa mata de pelo negro y ondulado, gracias a su habilidad con el cepillo, la peinilla y las hebillas. "Y te voy a dar un permanente gratis."
Edna, que se pasaba horas enteras dibujando mujeres curvilíneas en extraños conjuntos de ropa, añadió: "Yo voy a tener una tienda de ropa y tú vas a poder conseguir toda la ropa que quieras, gratis."
"Anda, yo voy a ser una vieja rica," reía Mami y a todos nos divertía imaginarnos a Mami vieja. Era imposible pensar que Mami se viera distinta a como era entonces, su pelo revuelto, los rizos negros acariciándole las mejillas pecosas.
Cuando hablábamos así, Don Julio y Tata nos miraban pensativos, como si pudieran ver el futuro y supieran cómo serían en realidad nuestras vidas. A diferencia Mami, ellos eran viejos, pero aun a través del humo de cigarrillo que los envolvía y del hablar enredado después de mucha cerveza o vino, parecían tener una sabiduría que Mami no tenía.
"No canten victoria," empezó Tata y no tuvo que terminar para confirmar lo que yo sabía, que anunciar el porvenir era hacerle mal de ojo.
Mami, Tata y Don Julio con frecuencia me decían lo inteligente que era, pero yo interpretaba sus halagos, más como un anhelo de ellos que como una realidad. Mis notas estaban entre regulares y flojas y me había colgado en Geometría, lo que significaba tener que coger clases de verano y no poder trabajar. Había aprendido el inglés rápidamente. Pero, eso no era raro, puesto que en Performing Arts analizábamos, memorizábamos y recitábamos algunos de los mejores textos escritos en lengua inglesa. Mis hermanas y hermanos, que no tuvieron el beneficio de ir a Performing Arts, hablaban el inglés tan bien como yo, sólo que con acento de Brooklyn.
Mami estaba orgullosa de que yo fuera a la ciudad todos los días, regresara a la hora que debía y que estuviera pendiente de que no me pasara nada. Nunca le confesé cuán asustada me sentía caminando por las calles oscuras tan temprano en la mañana hasta llegar al _subway_. Nunca le mencioné nada sobre los hombres exhibicionistas o los que se aprovechaban de que el tren estuviera lleno para apretujarse contra mi cuerpo o para manosearme las partes que nadie tenía derecho a tocar a menos que yo se lo permitiera. No le informé nada sobre la vez que una mujer, zarandeándo una sombrilla, me corrió desde la estación del tren hasta la puerta de la escuela gritando: "spic asquerosa, maldita spic asquerosa, salte de mi calle". Nunca le dije a Mami que me avergonzaba de donde vivía, que en el _Daily News_ y en el _Herald American_ , los oficiales de gobierno le llamaban a nuestro vecindario "el ghetto" y a nuestro edificio de apartamentos, vivienda para personas de escasos recursos. Me tragaba la humillación cuando esos mismos periódicos, si incluían alguna historia que tuviera el término "puertorriqueño," generalmente estaban describiendo a un criminal. No le dije a Mami que aunque ella tenía grandes expectativas para nosotros, fuera de nuestra puerta esas expectativas eran muy bajas, que el resto de Nueva York nos veía como "spics" asquerosos, asaltantes potenciales, traficantes de drogas, prostitutas.
Mami se sentía feliz de que yo a los dieciséis años, y ya "casi mujer," no mostrara ningún interés en los muchachos.
"Ella es demasia'o lista para meterse con esos bambalanes de por ahí," aseguraba cuando sabía que yo la estaba oyendo.
Y yo no discutía, aunque las cualidades de los muchachos no era el problema. No habia muchachos de mi edad en nuestro vecindario, y en la escuela algunos de los muchachos eran homosexuales y los que no lo eran no tenían ningún interés en una muchacha como yo. Era pobre, con el talento suficiente para no hacer una pachotada en escena, pero buena sólo para representar a Cleopatra y otros personajes exóticos. Cuando surgió el tema de las citas con muchachos en la clase de Estudios Sociales, yo admití que mi mamá no me dejaba salir con muchachos a menos que fuera con chaperona. Eso me aseguraba que ningún muchacho de toda mi clase me invitaría a salir. ¿Para qué? Si le pedía permiso a Mami para salir, me daría un sermón de cómo los muchachos sólo quieren una cosa y yo no estaba dispuesta a dársela. Todo lo que tenía que hacer era mirar a mi alrededor para saber lo que le pasaba a la muchacha que dejaba que un hombre tomara el lugar de una educación.
En el apretujado y ruidoso apartamento donde mi mamá luchaba para mantenernos seguros, donde mi abuela trataba de apaciguar su dolor con el alcohol, donde mis hermanas y hermanos planificaban e inventaban su futuro, yo improvisaba. Cuando me dolía, lloraba lágrimas silenciosas, y cuando llegaban cosas buenas a mi vida, las aceptaba agradecida, pero sin aspavientos, temerosa de que si las disfrutaba demasiado pudieran desvanecerse como una gota de agua en el desierto.
# "A mí no me importa que vaya el mundo entero."
#
Mami fue saliendo del luto gradualmente. Un día se hizo rolos, otro se puso una blusa gris en vez de negra. Poco a poco fue abandonando los colores grises por el azul oscuro y el marrón. Entonces, una pieza a la vez, fue sacando los estampados de flores y los diseños audaces que tanto le gustaban. Reaparecieron los tacos altos, los lápices de labios brillantes, las pantallas largas, los collares y el esmalte de uñas. Volvieron sus sonrisas. Breves, tímidas al principio, luego más amplias, iluminándole todo el rostro, como si se estuviera probando su antiguo yo, poquito a poco, a ver si todavía le servía.
Fuimos acomodando nuestro luto a sus reacciones. Poníamos el radio bajito y si ella no decía nada, le subíamos el volumen. Bailábamos por el apartamento o cantábamos boleros suavecitos en la ducha y cuando no protestaba, pasábamos a merengues y a rancheras mexicanas. Las visitas a los parientes se fueron haciendo más frecuentes y más largas. La Muda venía con Luigi que se veía cada vez más triste a pesar de que él y La Muda ahora estaban viviendo juntos en un apartamento en el edificio de la mamá de ella. Luigi decía que no le gustaba Nueva York. No encontraba trabajo y se quejaba de que los inviernos tan fríos le producían artritis. Y era cierto. Sus dedos huesudos que antes barajaban las cartas con la velocidad del rayo, ahora estaban torpes, impedidos por los bultos y chichones alrededor de los nudillos. Ya no nos hacía trucos de magia cuando venía de visita, sino que se sentaba callado con las manos en la falda.
Tío Chico desaparecía durante semanas hasta que un buen día, aparecía a media mañana, a veces sobrio, pero la mayor parte de las veces borracho. Tata lo aseaba y le preparaba un asopao sustancioso y café negro fuerte, para ayudarlo a pasar la juma. Se quedaba un par de días con nosotros, durmiendo la mayor parte del tiempo y después nos preparaba un sancocho con mucha carne o un gallo guisado con vino tinto y mucho culantro. Era un excelente cocinero y él, Tata y Mami, cocinaban cada uno un plato diferente para la comida del domingo y después simulaban que estaban discutiendo sobre quién cocinaba mejor. La vecina de abajo, que al principio criticaban muchísimo el ruido que hacíamos, subían ahora todos los días a visitar a Mami y a Tata y, con frecuencia, se quedaban a comer. Su hijo mayor, Jimmy, era un poquito menor que yo. Tenía la cara larga y llena de granitos, el pelo con un recorte bien pegadito, un bigote espelusa'o y las orejas grandes. Mis hermanas y hermanos me fastidiaban diciéndome que yo le gustaba a Jimmy y cuando él subía, yo me quedaba en mi cuarto para que me dejaran quieta. Siempre que oía pasos bajando, Jimmy, asomaba la cabeza a ver si era yo, entonces decía que él también iba a salir y me acompañaba a la parada, donde yo cogía la guagua para ir a mis clases de verano. Casi todos los días, cuando regresaba, lo encontraba en la esquina de la Rockaway, esperando para acompañarme a casa.
"Mami, ¿puedo ir a casa de Alma y de Corazón cuando salga de la escuela?" le pregunté un día. Dijo que sí y después de eso, visitaba a mis primas todos los días para ahorrarme la cara anhelante de Jimmy en la parada de guaguas, después de haberme pasado toda la mañana en clase batallando con los teoremas de los triángulos congruentes.
Los fines de semana, Mami nos llevaba a la playa en Coney Island. Cargados con mantas; neveritas llenas de hielo, bebidas y comidas; un montón de toallas y un par de cubos y palas de plástico, marchábamos en tropel hasta el _subway_ , lleno de gente con cargas parecidas a las nuestras. Una vez el pasadía empezó allí mismo, cuando una nena se quejó de que tenía hambre y en un dos por tres, todo el mundo empezó a sacar pollo frito y ensalada de papa y a ofrecerle a gente que no conocían pero que estaban deseosas de compartir también su ensalada de repollo y sus sándwiches de jamón y queso.
La larga calle que llegaba hasta la playa estaba bordeada de kioscos donde se vendían _hot dogs, hamburgers_ , refrescos, helados, periódicos y revistas, protector solar y peluches. Había un ancho paseo tablado con juegos y más puestos de comida y lo mejor, un parque de diversiones con excitantes machinas y la mundialmente famosa montaña rusa.
No me dejaban comprar nada en los kioscos porque eran muy caros, ni podíamos ir al paseo tablado porque nos podía pasar "algo," ni al parque de diversiones, de donde salían gritos de terror cada dos o tres minutos cuando el carrito de la montaña rusa subía y bajaba por los rieles. Agarrados de la mano, luchábamos con el gentío hasta llegar a la playa y una vez allí, nos íbamos empujando hasta encontrar un pedazo de arena lo suficientemente grande como para acomodar las cosas y para abrir un par de toallas donde cupiéramos un adulto y siete niños.
Ninguno de nosotros sabía nadar, así es que buscábamos un sitio cerca del salvavidas, a pesar de que nos preguntábamos cómo podría darse cuenta de quién se estaba ahogando, entre las miles de personas que gritaban dentro y fuera del agua, porque era divertido gritar cuando se entraba y se salía del mar.
Para que los más chiquitos pudieran jugar en las olas, alguien los velaba mientras uno de nosotros se quedaba en la manta, cuidando de que nadie se fuera a llevar la neverita de comida, el monedero de Mami o nuestra ropa. Generalmente, me ofrecía yo para hacerlo porque la playa, con sus olas batiendo interminablemente, me aterrorizaba. La única vez que me había metido en el mar helado, a brincar las olas con Delsa y Norma, una marejada enorme me había empujado al fondo y me había arrastrado hacia adentro. Me rescató, no el salvavidas musculoso que nunca vio que me estaba ahogando, sino mi mamá y alguien que estaba cerca, que me sacaron escupiendo y tosiendo y casi muerta de la vergüenza.
Una vez, después de un día de playa, convencimos a Mami de que nos llevara al parque de diversiones. Empacamos las cosas, nos turnamos cargando la neverita y la frisa y fuimos de una machina a otra, tratando de decidir en cuál nos hubiéramos montado de haber tenido permiso, cuando de pronto, Mami se dio cuenta de que "algo" había pasado: faltaba Edna. Volvimos sobre nuestros pasos, la llamamos, buscamos en círculos cada vez más cerrados, alrededor de donde uno de nosotros se había quedado rodeado de nuestros tereques. Mami estaba histérica, llamando a un policía, pero no se veía uno por ninguna parte. Finalmente, apareció y, entre sollozos, le explicamos que Edna se había perdido, se la describimos y esperamos a que él la encontrara. Nos indicó que nos quedáramos donde estábamos, despareció por unos momentos y volvió para informarnos que ya había "dado parte", gestión que no satisfizo a Mami que gemía y le reclamaba que él no estaba haciendo nada, mientras su niña se encontraba en peligro mortal. Algunas personas se acercaron. Les explicamos lo que había pasado, cómo estaba vestida Edna y dónde la habíamos visto por última vez. Algunos hombres y muchachos salieron a buscarla, mientras sus esposas o novias se quedaron con nosotras pasándole la mano por los hombros a Mami y asegurándole que todo iba a salir bien.
De pronto, se abrió la multitud. Un enorme caballo castaño, montado por un corpulento policía, galopaba hacia nosotros. Sentada frente al policía, el rostro extasiado, venía Edna. El oficial se la entregó, a Mami, que la abrazó, la besó, le dio las gracias al policía, a la gente, a Dios y a las once mil Vírgenes por salvarle a la nena, mientras nosotros agobiábamos a Edna preguntándole cómo había sido montar a caballo.
_"It was fun,"_ dijo, "pero el pelo me hacía cosquillas en las piernas."
En casa, hicimos bromas y nos reímos de la aventura de Edna, pero durante algunas noches yo fantaseé con que un hombre guapo, en uniforme y montado a caballo me rescataba. Imaginaba el viento levantándome el cabello, sus brazos alrededor de mi cintura, el pelaje del caballo haciéndome cosquillas en las piernas desnudas. Me aferré a la imagen del policía y su caballo como si hubiera sido un regalo y dejé fuera las letanías de Mami, de todos los "algos" que hubieran podido pasar si no hubiéramos encontrado a Edna a tiempo.
Mami, definitivamente, había dejado el luto cuando quiso ir a bailar.
"Va un grupo de la factoría," le dijo a Tata.
"Jum," respondió Tata, lo que quería decir "a mí no me importa que vaya el mundo entero, tú no vas."
"Toca Tito Puente," añadió Mami medio indiferente. Tata chupó su cigarillo.
Desde donde yo estaba sentada leyendo, veía a Mami sacando y organizando medias y ropa interior del canasto de ropa limpia. A cada rato, levantaba la vista y miraba a Tata como para tantear cómo estaba de humor. Era gracioso verla comportándose como lo hacía yo cuando quería algo: las indirectas no muy sutiles, la justificación, "todos mis amigos lo hacen," la mención de alguna celebridad. Tata quedaba tan poco impresionada con las técnicas de Mami, como Mami con las mías.
Después de un ratito, empezó de nuevo. "A las nenas hay que exponerlas a estas experiencias pa' que aprendan cómo comportarse."
Tata viró la cabeza lentamente hacia Mami y la miró fijamente. "¿Tú quieres exponerlas al ambiente de un _nightclub_ para que aprendan a comportarse?" preguntó Tata, cada palabra enunciada con tal claridad que hubiera podido ser una de las alumnas estrellas de la clase de Voz y Dicción de la Dra. Dycke, si la Dra. Dycke hubiera hablado español.
"Negi está estudiando pa' artista, debería conocer otros artistas," dijo Mami, inspeccionando un par de medias.
"Esos sitios no son pa' mujeres decentes," concluyó Tata después de un ratito y pareció que sería el fin del asunto porque Mami se levantó, metió las medias enrolladas como bolas en la canasta y se fue a distribuirlas por los gaveteros correspondientes.
No se dio por vencida. Durante muchos días jeringó a Tata hasta que ella aceptó quedarse con los nenes. Tata vivía con nosotros y casi nunca salía del apartamento, así es que no es que fuera a tener planes para el sábado por la noche. Pero Mami nunca dio por sentado que por estar allí, Tata fuera nuestra niñera, y nunca salió del edificio sin comunicarle a Tata para dónde iba y a qué hora regresaría.
El sábado por la noche, mientras nos estábamos arreglando, mis hermanas y hermanos entraban y salían del cuarto, dándonos opiniones que nadie les había pedido, sobre qué hacer con el pelo, el maquillaje y la ropa.
Estaban tan excitados como nosotras, como si el ver a Mami tan contenta y arreglada, por primera vez en dos años, fuera motivo de celebración. A la hora de irnos, Don Julio insistió en acompañarnos a la estación. Nos veló mientras subimos la escalera hasta la plataforma del tren y esperó hasta que nos perdió de vista, para regresar a la Avenida Pitkin.
El Club estaba en el Upper West Side. Era un área parecida a nuestro vecindario, con negocios en la planta baja, al mismo nivel de la calle, y apartamentos en los pisos superiores de los edificios de cuatro y cinco pisos. Cuando salimos del _subway_ , oímos una música que salía de unas ventanas oscurecidas con tinte negro en los altos de un restaurante. Frente a la puerta que daba al Club, había unos hombres zanganeando, vestidos con camisas planchadas metidas por dentro, pantalones de correa, costuras gruesas y filo bien marcado. Nos miraron de arriba a abajo y mascullaron unos piropos. Mami me agarró de la mano, me haló hacia ella y prácticamente me arrastró hacia adentro por una empinada escalera, hacia una música ensordecedora. Nos selló las manos una mujer gruesa en un vestido corto, estrecho y escotado, que dejaba ver más carne que la que tenía yo en mi cuerpo entero. Al entrar al Club, Mami giró el cuello para todas partes con una expresión de pánico en la cara, como si después de haber llegado tan lejos, tuviera dudas de que fuera tan buena la idea. Agarrándomela con tanta fuerza que la mano se me adormeció, me remolcó por entre el revolú de gente, buscando a sus amistades. Cuando alcanzó a verlas, me aflojó un poco la mano y yo estiré los dedos hasta que logré volver a sentirlos.
Nunca había estado en un salón tan grande, con tanta gente, con tantos perfumes y colonias para después de afeitarse, mezcladas con el agrio olor a humo de cigarrillos, a "esprei" de pelo, a ron y sudor. Las mujeres tenían puestos trajes de brillo y los hombres usaban el pelo liso y reluciente. Las prendas resplandecían en la oscuridad. El vaporizo caliente de tanto cuerpo junto mareaba.
La pista de baile quedaba en el medio del salón. Estaba atestada de hombres y mujeres cuyas caderas parecían estar desprendidas de sus torsos y cuyos brazos ondulaban para dentro, para fuera, alrededor de los cuerpos, como serpientes en su nido. Mami me presentó a sus amigos, pero la música estaba tan alta que no entendí ninguno de los nombres. Parecía ser que la mujer del traje de lentejuelas verdes estaba con el hombre de la guayabera crema y la mujer del vestido de tafeta rosa estaba con el hombre trigueño del traje azul celeste.
Tan pronto nos sentamos, dos hombres extendieron sus manos frente a nosotras. Miré a Mami para estar segura de que podía aceptar y ella me dio permiso con la cabeza y se levantó a bailar con su pareja, un hombre bajito y redondo con una herradura de pelo alrededor de una coronilla brillosa. Mi pareja era más joven, flaco y olía a cigarrillo y colonia dulce.
Nunca había visto bailar a Mami, ni tenía idea de dónde había aprendido, pero lo hacía bien. Con los labios entreabiertos en una media sonrisa, los ojos ardientes y las mejillas encendidas, daba vueltas y giraba, según su pareja la iba guiando, de aquí para allá, la traía hacia él y la volvía a hacer dar vueltas en un apretado círculo. Me confundía verla sonreírle a hombres que no conocía, que le cogían la mano, le ponían la suya en la espalda y la tomaban por el codo para guiarla hasta la mesa.
Si ver a Mami bailando era nuevo para mí, estar tan cerca de hombres extraños era aún más nuevo. A pesar de que tenía dieciséis años y era casi una mujer, nunca había tenido novio ni me había besado nadie que no fuera pariente. No me consideraba fea, pero nadie me había dicho linda. En casa, a mis hermanas Delsa y Norma, con frecuencia les decían que eran bonitas mientras que a mí me decían que era "inteligente."
Pero, en la pista de baile, la mujer que sabe bailar es hermosa y el hombre con caderas sueltas y sabor, es guapo, independientemente de sus facciones o del cuerpo que tenga. Cuando mi pareja me sacó y me guió por los complejos pasos de una salsa, me sentí bella por primera vez en mi vida. No era por lo que tenía puesto, ni por la cantidad de maquillaje que logré ponerme sin que Mami protestara. La sensación me venía del calor que generaba el propio baile y no tenía tanto que ver con cómo me veía, sino, con cómo me movía. Me convertí en los ritmos complejos, consciente solamente de la alegría de moverme libremente, con gracia, dentro y fuera de los brazos de un hombre que nunca había visto, al ritmo de una música que nunca había oído.
Bailé con muchos hombres: bajitos, altos, flacos, viejos, jóvenes, trigueños, pálidos. Mami también. A veces, el hombre que me sacaba a mí, la sacaba a ella para la próxima pieza. O bailaba con ella y después me sacaba a mí y a través de gestos con la mano y movimientos exagerados de los labios, que me recordaban a La Muda, me felicitaban por tener una mamá tan linda y a ella, por tener una hija que bailara tan bien.
La banda tocaba _sets_ largos, a todo volumen. Me sorprendió que fuera realmente Tito Puente. Yo creía que Mami se lo había inventado para impresionar a Tata, que era fanática suya. Cuando la orquesta de Tito Puente cogió un descanso, otra orquesta tocó unas piezas más suaves como para darnos un descanso a los que estábamos bailando, con dos o tres boleros, antes de que volviera a empezar la salsa.
El baile duró hasta la madrugada y cuando salimos, yo estaba prácticamente sorda y tan sedienta que la lengua se me pegaba al cielo de la boca. Los amigos de Mami nos invitaron a una cafetería que abría toda la noche, un poquito más abajo del Club, y nos acomodamos en dos mesas con vista a Broadway y comimos huevos fritos, _pancakes_ con sirop y mucho café. A cada rato Mami me miraba para ver si me veía demasiado extenuada, pero yo estaba tan contenta, que mi única preocupación era que fuéramos a parecer unas jíbaras, vestidas de fiesta, en el _subway_ , a las seis de la mañana.
Nos despedimos de los amigos que nos informaron que habría otra fiesta la semana siguiente en otro lugar. "Ya veremos," respondió Mami.
Llegamos a casa cuando los demás se estaban levantando, con sus preguntas de cómo lo habíamos pasado y si yo había encontrado novio. Tomamos más café con los muchachos y con Tata, que miraba enojada nuestro pelo revuelto, el maquillaje corrido y la ropa sudada.
"Sí, lo pasamos bien," contestó Mami, "la próxima vez, Delsa y Norma deberían venir también, ¿no te parece?"
Le dije que estaba segura que lo disfrutarían, pero que yo esperaba que por ser la mayor, me llevara a mí también. Sonrió y se arrastró hasta la cama. Yo me quedé con Delsa y Norma, contándoles los más mínimos detalles de todo lo que había visto y hecho esa noche. Estuvimos de acuerdo en que, si Mami nos iba a llevar a bailar, teníamos que practicar en casa y les prometí que les enseñaría los pasos nuevos que había aprendido con mis parejas. Cuando finalmente me acosté, estuve despierta mucho rato, repasando cada momento de esa noche, mientras mi cuerpo se contraía con movimientos involuntarios de energía contenida. Me dormí arrullada por el recuerdo de la melodía de un bolero, segura de que nunca había sido tan feliz como esa noche.
Ahora íbamos a bailar casi todos los sábados. Aunque Delsa y Norma tenían solo catorce y trece años, no era raro que hubiera muchachitos más pequeños en los clubes. La gente traía a la familia completa: mamás, papás, abuelas, nenes tan chiquitos que andaban dando pinitos y moviendo sus culeritos al son del merengue, mientras los demás los animaban con palmadas.
Se pagaba medio precio por los niños menores de dieciocho años y para los menores de doce, era gratis. Así era que Mami podía traernos a las tres mayores y hasta a Héctor, de vez en cuando. Las bebidas se vendían a la carta o se podía pedir un "servicio" que consistía de una botella de ron, dos _Coca-Colas_ , una hielera, vasos plásticos y unos pedacitos de limón. Había unas mesas largas acomodadas alrededor de la pista de baile, ocho sillas plegadizas por mesa, un montón de servilletas de coctel y dos ceniceros de aluminio en cada una. Algunos sitios vendían frituras como alcapurrias y pastelillos, o bolsitas de papitas fritas y _cornchips_. Pero la mayoría, sólo servía bebidas. La única manera de garantizar que conseguiríamos una mesa era ordenando un servicio, así es que, si veníamos solas, lo pedíamos, nos tomábamos las Coca-Colas y le llevábamos el ron a Tata y a Don Julio.
Sólo íbamos a los clubes donde hubiera música en vivo, generalmente en el Upper West Side o en el Barrio, pero nunca nos aventurábamos por el Bronx o Queens porque Mami no conocía bien esos precintos. En algunos bailes nos encontrábamos con gente que habíamos conocido en otros clubes y a veces nos topábamos con algún muchacho que bailaba particularmente bien. Delsa, Norma o yo, le decíamos dónde íbamos a estar la semana siguiente. No los considerábamos novios, las únicas veces que los veíamos era en los clubes y esa relación era monitoreada por Mami, quien con una mirada o un gesto de la boca nos hacía saber que nos estábamos poniendo demasiado zalameras, y que era mejor que nos recogiéramos o nos íbamos para casa.
Aunque Tío Chico me había tocado un seno una vez y había visto varios penes desvalidos colgando de las braguetas abiertas de los exhibicionistas y el pene triunfalmente erecto en la falda de un camionero descarado, nunca había estado tan cerca de los hombres como lo estaba en la pista de baile. Algunos bailaban tan pegados que tenían una erección. Cuando nos enfrentáramos a esa situación, se suponía que les diéramos el beneficio de la duda. Si se separaban abochornados, se trataba de un accidente. Si se pegaban más, eran unos frescos. Les decíamos "rompemedias" porque bailaban tan pegados que nos rompían las medias. Un hombre que se pusiera fresco se corría el riesgo de que lo dejaran plantado en la pista. Un hombre solo en la pista sería notado por todo el mundo y de seguro, tendría problemas para conseguir otras parejas. Así es que la mayoría de los hombres eran corteses y mantenían la distancia mientras lograban también bailar un bolero con el fuego suficiente para hacer ruborizar a una puta.
Estaban también los pulpos, cuyas manos, en vez de guiarnos en complicadas combinaciones de baile, se paseaban por nuestras espaldas, bajaban hasta las nalgas, subían por debajo de nuestros brazos, cerca de los senos, mientras sus piernas trataban de insinuarse entre las nuestras. A estos hombres, también había que evitarlos.
De vez en cuando, no me retiraba cuando un hombre se excitaba. Estaríamos bailando una pieza suave y si sentía que se le estaba parando, me le pegaba más para ver su reacción. Si se me tiraba encima agresivamente, o si sus manos y piernas de pulpo se extraviaban hacia donde no debían, me separaba, porque ya no me parecía que le estaba dando algo, sino que él se creía con derecho a ello. Me gustaba el hombre que se quedaba sin aliento de la sorpresa, que me atraía hacia él con ternura, y en círculos lentos y discretos, movía sus caderas pegadas a las mías, sin perder el paso. Saboreaba el poder que me daba el ser capaz de excitar a un hombre, el sentir su aliento cálido junto a mi oreja, lento al principio, después más intenso, más caliente, nuestros cuerpos pegados en un todo sinuoso que se movía rítmicamente a través del salón caliente. Perdía todo sentido del tiempo, abrazada y abrazando, hermosa, elegante, temblando con unas sensaciones que sólo eran posibles de esta manera, en este lugar.
Al terminar el bolero, mi pareja quería quedarse siempre para la próxima pieza, pero yo insistía en que me llevara a la mesa. No confiaba en los sentimientos que me hacían bailar de ese modo, me avergonzaba de haberme dejado ir tanto. Rehusaba mirarlo a los ojos, temerosa de lo que vería en ellos. Si me invitaba a bailar de nuevo, me negaba o le decía que sólo bailaba piezas movidas. Nunca reconocía haber tenido parte en lo que habíamos hecho.
Más tarde, la vergüenza era sustituida por el placer de su cuerpo contra el mío, su rostro, una mancha anónima, hasta que todo lo que quedaba era el estremecimiento de la piel, el fogaje entre mis piernas y el ritmo lento del bolero.
A veces, a pesar de los esfuerzos de Mami de protegernos de la violencia del mundo, "algo" sucedía. Sufrimos el asesinato del Presidente Kennedy con el resto del país y sollozamos cuando John John saludó el paso del féretro de su padre. El radio y la televisión nos trajeron la noticia de cómo por lo menos treinta vecinos escucharon los gritos de Kitty Genovese mientras la mataban a puñaladas y nadie salió en su auxilio. Durante semanas, Mami entraba en crisis sólo con que bajáramos a la pizzería. Pero, ella no era la única que se preocupaba. Cuando se desmontaba del tren al regresar del trabajo, allí estaban Don Julio o Héctor, esperando para acompañarla a casa.
El suceso más alarmante del verano del 1964 fue que vecindarios completos como el nuestro se volvieron en contra de ellos mismos. Lo leíamos en el periódico, lo oíamos por radio, veíamos las imágenes borrosas en negro y blanco de gente que se parecía a nosotros, corriendo por unas calles semejantes a las nuestras, prendiendo fuego, golpeándose unos a otros, siendo perseguidos por policías —algunos montados a caballo como el que rescató a Edna. Los oficiales —todos hombres blancos— arrastraban a los amotinados de piel oscura por las aceras repletas de basura y vidrios rotos. Los apaleaban con sus macanas, los metían a empujones en los carros patrulla y se los llevaban, mientras una multitud que gritaba y maldecía los seguía, sus caras retorcidas en una mueca.
Mami se negaba a ir a trabajar después de que se reportaba un motín y yo no iba a mis clases de verano. Nos ahogábamos de calor en el apartamento, pero no nos dejaban salir. Don Julio, cuyas hijas ya adultas vivían en otra parte de Brooklyn, nos traía noticias de edificios quemados y de turbas que rompían los cristales de las tiendas y cargaban con todo lo que se podían llevar.
"Yo vi a un hombre salir corriendo con un televisor a color," decía, "y una mujer y tres muchachitos arrastraron un sofá de una mueblería y entraron de nuevo a coger una mesa y unas sillas."
A nosotros nos parecía comiquísimo, pero a Mami no. "Desordenados," refunfuñaba, "si yo llego a coger a alguno de ustedes haciendo algo así..."
Una noche calurosa, acabándonos de acostar, escuchamos gritos, cristales rotos y alarmas activadas.
"¡Quítesen de ahí! ¡Apaguen las luces!" gritó Mami cuando nos vio asomados por la ventana averiguando qué pasaba. Dos cuadras más abajo una turba corría hacia la Avenida Rockaway, armada con bates y aros de goma, dándole a todo lo que encontraba a su paso. Mami nos empujó hacia el medio del cuarto, se aseguró de que las puertas estuvieran cerradas con seguro y levantó el teléfono.
"¿Llamamos a la policía?" pregunté, lista para servirle de intérprete.
"No," susurró. "Quería estar segura de que funciona."
Nos apiñamos en la oscuridad, pendientes por si oíamos pasos por la escalera, el crujir de madera astillada, alguna explosión o cualquier cosa que nos avisara que la violencia había tocado a nuestra puerta. Cuando oímos las siernas de la policía, Héctor se arrastró hasta la ventana, se asomó con cuidado y gateó de vuelta.
"Los cogieron allá abajo. No está pasando na'."
Después de un rato, Mami fue a averiguar. Yo la seguí, a pesar de que nos susurró que nos quedáramos donde estábamos. La calle estaba desierta. A un bloque de distancia un par de carros patrulla se había estacionado en el medio de la calle con los biombos encendidos y los radios prendidos en su cháchara monótona. Para el otro lado había más carros patrulla, pero no había gente. Las aceras estaban asquerosas de la basura de los zafacones que la gente había volcado y de relucientes fragmentos de vidrio. Mami cerró las ventanas y las cortinas.
"To' el mundo pa' la cama," ordenó.
Fue imposible dormir. Durante horas, el chirrido de las alarmas de las tiendas nos mantuvieron despiertos. También, la espera. Estaba segura de que si me quedaba dormida, me despertaría a mitad de noche en medio de un fuego o de una turba saqueando la farmacia de la planta baja. Pero, amaneció y no había pasado nada más. Los comerciantes cubrieron las ventanas rotas de sus negocios con planchas de madera donde le garabatearon o pintaron "abierto para negocio" o "cerrado hasta nuevo aviso," dependiendo de los daños. Cuando entrábamos a comprar algo, nos miraban con desconfianza, como si hubiéramos sido parte de la violencia, pero nosotros les devolvíamos la mirada, ofendidos de que no hubiera nadie immune a su sospecha y coraje.
"Tan pronto ahorre lo suficiente pa' los dos meses de renta y la fianza," suspiró Mami, "nos mudamos."
La noche antes del examen de geometría Regents de mi clase de verano, Ray Barreto tocaba en un club del Barrio.
"Quédate en casa y ponte a estudiar," me dijo Mami.
"No tengo que estudiar. Saqué buenas notas en todos los exámenes."
"Pero si te cuelgas en éste, tienes que volver a coger la clase."
"No me voy a colgar. Me lo sé todo."
"Bueno, está bien, si estás segura."
"Estoy segura."
Bailé hasta que me dolían los pies. Bailé hasta que me quedé ronca de tanto gritar por encima de la música para que me oyeran. Bailé hasta que me ardían los ojos del humo en el salón y del maquillaje derretido que me caía en ellos. Bailé hasta que los tímpanos me vibraban como las congas de Ray Barreto. Cuando terminó la música salimos con el grupo y caímos en medio de un motín.
Exhaustas y vestidas en trajes de noche, tacos altos y prendas de fantasía, pero vistosas, nos pegamos contra la pared del edificio y vimos pasar a un montón de hombres cargando bates, palos, tapas de zafacones. La entrada del _subway_ quedaba a medio bloque, en dirección contraria y tan pronto aminoró la turba, corrimos hacia la entrada y justo habíamos logrado bajar las escaleras cuando apareció por la esquina otro grupo furioso. La gente que había estado en el baile estaba ahora abajo, en la estación del tren, y había rumores de que la turba vendría por nosotros. Los hombres, en sus guayaberas y trajes color pastel, formaron una línea frente a las mujeres y a los niños, quienes se concentraron en la punta más lejana de la plataforma y buscaban con ansiedad alguna señal del tren en el oscuridad del túnel. Encima de nosotros, las alarmas chillaban, la gente gritaba y maldecía, los carros se detenían de golpe, el cristal se quebraba y objetos pesados caían al suelo. Después de mucho rato, aullaron las sirenas. Cuando llegó el tren finalmente, corrimos a entrar. Nos empujamos hasta el rincón más distante y no nos sentimos seguras hasta que se cerró la puerta. Tan pronto se movió el tren, todo el mundo se relajó y nos reímos de la tontería de huir de una gente que no tenía ningún interés en nosotros. Pero era una risa forzada; nuestro miedo era real; y aunque Mami, Delsa y yo, nos reímos como todo el mundo, en casa no mencionamos nada de lo que nos había pasado. Nos había tocado demasiado de cerca como para cogerlo a broma.
Tres horas después de regresar a casa de nuestra noche de Ray Barreto y su secuela, mordía yo un lápiz, tratando de recordar qué eran postulados y por qué x era igual a z. Pero, fue inútil. Las fórmulas, los teoremas y las hipótesis, habían huido. Fracasé el examen Regents por segunda vez, lo que quería decir que tendría que repetir geometría por tercera vez en dos años.
Una vez empezaron las clases, se acabaron nuestros fines de semana de bailes. Tomaba demasiado tiempo y dinero preparar a siete niños para la escuela. Compartíamos la ropa y nuestros parientes nos traían la que ya no usaban, pero Mami siempre nos compraba ropa y zapatos nuevos para nuestro primer día de clase. A nosotras nos compraban bultos para los libros y nos mandaban a hacer peinados nuevos en el "biuti." A los varones los mandaba a recortar y les compraba pantalones nuevos, camisas blancas y corbatas para las asambleas. Pero, la ropa no era el único gasto. Había que comprar lápices, plumas, papel de rayas, cartapacios, papel de construcción, Crayolas, cinta adhesiva y pega. Las maestras mandaban listas con los demás materiales que necesitábamos; uniforme de educación física, tenis, mapas, reglas, libretas de dibujo, diccionarios. A eso había que añadirle el dinero del pasaje para mí y para Delsa, que acababa de entrar a escuela superior, y una pequeña mesada para un refresco o una taza de café. Según enfriaba el tiempo, necesitábamos abrigos, guantes, botas, gorros.
A veces, ni con un trabajo a tiempo completo, ni con las horas extra que lograba acumular, alcanzaba Mami, a cubrir todos los gastos del regreso a la escuela. Además de renunciar a nuestros bailes los fines de semana, renunciamos al teléfono, a las comidas enlatadas, a los dulces. Y cuando todavía no nos alcanzaba, fuimos hasta las oficinas del _welfare_ a solicitar ayuda de emergencia para ropa de invierno o la electricidad o para pagar un par de meses de renta en lo que las cosas se normalizaban. Mami tuvo que perder un día de trabajo y la paga de ese día para que le dijeran que el _welfare_ no le daba a menos que no estuviera desempleada. Una vez, un empleado del _welfare_ me preguntó por qué yo no le ayudaba a mi mamá.
"Estoy en escuela superior," le contesté sorprendida.
"Puedes trabajar a tiempo parcial."
"No tengo tiempo, la escuela me queda a hora y media de casa."
"¿Qué es lo que pasa?" preguntó Mami, cuando se dio cuenta de que el empleado había dejado de hacerle caso a ella para discutir conmigo.
"Dice que debo conseguirme un trabajo."
_"She job school,"_ le informó Mami al empleado. De vez en cuando decía frases muy efectivas en inglés. Cuando lo hacía y la persona con quien estaba hablando la entendía, resplandecía de orgullo, pero Mami no seguía para no dañarlo. Cuando regresábamos a casa le comenté a Mami que quizás debía buscarme un trabajo después de clase.
"No te quiero en la calle después que oscurezca."
"Pero, Héctor trabaja."
"Él es casi un hombre. No es lo mismo."
Héctor tenía doce años, era largo y flaco, y por lo que yo podía ver, no era casi un hombre. Pero era macho y yo era hembra y ahí estaba la diferencia. Con todo lo que se necesitaba el poco o mucho dinero que yo hubiere podido aportar, no valía la pena que me arriesgara a estar lejos de casa después del anochecer. "Algo" me podía pasar.
# "Ella no es exactamente Método."
#
Quise hacer de Scout en _To Kill a Mockingbird_ de Eliza en _Pigmalion_ , de Laura en _The Glass Menagerie_ y la Antígona de Sófocles. Pero, una vez más me asignaron el papel de Cleopatra, esta vez con William haciendo de Julio César.
Harry, mi primer César, caracterizaba a su personaje como un soldado, con mucha fanfarronería y pavoneo de macho. El César de William era el emperador acostumbrado a ser obedecido. Saqué el vestido de mantel amarillo y traté de buscarle un nuevo ángulo al mismo rol. Mi primera Cleopatra había sido mimosa y coqueta; ésta, decidí, sería una reina astuta, "artimañosa," enfrentando a un oponente igual de calculador.
Se me hacía difícil entrar en personaje. Las pocas veces que había tratado de salirme con la mía y hacer lo que no debía, me cogieron; así es que la astucia no se me daba con facilidad.
"Ahí es donde entra la actuación," afirmó Laura Figueroa cuando le expliqué mi dilema. Era una de las mejores actrices en la clase, capaz de hacer cualquier papel, cualquier acento, desde clásico hasta contemporáneo. Su especialidad, sin embargo, eran las viejitas. No es que las prefiera, pero cada vez que nos asignaban un papel, yo era Cleopatra y ella era Vieja. "A lo mejor puedes moldear su carácter usando a alguien que conozcas." Sacudí la cabeza. "Bueno, pues, una combinación de gente."
"Me hubiera gustado ver la Cleopatra de Elizabeth Taylor."
"Te podría servir, pero ella no es exactamente Método."
En Performing Arts, despreciábamos a los actores de cine. Se decía que "indicaban," que sus actuaciones dependían de miradas y pequeños gestos faciales que con frecuencia no eran más que manerismos. Apenas trabajaban con sus voces y parecían preocuparse más por cómo se veían que por crear un personaje.
Era crucial para nuestro desarrollo como actores, se nos decía, aprender a diferencia entre estar pendiente de uno mismo y ser consciente de uno mismo. Un actor pendiente de sí mismo era demasiado intenso; demasiado vigilante de su propia actuación. Los actores conscientes de sí mismos confiaban en que las semanas o meses de preparación para un papel les ayudarían a convertirse en el personaje, a la vez que mantenían también el nivel de agilidad mental que les permitiría reaccionar a los demás actores y a la situación particular que se estuviese representando. Una representación, nos decían, es algo vivo que va cambiando y desarrollándose cada vez que el actor pisa el escenario.
Entendía los conceptos, observaba y guaradaba situaciones y momentos en mi "memoria de los sentidos" para usarlos más tarde, cuando tuviera que recurrir a ellos. Pero estaba convencida de que mi vida no me proveía variedad suficiente para hacerme una buena actriz. ¿Cómo me la iba a proveer, si cada paso que daba estaba monitoreado por Mami? Y las veces que yo nada más que consideraba ir en contra de su voluntad, una vocecita se activaba en mi cabeza para recordarme que entre ella y el resto del mundo no había nada más que miradas hostiles y pocas expectativas. Si caía, sólo mi madre estaría allí para recogerme. Sí, tenía siete hermanas y hermanos, pero eran más jóvenes e indefensos. Sí, estaba Tata, pero con frecuencia estaba borracha. Estaban los demás parientes; pero ellos tenían sus propios hijos e hijas, sus propias vidas, sus propios problemas. Estaba mi padre, lejos en Puerto Rico, con su nueva esposa, sus nuevos hijos, su nueva vida. Y a los diecisiete años no quería estar sola. Todavía no.
Mi maestra de Drama, Misis Provet, me llamó a la oficina un día para preguntarme si me interesaba trabajar como ujier en un teatro durante los fines de semana.
"No puedo trabajar de noche."
"Es los domingos por la tarde."
Acepté, contenta de que finalmente ganaría dinero, ansiosa de exponerme al teatro, aunque fuera de ujier. Vería actores trabajando, estudiaría sus técnicas y quizás aprendería algo nuevo.
El empleo no requería entrevista. Tenía que presentarme a trabajar a medio día el domingo siguiente. La dirección era en "Loisaida." Salí de la estación del tren y caí frente a una fila de edificios deteriorados de dos y tres pisos; no había una sola marquesina de teatro por todo aquello. Me orienté, volví a leer la dirección, caminé de arriba a abajo por el bloque hasta que encontré un toldo todo deshilachado sobre una entrada oscura que daba a una escalera. Estaba bien nerviosa, renuente a entrar, insegura de dónde estaba. Los "algos" que podían pasarme en los pasillos oscuros de edificios extraños resonaban en mi cabeza, pero los acallé, cogí aire y subí. Al final de la escalera había tres puertas y la ventanilla de la boletería. Toqué a la puerta donde decía "Oficina" y me recibió un señor de barba, vestido de negro.
"Soy de Peforming Arts," me presenté. "La ujier."
"Ah, sí," me dijo. "Venga conmigo." Me condujo por el pasillo. "Soy el Sr. Rosenberg," dijo, mientras abría las hojas de las puertas centrales.
"Voy a estar vendiendo los boletos y usted va a estar aquí mismo." Se dio cuenta de que vacilé, miró hacia adentro y notó que el cuarto estaba oscuro. "Ay, perdón," dijo, encendió la luz y nos encontramos en la parte de atrás de un pequeño teatro.
"Los números están en los brazos de las butacas. ¿Ve? Acompañe a las personas hasta sus asientos y asegúrese de que tengan el programa." Rebuscó por detrás de la cortina que estaba contra la pared de atrás y sacó una caja llena de papeles escritos en una letra que no entendía, pero que sabía era hebreo. Los caracteres negros eran similares a los que había en las vitrinas de las tiendas por todo Nueva York, y había aprendido que significaba que el establecimiento servía comida _Kosher_ , aunque no estaba muy segura de qué cosa era comida _Kosher_.
"Dóblelo por la mitad, así," me enseñó. "La obra empieza dentro de una hora. Puede verla desde cualquier butaca que esté libre acá detrás." Sacó una linterna de detrás de la cortina y me la entregó. "Si alguien llega tarde, espere a que haya un cambio de escena para traerlo hasta su asiento. Al final, asegúrese de que no olviden sus pertenencias."
Me senté en una de las butacas de madera que cerraban hacia arriba y doblé los programas. Cuando terminé, bajé por el pasillo escalonado hasta llegar al pie del escenario. No era muy alto, estaba, más o menos, al nivel de mi rodilla. La escenografía era la de una cocina en un apartamento. La puerta lateral derecha conducía a una escalera que bajaba fuera de escena, una ventana lateral a la izquierda llevaba a la escalera de escape. Una mesa cubierta con un mantel a cuadros, tres sillas, una estufa, cortinas en las ventanas y unos platos completaban la escenografía. Una bombilla sola iluminaba el escenario y sentí una emoción súbita. Éste era un escenario de verdad, en un teatro de verdad y estaba a punto de ver una representación en vivo, con actores de verdad.
Unos momentos después, se escucharon pasos por la escalera. El Sr. Rosenberg abrió las puertas y yo corrí a mi puesto en la parte de atrás del teatro, agarré un montón de programas y se los fui dando a la gente según me fueron enseñando sus boletos. La mayoría eran personas mayores, muy ordenadas y correctas, y muy agradecidas de que se les acompañara a sus asientos, aunque era obvio que estaban familiarizados con el arreglo de las butacas y sabían para dónde ir. Todo el mundo estaba tan empaquetado como nos empaquetábamos Mami, mis hermanas y yo cuando salíamos a bailar. Las mujeres tenían puestas pelucas, prendas, pieles. Los hombres usaban trajes y sombreros que colocaban en sus rodillas tan pronto se sentaban. Había un fuerte olor a bolas de naftalina, a cigarrillo y perfume.
Cuando bajaron las luces, me paré en la parte de atrás del teatro. Los actores entraron vestidos con la ropa que yo asociaba con los dueños de las tiendas de mercancía de segunda mano y los _deli's_ de la Avenida Graham. Hablaban yídish, un idioma que me resultaba familiar porque lo había escuchado en la _Marketa_ , en el sitio de cambiar cheques y en las calles de Williamsburg.
A pesar de que no entendía ni una palabra, me cautivó la acción en el escenario. El drama giraba en torno a una familia cuyo hijo se había alejado de las tradiciones que habían traído de su patria hasta los Estados Unidos. Se había enamorado de una muchacha norteamericana y su familia se negaba a conocerla.
Al final de primer acto, aplaudí vigorosamente con el resto del público. Una señora que estaba delante de mí, se viró y sonrió en dirección mía. Cuando me pasó por el lado durante el intermedio, me preguntó si entendía algo.
"No sé la lengua, pero puedo seguir la acción. Los actores son muy buenos."
"¡Estupendo!" me dijo y me dio una palmadita en la mano.
Cuando empezó el segundo acto, un nuevo personaje fue presentado y me sorprendió ver al Sr. Rosenberg en el escenario. Hacía de abuelo o de algún otro pariente mayor, y tuvo varias alocuciones muy vehementes. Al final del segundo acto pronunció un largo monólogo que provocó que al final, el público se pusiera de pie y dejó a todo el mundo, incluyéndome a mí, con lágrimas en los ojos.
En el tercer acto, el joven decidió no casarse con la muchacha americana y la obra terminó con la familia entera en escena, alrededor de velas encendidas y cantando un himno hermoso y solemne. En este punto, yo estaba ya sollozando y la señora que me había hablado antes se me acercó con un Kleenex.
"Teatro Yídish," dijo, abriendo sus brazos en un gesto dramático. "¡El mejor!" Le di las gracias, asentí con la cabeza y traté, lo mejor que pude, de vaciar el teatro, pero estaba tan afligida que me quedé sentada en la fila de atrás, llorando y sintiéndome estúpida porque no podía parar.
El Sr. Rosenberg salió por el lado derecho del escenario, saltó hasta el piso y vino hacia mí.
"Lo siento tanto," le dije. "No sé qué me pasó. Fue tan hermoso. Su actuación. La canción al final. Y las velas. No tengo idea de lo que usted estaba diciendo..." Seguía llorando a lágrima viva y me limpié la cara con el Kleenex, que para entonces estaba hecho trizas entre mis dedos.
"No es nada. Ya pasó," me dijo. "Es muy halagador," añadió con una sonrisa. Me ofreció presentarme a los actores y me llevó trasbastidores hasta sus camerinos. Le di la mano a cada uno y les dije cuánto había disfrutado la obra. Me miraron con curiosidad y la mujer que hacía de Mamma, me acarició el cachete. Estuve a punto de empezar a llorar de nuevo, pero el Sr. Rosenberg me sacó afuera.
"Regrese en dos horas," me dijo.
Fui ujier en otra función ese día y la obra volvió a conmoverme profundamente. Durante cuatro domingos corridos, vi a esos actores representar la misma obra, una función en matinée y la otra por la tardecita. Ninguna función era igual a la otra. Sus voces, sus gestos, el nivel de concentración, cambiaba la dinámica cada vez que decían sus líneas, lo que hacía que le obra variara y se renovara continuamente.
Hasta ese momento, la experiencia teatral era un concepto que se enseñaba en Performing Arts, no algo que yo hubiera vivido. Pero, ahora entendía, por fin, por qué mis maestros en Performing Arts, amaban tanto el teatro y por qué insistían en que los sacrificios valían la pena.
Después del sexto domingo, el Sr. Rosenberg me dijo que ya no me necesitaría más. "Estaremos ensayando durante un par de meses," me dijo, "entonces abriremos con otra obra." Me dio pena y le dije que llamara a la escuela cuando me necesitara. Un par de meses más tarde, lo hizo.
"Tráigase un pañuelo," me dijo, antes de despedirse. Lo hice.
A Mami le avisaron de un baile en la Armería de Park Avenue. "Hace tiempo que no vamos," pensó, "y las Navidades ya están ahí."
Guardé el dinero que me gané de ujier para un vestido y me dejaron ir de tiendas sola.
"Pero no llegues a casa con nada estrambótico," me dijo Mami.
"No quiero parecer una payasa... A lo mejor algo en negro."
"¡Ay, no! Negro no." Mami había salido de su ropa de luto hacía poco. Tenía miedo de que le trajera mala suerte si se quedaba con ella. Cuando le sugerí que la quemara, se azoró y entendí que el fuego implicaría cosas terribles para el pobre, difunto Francisco. Si le regalaba la ropa a alguien que no estuviera de luto, le traería mala suerte a la que la recibiera. Así es que la atacuñó en una bolsa, la amarró con unos cuantos nudos para que la mala suerte de la ropa no fuera a escaparse y la puso afuera con la basura.
"Rojo entonces, ya que es Navidad."
"¡Dios te libre!" saltó Tata. La ropa roja, según ella, traía periodos abundantes y en las mujeres en edad reproductiva, ocasionaba abortos.
"No tengo planes de salir encinta en buen tiempo," le recordé.
"Aun así."
"Cómprate algo con todos los colores del arcoiris," sugirió Edna.
"Olvídenlo, ya veré qué hago," contesté. Usé mi traje como excusa para ir de compras a Manhattan.
"¿Por qué no puedes comprarlo por aquí?" me preguntó Mami, "¿O en la Avenida Flatbush? Tienen montones de cosas lindas."
Pero, yo no quería comprarlo en Brooklyn. En Performing Arts había aprendido que Brooklyn no era la Ciudad de Nueva York. Hasta el propio alcalde lo llamaba "un distrito exterior." Manhattan era el centro financiero, teatral y artístico de los Estados Unidos. Quería estar en él, moverme de los márgenes al centro. Quería subir a la punta del Empire State Building, contemplar la ciudad y mirar más allá, al vasto horizonte que yo sabía que existía, pero que no podía ver desde las aceras de Brooklyn.
De todos los bailes a los que habíamos ido, el de la Armería fue el mejor. Tres bandas tocaron sin parar y había más puertorriqueños de los que nunca había visto en un salón tan grande. Después del baile, Mami, Delsa, Norma y yo caminamos por Park Avenue, buscando algún sitio dónde comer algo, pero no encontramos. La calle, dividida por una isleta con arbustos bajos y árboles flacos, era casi toda residencial y los negocios abrían sólo de día. Hambrientas como estábamos, nos gustó, sin embargo, caminar por Park Avenue y, con la pavera que traíamos del baile, nos reímos, bromeamos y jugamos a ser señoras ricas paseando por el vecindario. Dejamos la Armería atrás y caminamos con los abrigos sobre los hombros, como si fueran de visón, los tacos repicando y nuestras manos colgando lánguidamente de las muñecas, extendidas hacia un caballero invisible.
Unas luces brillaron detrás de nosotras. Un carro patrulla se detuvo, se bajó un policía y se nos acercó caminando to' espatarra'o por el medio de la calle, como un sheriff de película de vaqueros a punto de retar al ladrón de bancos a un tiroteo.
"Buenos días, señoras," nos dijo. "¿Puedo dirigirlas hacia donde van?" Su voz, con un tono de corrección fingida, le salió con una sonrisita burlona y sus ojos, invisibles bajo la gorra, eran como manos toqueteando cada pulgada de nuestros cuerpos.
Mami se colocó entre nosotras y el guardia, pero necesitaba que alguna de nosotras le tradujera. Su rostro cambió, en segundos, de la alegría al pánico, a la expresión dura y seria que ponía cuando estaba asustada, pero tenía que mantenerse fuerte por el bien de sus hijas. "Dile que sólo estamos buscando un sitio donde comer," dijo dirigiéndose hacia Delsa, Norma y a mí, a la primera que hablara. De un salto, quedé frente a ella, sonreí con mi sonrisa más encantadora y me convertí en Cleopatra, Reina del Nilo.
"Estábamos en un baile en Armería, oficial" enuncié claramente. "La noche está tan bonita, que decidimos volver caminando a casa."
"¿Ustedes viven por aquí?" Trató de intimidarme con la mirada; pero, aunque las rodillas me temblaban, mantuve mi noble porte.
"Sí, ahí mismo," señalé con todo mi brazo, con todo mi cuerpo, a un lugar lejano, mi palacio, hacia Brooklyn. Mami, Delsa y Norma me miraron como si me hubieran crecido cuernos y un rabo. Se ajustaron y abotonaron los abrigos y se quedaron paradas, humildemente, frente al policía, rogando que no hubiéramos hecho nada ilegal y que yo no lo estuviera empeorando, mientras yo rogaba que no me fuera a pedir una dirección.
"Estaban haciendo mucho ruido," su voz se suavizó. "Éste es un vecindario residencial." Ahora sí que descubrió a América el muy zángano, pensé.
"¿Qué?" gritó Mami en inglés, como si al hablar alto la fueran a entender mejor. _"What the matter?"_ Sonaba asustada, definitivamente no como alguien que viviese en Park Avenue.
"Nos denunciaron por hacer mucho ruido," dije en español, mi voz baja y tranquila." ¿Uno no puede hablar y reírse en Park Avenue?" replicó Delsa en español y yo le lancé una mirada de "cierra el pico." Me volví hacia el policía.
"Bajaremos la voz." Yo, la astuta Cleopatra, batí los párpados y sonreí altiva. "Sentimos tanto haber importunado a los vecinos. Vamos chicas." Les hice un gesto con la mano, empecé a caminar y le pasé por el lado. "Gracias," le dije, cuando el agente se movió para dejarnos pasar; Cleopatra, Reina del Nilo y sus fieles servidoras, que la seguían confundidas, no muy seguras de que la ley ya las hubiera despachado. Seguí caminando hasta que sentí que se cerró la puerta del carro patrulla y lo vi pasarnos por el lado y alejarse por Park Avenue. Tan pronto desapareció, estallamos de la risa.
"¿Cómo te atreviste?" se reía Mami.
"Yo no sé, no lo pensé... Lo hice." Doblamos en la esquina, para salirnos de Park Avenue, por si acaso el policía volvía a pasar a cotejar si entrábamos en alguno de los edificios del vecindario.
" _¡Wow!_ Negi, te la comiste, esas clasecitas de actuación te están sirviendo pa' algo," me dijo Norma. "¡Le hiciste creer que vivíamos en Park Avenue!"
Estaba alborozada. Acababa de actuar frente al público más critico y exigente que encontraría jamás y había recibido una crítica apoteósica.
Empezamos a salir a bailar de nuevo, pero no tan frecuentemente como en el verano, dos veces al mes, más o menos. En uno de los clubes, Mami conoció, bailó con y se enamoró de Don Carlos. Era flaco, de piel chocolate, sonrisa tímida, voz suave. Usaba siempre traje oscuro, camisa blanca, una corbata estrecha y unos espejuelos de carey rectangulares con lentes verdes. Yo pensé que era ciego. ¿Por qué otra razón usaría alguien espejuelos oscuros en un _nightclub_ , que ya de por sí tenía poca iluminación?
Bailó conmigo, con Delsa y con Norma, guardando siempre más de la distancia requerida, mirando por encima de nuestras cabezas en todo momento, como si mostrar cualquier interés en nosotras, más allá del mínimo que requería la cortesía, estuviera prohibido. Después nos llevó a cenar; nos preguntó de la escuela, lo que queríamos hacer cuando fuéramos grandes; las preguntas esperadas que los adultos que están tratando de congraciarse con los jóvenes siempre hacen. Norma y yo le preguntamos qué hacía él (contabilidad), dónde trabajaba (Xerox), si estaba casado (divorciado), cuántos hijos tenía (tres) y si alguna vez se enfermaba (muy rara vez), mientras Mami nos pateaba por debajo de la mesa por impertinentes. Pero éste era el único hombre por quien Mami había mostrado algún interés desde la muerte de Francisco.
Mami le debe haber dicho a dónde iríamos a bailar porque después del primer encuentro, cada vez que entrábamos en un club, ahí estaba Don Carlos. Delsa, Norma y yo, nos dimos cuenta enseguida de lo que estaba pasando, especialmente porque Mami no bailaba con nadie más. Don Carlos se sentaba en nuestra mesa y aunque la música era ensordecedora, él y Mami mantenían unas conversaciones animadísimas durante toda la noche, aun cuando estaban bailando.
Delsa, Norma y yo, relajábamos a Mami con que tenía novio y ella se ruborizaba y nos pedía que no le dijéramos nada a Tata. Nos gustaba compartir un secreto con ella, saber algo de su vida que nadie más sabía. Pero no nos gustaba que Don Carlos se sentara con nosotras. Los hombres creían que era nuestro papá y no nos sacaban a bailar.
"Se sienta ahí, con esos espejuelos oscuros, como un gángster o algo así y espanta a los muchachos," nos quejábamos con Mami.
La vez siguiente usó unos lentes regulares, gruesos como culo de botella, por los que miraba frunciendo los ojos como si la receta no fuera la suya. Tampoco ayudó. Ahora los hombres creían que Don Carlos estaba vigilándoles cada uno de sus movimientos.
Después de dos semanas de noviazgo, Mami invitó a Don Carlos, un domingo, a comer. Cuando Mami le anunció que venía un hombre de visita, Tata entrecerró los ojos, frunció la boca y se alejó de Mami sin decir palabra.
"Me alegro por ti," le dijo Don Julio. "Tú todavía eres una mujer joven. Debes tener un esposo."
"Es sólo un amigo," le contestó Mami, pero la cara se le puso colorada.
Llegó el domingo y la casa estaba inmaculada. Se habían bajado los tenderetes de ropa, los pañales estaban doblados y guardados, los pisos estaban relucientes, todas las camas estaban hechas y con sus colchas de chenille y las almohadas formaban unas discretas lomitas en las cabeceras. Mami tenía miedo de que Tata estuviera de mal humor y la fuera a abochornar delante de Don Carlos, pero Tata se vistió temprano, ocupó su lugar en la estufa junto a Mami y le ayudó a cocinar —y en un momento, hasta le dijo que ella se hacía cargo de terminar para que Mami pudiera arreglarse.
Don Carlos se apareció horas después de lo esperado, con una caja de galletitas italianas para nosotros y una botella de vino para Tata. No le trajo nada a Mami, quien estuvo fría y correcta con él, su rostro hecho una rígida máscara. Tenía puestos sus lentes oscuros de nuevo y no se los quitó en todo el tiempo que estuvo con nosotros. Nunca se disculpó por haber llegado tarde, ni ofreció excusas. Se quedó todo el tiempo en la mesa de la cocina, hablando y bebiendo con Tata y Don Julio, mientras mis hermanas y hermanos entraban y salían, le pasaban revista y volvían corriendo al cuarto para comparar notas. A punto de irse, le pidió a Mami que lo acompañara hasta la puerta de salida, dos pisos más abajo. Le dio la mano a todo el mundo, hasta a los nenes, y bajó las escaleras detrás de Mami. Tan pronto salió, empezamos a hablar de él.
"Es un hombre inteligente y bien educado," fue la evaluación de Don Julio.
"Gana un montón de chavos," le informó Delsa a todo el mundo. "Cuando salimos a bailar, él paga y después nos lleva a desayunar y paga por eso también."
"Trata a Mami con respeto," señaló Norma.
"¿Ah, sí? Y se apareció cuando le dio la gana," nos recordó Héctor.
"A lo mejor tenía que trabajar," lo defendió Alicia.
"¿Domingo?" pregunté.
"Es bien alto," observó Raymond.
"¡Pero es tan flaco!" añadió Edna.
"Yo lo único que digo," habló Tata finalmente, "es que no confío en ningún hombre que no me mire a los ojos."
Se chavó, me dije.
Como pasó con Francisco, Tata y Mami discutían sobre si era apropiado o no traer un hombre a la familia. Tata acusaba a Mami de darnos malos ejemplos y Mami insistía en que a los treinta y tres años, todavía era una mujer joven y tenía derecho a vivir. Si a Tata no le gustaba, que se mudara. Don Julio, se puso del lado de Mami y Tata, en minoría, aceptó lo inevitable. Una noche, Don Carlos vino a comer y a la mañana siguiente, allí estaba todavía.
El tercer año en Performing Arts resultó ser mi mejor año; mi promedio era excelente, ayudado por mis notas, casi perfectas, en geometría, que finalmente, después de tres intentos, había logrado dominar. Mi Cleopatra astuta fue un éxito y el semestre de primavera, cuando trabajamos con caracterizaciones, me asignaron el papel de una de las malvadas hermanastras en unas escenas de _La Cenicienta_. Nos aconsejaron que usáramos algún animal como el modelo físico de nuestro personaje y yo escogí el camello, por su aire altanero y su andar desgarbado.
Era patrullera de pasillo y estaba encargada de cotejar que los estudiantes que anduvieran por los pasillos durante horas de clase tuvieran un pase firmado por alguna maestra. El pasillo que más me gustaba patrullar estaba en el piso del Departamento de Danza. Me sentaba donde pudiera observar la clase de ballet; donde podía ver a los bailarines lanzarse a través del espacio con un desenfreno controlado que hacía que mis propios músculos añoraran el movimiento. Tenía envidia del entrenamiento que los volvía tan gráciles y fuertes, de los pasos complicados que ejecutaban, según la maestra pedía cada movimiento en francés.
En Performing Arts, el entrenamiento para actores era en danza moderna; su idioma, el inglés; y el propósito, que no hiciéramos el ridículo si nos tocaba trabajar en algún musical. Pero, yo había llegado a preferir mi clase de baile a la de actuación. Aunque me daba cuenta de que no era una gran actriz, veía que era una de las mejores bailarinas del Departamento de Drama. Practicaba todo el tiempo. El semestre que estudiamos jazz y aprendimos a hacer _isolations_ , empecé a hacer diminutos movimientos con el torso, las caderas y la espalda, mientras esperaba el tren o mientras estaba sentada en la clase de Historia. En casa, no podía estarme quieta frente al televisor mientras veíamos _Candid Camera_ o _The Jackie Gleason Show_. Con un ojo en la pantalla, me estiraba, hacía _splits_ , contaba cien _pliés_ en primera, segunda, tercera, cuarta o quinta posición, mientras mis hermanas y hermanos se quejaban de que mis movimientos los distraían. Para recoger algo del piso, me doblaba desde la cadera, con la espalda recta para estirar los muslos y las pantorrillas. Usaba el _counter_ de la cocina como si fuera una barra, saltaba de cuarto en cuarto, levantaba y sostenía la pierna contra el cachete, como hacían las bailarinas de can-can en _The Ed Sullivan Show_.
Sabía que nunca sería bailarina; no era mi intención. En Performing Arts aprendimos que si los actores tenían que esperar diez años para poder empezar a vivir de su arte, las bailarinas tenían suerte si podían sacarle ese mismo número de años a su carrera. Para mí, el baile no era para ser compartido, sino para transportarme a un lugar que nada más podía transportarme. Bailaba para mí, aunque un experto bailarín estuviera guiándome sobre el piso brillante. No importaba que nadie me viera bailar. Lo importante era que podía hacerlo.
En el teatro yídish, hacía de ujier en dos funciones todos los domingos, por lo que me pagaban en unos billetes arrugados, al final del día. La compañía trabajaba por repertorio, alternando comedias con tragedias. Mientras estaban en ensayo, yo me quedaba sin trabajo. Cuando había función, un actor siempre administraba la boletería, lo que me hacía pensar que el Sr. Rosenberg sólo escogía obras en las que uno de los actores entraba siempre en el segundo acto.
Llegué a conocer a los miembros del público regular por sus nombres. El Sr. y la Sr. Karinsky ocupaban siempre las mismas dos butacas en la fila C, centro. La Sra. Shapiro y su hermana, la Srta. Levine, preferían la primera fila, centro, porque la Srta. Levine tenía problemas de audición. La Sra. Mlynarski, siempre le traía a los actores un _coffee cake_ , que me entregaba con mucha ceremonia y que se suponía yo llevara trasbastidores inmediatamente, mientras ella permanecía en la puerta —un enorme bulto inamovible— e impedía que nadie entrara hasta que yo regresaba a comunicarle las más expresivas gracias de parte del elenco. Casi todos los asiduos sabían dónde estaban sus asientos, así es que me preguntaba por qué me seguiría pagando el Sr. Rosenberg para servir de ujier; hasta que una noche al Sr. Aronson le dio un fuerte ataque de tos. Bajé por el pasillo con la linterna y lo ayudé a salir de la sala del teatro hasta el pasillo, seguida de la muy afligida y avergonzada Sra. Aronson y de otro señor que pidió que le trajera un vaso de agua.
"No se preocupe," dijo el señor, inclinándose sobre el Sr. Aronson, que se estaba poniendo azul, "soy médico."
No encontré un vaso de agua por ninguna parte, así es que corrí hasta el Deli que había por allí cerca y le dije al muchacho del counter que teníamos una emergencia en el teatro y que por favor, por favor, por favor, me diera un vaso de agua. Cuando regresé, estaban en el Intermedio y el Sr. Aronson estaba sentado en el piso con el doctor arrodillado a su lado. Había recobrado un poco el color y la tos se le había calmado. Se tomó el agua en sorbitos, mientras el público lo observaba y daba vueltas y comentaba sobre lo que estaba pasando.
"Te ves mejor, Morey," le dijo la Srta. Levine.
"Es la vesícula," diagnosticó el Sr. Klein.
"Muévanse, necesita aire fresco," le ordenó a todo el mundo la Sra. Mlynarski.
Cuando las luces llamando al segundo acto parpadearon, su esposa y el doctor ayudaron al Sr. Aronson a bajar la escalera y a salir del edificio.
Después que todos se sentaron y se reanudó la obra, yo empecé a sudar y a temblar de pensar que algún día, a alguna de las personas mayores, pudiera darle un ataque al corazón o un derrame durante una función y no hubiera un médico para ayudarle. Más tarde, cuando el Sr. Rosenberg me estaba pagando y yo le expresé mi temor, me tranquilizó.
"No te preocupes," me dijo, agitando la mano, "en esta casa siempre hay un doctor." Se rió, pero yo no cogí el chiste.
Mami decidió que necesitábamos un apartamento que nos diera a todos mayor privacidad. Encontró uno en el segundo piso de una casa de dos plantas, en una calle con una línea de árboles, y casas idénticas, con las entradas separadas de la acera por un patio de cemento detrás de una verja de hierro forjado. Ella y Don Carlos cogieron el cuarto de atrás. Tata, la muchachería y yo desparramamos nuestras cosas en tres cuartos, uno de los cuales, la sala, daba a la calle soleada. Los otros cuartos no tenían mucha luz porque las ventanas daban a un respiradero. Cuando estaba organizando mis cosas en el cuarto del medio, con Delsa y Norma, noté que un pasillo al lado de la cocina era lo suficientemente ancho para acomodar un catrecito de los de abrir y cerrar.
"¿Mami, puedo coger este cuarto?"
"Esto no es un cuarto, es un pasillo."
"Si cierro estas puertas," cerré las que daban al pasillo de afuera y a su cuarto, "todavía me queda la puerta que da a la cocina y puedo poner el catre aquí y una mesita y ya tengo mi propio cuarto."
Mami entró al cuartito que había creado. "Es tan oscuro."
"Tiene una luz. ¿Ves?" Halé la cadenita de la bombilla que colgaba del techo. "Este cuarto es inútil, no necesitamos otra entrada."
"Mmm," Mami lo pensó un momento y accedió. En la tienda donde se vendían cosas en segundas manos, encontramos una coqueta de cristal y metal dorado con un espejo ovalado y una silla haciendo juego, tapizada en _vinyl_ blanco. Arrastré el catre y lo metí en el pasillo, donde se daba un abrazo tan estrecho con las paredes, que la única manera en que podía meterme en la cama era subiéndome por los pies hasta llegar a la cabecera, que daba a la puerta del cuarto de Mami. La coqueta cupo pegada a la otra puerta de entrada. Atornillé dos o tres ganchos de colgar ropa en la pared y metí mi ropa interior en una canasta que iba debajo de la cama. Era como vivir en una caja larga, pero era privado, mi propio cuarto donde podía tener mis cosas y donde dormía sola, aunque cada vez que me viraba le daba a la pared con una pierna o un pie.
"Suena como si estuvieras peleando allá adentro," se quejó Tata una mañana cuando salí del cuarto con los codos y las rodillas magulladas. "Y ahí no te entra aire fresco. Te vas a enfermar."
"Quizás ese cuarto no fue tan buena idea na'," advirtió Mami.
"Tuve una pesadilla," mentí, "hay suficiente aire fresco." Esa noche empecé a adiestrarme para dormir boca arriba, perfectamente quieta: Cleopatra, rodeada de sus pertenencias, en su sarcófago.
Tan pronto terminaron las clases, contesté un anuncio clasificado para un trabajo de verano en una compañía de revelado de fotografía. Me entrevistó el Sr. Murphy, un hombre nervioso que me hacía preguntas, pero nunca me dejaba contestarlas.
Revisó la solicitud de empleo.
"Tú estás en Performing Arts, ¿verdad?"
"Sí señor," le dije sin poder disimular el orgullo en mi voz.
"¿Qué enseñan ahí?" Tenía un exquisito acento de Brooklyn.
Saqué mi habla estándar. "Estoy en el Departamento de Drama, así es que estudiamos actuación, voz y..."
"¿Qué? ¿Tú quieres ser estrella de cine?"
"Es una escuela con programa académico también."
"¿Ah, sí? ¿Tú viste _West Side Story_? Te me pareces a esa muchacha, como se llama, Mareer. Tú podrías hacer de ella."
"No canto..."
"No hay muchos papeles para puertorriqueños," volvió a interrumpir.
"Se nos entrena para hacer cualquier papel..."
"¿Qué es esto? ¿Teatro yídish? ¿Tú hablas yídish?"
"Yo era ujier..."
"¿Te botaron porque no sabías hablarlo?"
"No, es que ellos no tienen funciones durante el verano."
"Fue una peliculaza," reflexionó y me tomó un rato darme cuenta de que estábamos de nuevo en _West Side Story_. "¿Cómo se llama? Ganó un premio de la Academia, ¿verdad?"
"Rita Moreno. Es puertorriqueña."
"¿Puedes empezar el lunes?"
"¡Claro que sí!"
"A las ocho de la mañana. Te voy a tener una tarjeta junto al reloj para que ponches." Se levantó y me acompañó a la puerta. "Sí, Reeter, ese es el nombre." Me abrió la puerta.
Estaba contenta de haber encontrado un empleo, pero molesta por haberlo conseguido porque a mi jefe le encantó _West Side Story._ Odiaba esa película y no ayudaba nada el que cada vez que decía que era estudiante de drama, la gente esperara que me levantara la falta y rompiera a cantar, _"I feel pretty, oh so pretty..."_
A pesar de que no había visto nunca una representación teatral de _West Side Story_ , había leído que la María original fue representada por una actriz norteamericana, Carol Lawrence, mientras que Anita, la hizo Chita Rivera.
En la película, Natalie Wood hizo de María y Rita Moreno de Anita. Era una sutileza, pero no me pasó desapercibido, que la única virgen en toda la película —la dulce e inocente María— la representaba siempre una americana, mientras que la bola de fuego _sexy_ , era puertorriqueña. Pero, eso no era todo.
Los Jets tenían un lugar limpio, bonito y cálido donde reunirse, que recordaba la fuente de soda donde se pasaba Archie con Betty, Verónica y Jughead. El dueño era un viejito buenagente que aguantaba toda clase de pocavergüenzas, incluyendo la casi violación de Anita. Los Sharks tenían una azotea, ¿y qué hacían allí? Discutían si "América" era mejor que Puerto Rico.
"Es sólo una película," me recordó Laura Figueroa, una vez que yo me monté en tribuna con _West Side Story_.
"No es sólo una película," le argumenté, "es la _única_ película sobre los puertorriqueños que la gente ha visto. ¿Y cuál el mensaje? Las puertorriqueñas blancas se balancean de las escaleras de escape de incendios, cantándoles dulces canciones a los tipos italianos, mientras que las puertorriqueñas de piel oscura se acuestan con los novios. Oscuros también, que conste."
"Estás dándole demasiado color al asunto," insistía.
Cuando leímos _Romeo y Julieta_ en la clase de inglés y Misis Simmons nos dijo que _West Side Story_ estaba basada en una obra de Shakespeare, me desilusioné. Pensé que el Bardo hubiera podido hacer algo mejor. La escena de la muerte al final de la obra, era lo más necio que había visto. Durante la discusión, mis compañeros trataron de ayudarme a verlo de un modo diferente.
"¿Pero no entiendes?" me dijo Brenda, "murieron por amor."
"¿Pero, qué clase de razón más estúpida es esa?" le pregunté.
"No podían vivir el uno sin el otro," me explicó Ardyce.
"¡Por favor! Esa es la razón más ridícula para suicidarse."
"Obviamente, nunca has estado enamorada," suspiró Myra con desdén.
"Si lo hubiera estado, tampoco me hubiera matado por ningún muchacho."
"¿Aunque se pareciera a Richard Beymer?" preguntó Roger.
"Especialmente si se pareciera a Richard Beymer."
"Cleopatra se mató por Marco Antonio," me recordó Jay.
"No exactamente. Pensó que Marco Antonio estaba muerto y había perdido su aliado más importante. Sin él, los romanos la hubieran despojado de su rango." Nadie podía invocar a Cleopatra al lado mío y esperar que yo no tuviera los datos claros.
Misis Simmons levantó la mano para terminar la discusión. " _Romeo y Julieta_ es una de las más grandes historias de amor de todos los tiempos," concluyó, "pero, aparentemente, no lo es para todo el mundo." Sonó el timbre. "La semana que viene empezamos _Hamlet_." Me sonrió, "yo creo que ésa te va a gustar más," me dijo cuando salía del salón.
Mi trabajo de verano consistía en meter negativos y retratos dentro de sus sobres y enviárselos por correo a la gente que nos mandaba sus rollos para que los procesáramos. Las fotos se revelaban al lado, pero los gases se colaban por la pared hacia el cuarto donde yo trabajaba, que era oscuro y no tenía ventanas. Dos mujeres más trabajaban en unos escritorios, haciendo lo mismo que hacía yo, y una de ellas, Sheila, una mujer negra, no mucho mayor que yo, estaba encargada de enseñarme como hacerlo. La otra, una mujer asiática mayor, se pasaba murmurándose a sí misma todo el tiempo que estaba trabajando y casi nunca levantaba la vista de su estiba de sobres, negativos y fotos.
"Eso es lo que pasa cuando una trabaja aquí mucho tiempo," dijo Sheila, inclinando la cabeza hacia Mimi. "Te tuestas. Todos esos químicos." Se rió y yo supuse que tenía que estar bromeando.
"¿Cuánto tiempo llevas trabajando aquí?"
"¿Yo? Como siete meses. Tengo dos nenes que mantener, tú sabes. No soy como tú que te quedaste en la escuela y eso." Sheila trabajaba tres días a la semana; los otros dos estaba matriculada en un programa de adiestramientro para asistenta de enfermería. "Tenía que sacar mi GED," me dijo, "entonces, me hicieron coger Biología y Química y toda esa baba que había pasado durmiendo la primera vez que la cogí. ¿Tú tienes que estudiar todo eso en tu escuela?"
"Sí."
"Y a ti te gusta, ¿verdad?"
"No me molesta."
"Ojalá yo me hubiera quedado en la escuela. Ahora tengo dos nenes que mantener. Tú no vayas a pensar que dejar la escuela es bueno."
"Mi mamá no me dejaría."
"Estoy de acuerdo con ella." Rebuscó entre algunas fotos. "¡Mira este bobo en este retrato! ¿Qué es lo que tiene en la cabeza?"
"Parece un racimo de guineos."
"Aquí en este trabajo se ve la gente más tonta. Mírate ésta... Ella cree que se ve bien."
Durante horas, metía en sobres los recuerdos de la gente, al son del chachareo de Sheila y de los murmullos de Mimi. Por las tardes, al salir a la acera, respiraba el aire de Brooklyn, fresco y limpio en comparación con el que había en el edificio, donde pasaba ocho horas diariamente. Me iba a casa, me cambiaba y me encerraba en mi cuarto a leer o a escribir largas divagaciones en el diario que me había regalado La Muda cuando cumplí diecisiete años.
Me pagaban todos los viernes. En casa, le daba a Mami una parte de mi salario y le pagaba a mis hermanas y hermanos para que hicieran las tareas que me tocaban a mí pero que yo no quería hacer. El resto me lo gastaba en ropa para el semestre entrante, pero mi gasto mayor era en libros que no tenía que devolver a la biblioteca.
Mi primera compra fue _The Power of Positive Thinking_ del Dr. Norman Vincent Peale. Me encantaba su teoría de que los pensamientos negativos producen actos negativos. Mami, mis hermanas y hermanos, mis amigos de la escuela, me habían acusado más de una vez de tener una vena negativa y morbosa. Tenía la esperanza de que el libro del Dr. Peale me ayudara a pensar positivamente cuando la vida se tornara sombría, como estaba segura que sucedería.
Como sugería el Dr. Peale, hice una lista de las cosas buenas en mi vida:
(1) Había aprobado mi tercer Regents de Geometría con 96.
(2) Tenía un empleo.
(3) Mami tenía trabajo, estaba enamorada y contenta de nuevo.
(4) Delsa, Norma y Héctor también tenían trabajo.
(5) Con cinco personas trabajando en casa, ahora teníamos más dinero del que nunca habíamos tenido.
(6) Tenía mi propio cuarto.
(7) El pie de Raymond estaba completamente curado y el médico había dicho que ya no tenía que volver a citas de seguimiento.
Dr. Peale sugería diez, pero yo sólo pude pensar en siete, lo que me llevó a pensar que necesitaba el libro desesperadamente. Para ayudarme a entrar en una actitud positiva, memoricé canciones que hablaban sobre la buena vida. En la biblioteca escuchaba discos rayados de los musicales de Broadway y aprendí a cantar a toda boca _"Everything's Coming Up Roses"_ como Ethel Merman. Cantaba _"Luck Be a Lady Tonight"_ de _Guys and Dolls_ , todas las mañanas cuando me bañaba. Pero la canción que tarareaba en los momentos de duda venía de la odiada _West Side Story_. Me parecía insultante que lo único positivo en la vida de María fuera Tony, al pie de su escalera de escape. Pero, me encantaba su canción y me prometí a mí misma que todos los días pasaría algo bueno, y que tan pronto sucediera, lo sabría. A pesar de que en _West Side Story_ las cosas buenas estaban siempre a la vuelta de la esquina para los no-puertorriqueños solamente, me obligué a creer que en cualquier momento ocurriría un milagro, que se haría realidad y que me ocurriría a mí. Así es que, cuando llegaba a la esquina, reducía la velocidad y trataba de imaginar cómo se vería un milagro pitando por el río.
Después de que Don Carlos llegó a nuestras vidas, dejamos de ir a bailar tanto porque los fines de semana era cuando único Mami lo veía.
"¿No te parece raro," le decía Tata a Mami, "que no viva aquí con nosotros?"
"Es por su trabajo," le explicaba Mami. "Es demasiado lejos para él, ir de aquí hasta su trabajo en la ciudad."
"Tú y Negi van a la ciudad todos los días," le señalaba Tata.
"Es diferente. Él tiene dos trabajos. Uno de día y otro de noche."
Si fue o no que Tata le metió ideas en la cabeza a Mami, lo cierto es que para la mitad del verano, cuando Mami empezó a dar señales de que estaba encinta otra vez, empezó a preguntarle a Don Carlos dónde se metía durante la semana. Desde mi cuarto, al lado del de ellos, los oía discutir. O, mejor dicho, oía a Mami. Don Carlos respondía en voz muy baja, como si quisiera que sólo Mami escuchara sus excusas. A veces no le contestaba nada, lo que ponía a Mami furiosa y le lanzaba acusaciones a las que él, con terquedad, se negaba a responder. Cogía su maletín y se iba. Cuando Mami se calmaba, volvía y las cosas se arreglaban por un tiempito.
Nos había dicho que tenía tres hijos. Lo fastidiábamos pidiéndole que los queríamos conocer, pero él, por una razón u otra, siempre posponía la visita. Decía que trabajaba de contable durante el día y que por las noches trabajaba llevándoles los libros y los asuntos de impuestos a clientes privados. Durante el noviazgo, se lucía muchísimo pagando nuestros boletos y el desayuno después del baile. Pero, después que se mudó con nosotros, le daba trabajo abrir la cartera. No se ofrecía a ayudar con la compra, no nos daba ningún menudo que le sobrara, y no se ofreció a pagar la cuenta del teléfono cuando Mami no la pudo pagar y se lo cortaron. "Tacaño," fue la evaluación de Tata y se tocaba el codo con el puño. Nuestros parientes no eran ricos, pero no eran tacaños. Eran generosos con lo que tenían y la renuencia de Don Carlos a separarse de su dinero fue interpretada como una debilidad de carácter, una señal segura de que había otros rasgos más desagradables que todavía nos faltaban por descubrir.
# "Deja de pensar y baila."
#
En Peforming Arts estudiábamos a Shakespeare en la clase de Inglés, pero en el Departamento de Drama no nos asignaban escenas de sus obras hasta que no estábamos listos. Ahora que ya éramos _seniors_ y teníamos dos años completos de clases de voz, dicción y actuación a nuestro haber, finalmente, íbamos a poder montar algunas de las magníficas escenas del Bardo. Ya había manifestado mi disgusto por _Romeo y Julieta_ , así es que no fue sorpresa alguna que no me escogieran entre los Capuletos. Sería —qué iba a ser— Cleopatra en verso yámbico.
Me emparejaron con Northern Calloway (ninguna relación con Cab, solía decir), una de las estrellas del Departamento de Drama, que lo mismo hacía tragedias, que comedias, que musicales. Me caía bien, pero su franqueza y su sentido del humor medio perverso, me molestaba a veces. Cuando nos asignaron la segunda escena del primer acto, de _Antonio y Cleopatra_ de Shakespeare, tuve mis dudas de que pudiéramos trabajar bien los dos juntos; pero él era mucho más disciplinado de lo que esperaba. Me ayudó a descubrir aspectos de la personalidad de Cleopatra que yo no había percibido o destacado antes. Me recomendó que desechara el vestido del mantel amarillo porque había desarrollado manerismos a causa del poco movimiento que me permitía el traje.
Compré un par de cortinas de nilón y me hice un vestido transparente. Después de casi tres años de verme inventando vestuarios con sábanas, cortinas y retazos de tela, Mami ya no pedía inspeccionar todo lo que hacía. Pero, cuando vio la tela transparente, me advirtió: "Yo espero que tú pienses ponerte un refajo debajo de eso." Le expliqué que la Reina del Nilo no usaba refajo, pero, por aquello de la decencia, acepté usar el vestido sobre los _tights_ y el leotardo.
Estaba en el pasillo, leyendo el tablón de edictos del Departamento de Drama, donde se colocaban los recortes de periódicos de ex-alumnos famosos, con el año de su graduación anotado en una esquinita. En Performing Arts, la gente no se paraba, simplemente, sin hacer nada. Cada oportunidad para ejercitar el cuerpo o practicar las destrezas se aprovechaba, así es que, mientras leía, hacía _pliés_ en segunda posición.
Sentí a alguien parado detrás de mí y al virarme, me encontré cara a cara con un hombre de cabeza grande coronada de un pelo negro salvaje, una nariz prominente, penetrantes ojos negros bajo cejas bien proporcionadas y labios bien formados que no sonreían. Sabía que era uno de los maestros del Departamento de Baile.
"Usted debe ser una bailarina clásica india," declaró en una voz profunda con un dejo de acento extranjero.
"No señor. Soy una actriz puertorriqueña."
Pareció molestarse con la corrección. "No dije que es. Dije que debe ser. Pase a verme."
Me intrigó; imaginaba a los indios con tocados de plumas y mocasines, bailando _en pointe_ alrededor de la fogata. En un período libre corrí hasta la Oficina de Baile, pero no había nadie. Traté dos o tres veces más esa semana, pero nunca lo encontré.
Un día Miss Cahan, una maestra de baile del Departamento de Drama, me detuvo en un pasillo y me preguntó si quería hacer una prueba para una obra. "Es para una compañía de teatro infantil."
Me dijo que las audiciones serían más tarde en la semana y me dio la dirección. "Otros estudiantes van a audicionar," añadió. "No llegues tarde."
La dirección era en Madison Avenue. El portero me hizo esperar en lo que llamó y dio mi nombre y después de unos minutos, me indicó que subiera al quinto piso. El operador del ascensor —un hombre bajito, de piel aceitunada, en un elegante uniforme que le daba un aire de Napoleón extraviado en el siglo equivocado, no me miró mientras subimos. Señaló a la izquierda cuando llegamos a un sombrío pasillo alfombrado y esperó hasta que toqué el timbre que había debajo de la mirilla de la puerta del apartamento. Al abrirse la puerta, se cerró el ascensor. Miss Cahan, vestida en _tights_ y leotardo y una falda de baile larga me saludó y me condujo hasta un enorme salón con amplias ventanas de fondo.
Tenía diecisiete años y nunca había estado en una casa americana. Y aquí estaba, en un apartamento del Upper East Side —gruesas alfombras bajo mis pies, pinturas oscuras y tristes en las paredes, yardas de tela alrededor de las ventanas, dos sofás, butacas tapizadas, mesitas adornadas con figuritas de cristal y porcelana. Me moría de envidia.
Miss Cahan me presentó a Misis Kormendi, la autora y directora de la obra. Había otra estudiante de Performing Arts en la sala, Claire, quien, yo sabía, era una actriz de primera. Se había quitado los zapatos y estaba sentada con las piernas cruzadas, en el piso, al lado izquierdo de Miss Cahan. La sencillez de la blusa y el pantalón que tenía puestos, y su "Hola" tan natural, me hicieron pensar que vivía allí.
"¿Por qué no te quitas los zapatos y los pones debajo de ese banco?" Miss Cahan me señaló un mueble tapizado al que nunca le hubiera llamado banco, a pesar de que no sabía qué otro nombre darle. Tenía puesta una falda porque para Mami, las muchachas decentes no usaban pantalones a menos que no estuvieran montando a caballo.
"Quizás te debas quitar las medias también, para que no resbales," añadió.
_"Okay."_ Me viré de espaldas y discretamente me desabroché las medias de la fajita y me las bajé, preguntándome por qué tendría que desvestirme para audicionar para una obra infantil. Miss Cahan me leyó la mente.
"Debí haberte dicho que íbamos a bailar," me explicó, "para que vinieras preparada."
"¡Ah!" Me acerqué, hundiendo los dedos en la alfombra gruesa como los hundía en el suave y tibio fango de Puerto Rico, y me senté en el piso con las piernas dobladas, aunque había como diez butacas suntuosas donde, con gusto, me hubiera dejado caer.
Misis Kormendi me explicó que la obra estaba basada en una leyenda india. "India de la India," especificó, "no americana." Estaba buscando a alguien para el papel de la diosa Lakshmi quien, en la obra, era una estatua que se convertía en cisne. No pregunté cómo. La obra tenía mucho baile; por eso Miss Cahan estaba ayudando con la audición.
"Bien," dijo Miss Cahan. "Vamos a probar algunas cosas."
Sentada en el piso con su _clipboard_ en la falda, Misis Kormendi tomaba notas, mientras Miss Cahan nos dirigió a Claire y a mí en una serie de pasos que no se parecían en nada a lo que habíamos hecho hasta ahora en la clase de baile. Eran posturas estilizadas y dramáticas que requerían que nos moviéramos en un amplio _plié_ , segunda posición, nuestros torsos rígidos, los brazos y las manos en gestos que exigían coordinación y fuerza en unos músculos que yo nunca había usado. No podía seguir la coreografía y me detuve dos o tres veces avergonzada y frustrada, mientras Miss Cahan y Claire se deslizaban por el salón con soltura.
Miss Cahan me ajustó el paso. "Deja de pensar," dijo, "y baila. No te preocupes por recordar los pasos. Tus músculos se acordarán."
Éste era un concepto nuevo para mí. Trabajaba mucho en la clase de baile; me esforzaba para lograr saltos más altos, para estirarme aún más. Nunca dejaba que algo, meramente, pasara. Pero, confiaba en Miss Cahan, que como profesional, sabía de eso más que yo. Dejé de pensar. Lo próximo que supe fue que la audición había terminado y que Misis Kormendi prometió que se comunicaría con nosotras.
Claire y yo bajamos juntas en el ascensor. Aunque éramos compañeras de clase, no teníamos mucho que decirnos. Ella era una de las chicas listas, talentosas y populares que lograban siempre los mejores roles: Antígona o su hermana, Ismene, Julieta, Emily en _Our Town_ , Frankie en _Member of the Wedding_. Nos despedimos frente al edificio, pero, camino al _subway_ de regreso a Brooklyn, sabía que el papel era mío. Claire podría ser una ingenua angelical, pero yo había perfeccionado los personajes exóticos. Cleopatra, Reina de Nilo, estaba a punto de convertirse en Lakshmi, la Diosa Cisne.
Unos días después, Miss Cahan me pidió que me quedara después de la clase y me dijo que Misis Kormendi me quería en su obra. Estaba loca por llegar a casa para contarle a Mami que, a un año entero antes de mi graduación, ya tenía una parte en una obra. Sólo nueve años más de sacrificio y sería una estrella.
El primer ensayo de la obra de Misis Kormendi fue un sábado por la mañana en un estudio en Madison Avenue, no muy lejos de su apartamento.Estaban en medio de una clase de baile cuando llegué. Me asomé por la puerta abierta, pero la maestra, una mujer de rostro tirante y expresión agria, se acercó y me cerró la puerta en la cara. Me dio tanta vergüenza que se me salieron las lágrimas, pero me las tragué al oír pasos por el pasillo. Misis Kormendi apareció cuando estaba terminando la clase de ballet y el pasillo se llenó de bailarinas patilargas y altaneras.
Misis Kormendi besó a _Madame_ , la señora de cara agria, en los dos cachetes y conversaron mientras se vaciaba el salón. La maestra de baile miró con desdén hacia donde yo estaba y escuché a Misis Kormendi decir mi nombre. Me quedé en _tights_ y leotardo, pero no me atreví acercarme a las barras a hacer calentamiento estando _Madame_ allí. Su ojos me seguían y yo esperaba que se disculpara por su rudeza, pero no lo hizo.
Unos minutos después, apareció el resto del elenco; niños y niñas no mayores de doce años. Ellos también se desvistieron y se quedaron en _tights_ y leotardo, pero no tuvieron miedo de acercarse a la barra. Para entonces, la instructora se había ido y yo me uní a los chicos, muchos de los cuales estudiaban ballet desde antes de que pudieran pararse. Hicieron ejercicios y estiramiento, que traté de imitar, pero no podía seguirlos.
Cuando llegó todo el mundo, Misis Kormendi repartió copias de la obra e hicimos la lectura. "Memorícense sus líneas. La semana que viene empezamos a marcar." Los que íbamos a bailar teníamos que llegar al próximo ensayo dos horas antes que el resto del elenco porque el coreógrafo tenía que trabajar con nosotros aparte.
El sábado siguiente, subiendo hacia el estudio, escuché una música extraña, un rítmico golpear de pasos y campanas tintineando con furia que venía de detrás de la puerta abierta. A pesar de que ya me habían tirado la puerta en la cara una vez, no pude aguantar la curiosidad. En el salón estaba el maestro que me había confundido con una india. Tenía puesta una sábana alrededor de la cintura y las piernas, y una túnica blanca con diseños bordados en el frente y las mangas. En los tobillos, tenía unas campanas. Sus movimientos eran intensos, pegados al piso en un _plié_ profundo, los dedos gordos de cada pie arqueados hacia el techo. Saltaba, sus brazos y piernas punzaban el aire, los ojos se le iban hacia atrás, la boca se torcía en un gesto malvado. Desde el tronco del cuello, la cabeza se movía rápidamente para atrás y para alante, y volvía a caer en el mismo _plié_ profundo con los dedos hacia arriba. Nunca había visto nada tan salvaje ni tan hermoso. No podía ser baile, pero tampoco podía ser otra cosa. Cuando terminó, la voz de Misis Kormendi me llamó desde adentro, donde estaba sentada en la única silla en el estudio, con su _clipboard_ en la falda. Con la mano, me pidió que me acercara.
"Tú conoces a Matteo, ¿verdad?" sonrió.
"Lo he visto en la escuela." La otra persona en el salón era Northern, mi Antonio, quien sonrió divertido cuando me vio tan sorprendida de verlo allí.
Misis Kormendi y yo observamos cómo Matteo le iba enseñando a Northern los gestos estilizados y las expresiones faciales que acababa de ejecutar. Cuando llegó el resto de la clase, Matteo nos dio nuestra primera clase de baile clásico indio, que no tenía nada que ver con tocados de plumas y mocasines. Era una forma de danza antigua, cada una vinculada a un lugar diferente de la India y cada una con su música distintiva, su coreografía, sus posturas y vestuarios. El baile que le estaba enseñando a Northern se basaba en Kathakali el teatro de danza de Kerala, mientras que el resto íbamos a aprender Bharata Natyam, asociada con la región sur de la India. Matteo nos mostró algunas de las formas en que los bailes diferían en estilo y en el tipo de historias que los bailarines contaban con sus cuerpos. Nos explicó que, históricamente, Kathakali lo bailaban los hombres y Bharata Natyam, las mujeres. Hablaba reverentemente de una coreografía que había pasado de generación en generación por bailarines que muchas veces eran marginados por su dedicación al arte. Nos mostró fotos de esculturas basadas en los movimientos que estaba a punto de enseñarnos.
Nos pasó por la clase más exigente que habíamos tomado jamás. No era sólo la exigencia física del baile lo que resultaba retante. Es que lo que estábamos aprendiendo era más que teatro y más que baile. Era una forma de arte que combinaba teatro, baile, música y espectáculo. Tenía su propio lenguaje, único; cada gesto tenía un nombre, cada emoción un gesto. Cuando miré hacia el enorme espejo en la pared del estudio, vi lo que Matteo debió haber visto el día que yo estaba haciendo _pliés_ frente al tablón de edictos de Peforming Arts. No parecía una actriz puertorriqueña de Brooklyn. Parecía una bailarina clásica india.
Matteo daba clases en un estudio en el Upper West Side. Cobraba más por clase de lo que yo me ganaba de ujier. Llamé al Sr. Murphy, el de la compañía de revelados de fotos, y me ofreció trabajo los fines de semana y cuando pudiera venir. El problema era que entre la escuela, los ensayos y el trabajo para pagar las clases, no tenía tiempo de ir al estudio de Matteo y él tenía muy poca tolerancia con bailarines que no estuvieran comprometidos con su arte. Tomé dos o tres clases con él, pero lo que hacía mayormente era prestar atención a lo que enseñaba durante los ensayos, y venir, aunque no le tocara ensayar a mi personaje. Pronto me aprendí todos los bailes de la obra, incluyendo el cuento bailado de las damas acompañantes y el feroz baile del diablo, que realizaría Northern.
Según evolucionaron los ensayos, fui abandonando la fantasía de ser transportada por los aires, con alambres, a lo Mary Martin en _Peter Pan_. Lakshmi se pasaba toda la primera escena parada en un solo pie dentro de un templo mientras la Princesa lloraba y rezaba desesperada porque estaba enamorada de un príncipe pero comprometida con un rajá, que en realidad, era un demonio. Al comienzo de la segunda escena, un emocionante sonar de cítara me anunciaba que era el momento de comenzar mi transformación de piedra a cisne. Mis dedos temblaban, mis ojos se movían de un lado a otro, mis brazos se suavizaban y aleteaban. El volar se simulaba por _mudras_ , unos gestos de mano, lentos y vacilantes al principio, luego, plenamente convertidos en los sinuosos movimientos de una criatura que descubre que ya no es piedra dura, sino un suave y grácil pájaro. En las presentaciones, cuando mis dedos corbraban vida, el público quedaba sin aliento y ya al final del baile estaban de pie, aplaudiendo.
La Muda vino a la última presentación. Mi baile, tan parecido a su idioma sin palabras, era mi actuación mejor lograda hasta ahora. Según me fui transmutando de piedra silenciosa en diosa efusiva, yo era La Muda, atrapada en el silencio, pero ávida de comunicarme, hablando con el cuerpo, porque la voz me fallaba. Al bailar, no tenía lengua, pero era capaz de todo. Era un cisne, era una diosa, vencía a los demonios.
A los cinco meses de embarazo, Mami averiguó porqué era que a Don Carlos nunca le sobraba el dinero y tampoco regresaba todas las noches a casa. Nos había dicho que era divorciado, pero la verdad era que cuando no estaba con nosotros, estaba con su esposa en el Bronx. Mami se enteró cuando la llamó la esposa y la puso de vuelta y media, a ella, a toda su parentela y a las generaciones venideras. Cuando Mami lo confrontó, Don Carlos admitió que no estaba técnicamente divorciado, pero insistió que era sólo porque no se había terminado el papeleo. Ni Mami, ni Tata, ni Don Julio, ni ninguno de nosotros, le creyó. Desde mi punto de vista, el Don Carlos caballeroso y de hablar suave se volvió otro sinvergüenza más, que prometía más de lo que estaba dispuesto a cumplir.
Nuestro trato respetuoso dio paso a insolencias agresivas y boconas que Mami castigaba. Sus amenazas y bofetones y la insistencia de que le debíamos cortesía y deferencia a Don Carlos porque era una persona mayor y el papá de nuestro futuro hermano o hermana, no mejoraban nuestro comportamiento. En su lugar, un resentimiento amargo surgió donde una vez hubo afecto, y aunque eventualmente, Don Carlos sí se divorció y nos presentó a sus hijos y aflojó el bolsillo, para mí, por lo menos, el daño estaba hecho. Nunca le perdonaría que volviera a abrir las heridas, aún frescas que nos causó Papi cuando nos abandonó a la suerte americana y la muerte de Francisco.
Pero, lo que más me asustó de la traición de Don Carlos, fue darme cuenta que Mami no era immune al poder seductor de un hombre de palabra dulce y modos tiernos. "Los hombres quieren sólo una cosa," nos dijo tantas veces, que no podía mirar a un hombre sin oírlo. Si ella podía sucumbir ante el encanto, ¿cómo podría yo, más joven e inexperta, evitar el mismo destino?
Mami trabajó hasta un par de meses antes de dar a luz y entonces volvimos a humillarnos en la oficina del _welfare_. Después de explicarle nuestra situación, la trabajadora social vino al apartamento, sin avisar, para asegurarse de que Don Carlos no estuviera escondido detrás de la cortina de baño o en el _closet_.
Como no había avisado, el apartamento tenía el revolú caótico habitual, que yo encontraba cómodo, pero embarazoso porque sabía que la gente no debía vivir así. Las camas no estaban arregladas porque nos servían de asientos cuando hacíamos las asignaciones y veíamos televisión. Los trastes estaban sin fregar porque me tocaba a mí hacerlo y yo siempre esperaba hasta lo último, a ver si convencía a alguno de mis hermanos de que lo hiciera. Mami no había ido al _laundromat_ , así es que había una pila de ropa saliéndose del canasto de la ropa sucia. El baño estaba decorado con brasieres, panties y medias que estaban secándose, además de algunas camisas y blusas que se lavaban a mano y se tendían en ganchos. Franky tenía la nariz mocosa y nadie lo había ayudado a limpiársela. A Mami le dolía la espalda y se había quedado acostada; otra de las razones por las que el apartamento estaba tan regado. A Tata le dolían los huesos, así es que había empezado a beber temprano y ahora estaba sentada en la cocina fumando, sus ojos caramelos fijos en la trabajadora social pastosa que fue de cuarto en cuarto, abriendo cada gabinete y gaveta.
Mientras estuvo allí la trabajadora social, estuvimos quietos, sin atrevernos a mirarla, como si hubiéramos hecho "algo" y ella nos hubiera pillado. Mami iba detrás de ella con Delsa y conmigo de intépretes. Los nenes estaban sentados en la cama, haciendo que leían porque, mientras yo interpretaba en la cocina, Delsa les había ido a advertir que se quedaran quietecitos. Don Julio estaba al llegar y nos preocupaba que la trabajadora social fuera a pensar que vivía con nosotros, cosa que no era cierto. Pero, aún así, nos parecía que no debería visitarnos, que no debiéramos conocer a ningún hombre.
La trabajadora social fue minuciosa. En los símbolos crípticos de la taquigrafía, tomó notas en una libreta; se acomodó los espejuelos, abrió la nevera, apuntó algo, inspeccionó el horno. Cuando preguntaba algo, no estábamos seguras si estaba meramente conversando o si quería entramparnos para que admitiéramos que había un hombre debajo de la cama o detrás de la puerta, aunque nosotras sabíamos que no había ninguno.
Después que se fue la trabajadora social, el apartamento lucía más pequeño y más sórdido que cuando llegó. Había una cucaracha muerta en una esquina. El zafacón estaba lleno. Había grasa coagulada en los trastes sucios. Las paredes estaban descascaradas, la madera oscura se asomaba por debajo del linóleo. Las cortinas eran demasiado pesadas para los ganchos. Todo se veía peor, lo que supongo contribuiría a que nos viéramos realmente necesitados.
Esa trabajadora social callada y comedida fue la primera americana que vio cómo vivíamos; su visita, una invasión a la poquita privacidad que teníamos. Esa visita nos acentuó cuán dependientes éramos de la opinión de una persona totalmente extraña a nosotros, que no hablaba nuestro idioma, cuya vida era, claramente, mejor que la nuestra. ¿De qué otra manera podría pasar juicio sobre ella? Estaba que hervía, pero no tenía por dónde dejar escapar mi rabia y esa sensación de que mientras viviera bajo el amparo de Mami, mi destino estaría en manos de otra gente, cuyo poder era absoluto. Si no en las de ella, en las del departamento del _welfare_. Me encerré en mi cuarto y lloré escondida en la almohada, mientras mi familia hacía bromas, reía e imitaba la voz nasal de la trabajadora social y la manera en que se asomaba dentro del gabinete debajo del fregadero, como si allí cupiera un hombre. No tenía ninguna gracia reírse de uno mismo o de la gente que tenía nuestro destino en sus manos. Era patético.
Me quedé dormida bañada en llanto y no oí los gritos cuando el mundo se volvió negro, no oí a Mami arrastrar los pies desde su cuarto hasta la parte de al frente del apartamento, tropezando con los muebles según iba contando cabezas para asegurarse de que todos estuviéramos juntos en la absoluta oscuridad de Brooklyn. No la oí llamarme, mientras Tata y ella le decían a los nenes que se mantuvieran juntos hasta que lograran descifrar qué era lo que estaba pasando. Cuando desperté estaba ciega y el abrir los ojos no hizo ninguna diferencia. Pensé que me había muerto, pero sentía. Grité el nombre de Mami y eschuché: "¡Estamos aquí al frente!" Caminé a tientas desde mi cuarto, por la cocina hasta las ventanas abiertas de la sala, donde toda mi familia estaba apiñanda, unos con otros. Había gente en la calle, hablando en voces bajas e íntimas. La tibia luz amarilla de las velas oscilaba en las ventanas de los vecinos.
"¿Qué pasó?" pregunté.
Delsa me mandó a callar. A través de la estática de su radio de baterías, oíamos la noticia. En Nueva York y en todo el Noreste había un apagón. Por encima de los árboles escuálidos, sobre la línea plana e irregular de los edificios, diminutas lucesitas guiñaban y bailaban, las primeras estrellas que veía desde que llegamos a Brooklyn.
# "Tiene que ser pecado faltarle el respeto así a la Virgen."
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En medio del invierno nos mudamos a una casa de una sola familia en la Calle Stanhope. Ya no tenía un cuarto para mí sola, sino que compartía uno con Delsa, Norma, Alicia y Edna, mientras que los varones —Héctor, Raymond y Franky— dormían en otro. Mami puso la cunita del bebé por nacer, su cama doble y los gaveteros en el cuarto del medio, el único que tenía puerta. La planta baja tenía una sala pequeña, un área de comedor y una cocina bastante grande. El catre de Tata lo pusieron en la alacena, frente al arroz, a la harina, las habichuelas secas y las latas de salsa de tomate.
El tener una casa para nosotros solos nos hacía sentir ricos. Cero vecinos en el piso de abajo, golpeando el techo si hacíamos demasiado ruido. Nadie en el piso de arriba caminando duro y haciendo temblar las lámparas. Pero también quería decir que si se dañaba el inodoro, no había _"super"_ y cuando no se sentía suficiente calor, era porque no habíamos pagado la cuenta, no por que el casero fuera tacaño.
Estábamos cerca de nuestros parientes de nuevo y podía visitar a Alma y a Corazón cuando regresaba de la escuela por la tarde. Ellas también se habían mudado a un apartamento más espacioso, pero más oscuro, en Flushing Avenue, como a medio bloque del tren elevado. Alma se había graduado de Escuela Superior y trabajaba de secretaria en un negocio de ventas de medias al por mayor. Su oficina quedaba cerca de Performing Arts y todas las semanas nos encontrábamos en la ciudad para cenar. Nos llevábamos tan bien que enseguida empezamos a hacer planes de compartir un apartamento tan pronto me graduara de escuela superior y encontrara trabajo.
"Tiene que ser un apartamento de dos habitaciones," decía Alma. "Yo necesito privacidad."
"Sí. Y yo espero que tú sepas que yo no sé cocinar."
"Ni yo tampoco. Comemos fuera," sugirió.
Repasamos los clasificados para tener una idea de cuánto sería el alquiler en el Upper East Side, nuestra primera opción. Alma había leído en algún sitio que no deberíamos pagar más del equivalente de una semana de salario y pronto se hizo evidente que un apartamento de dos habitaciones en Manhattan sería muy caro.
"A lo mejor tenemos que buscar en Queens," surgirió, a lo que yo protesté. "Yo no quiero vivir en los precintos de las afueras."
"Seguimos buscando, entonces." La semana siguiente volvimos a revisar los clasificados. Calculamos cuánto teníamos que separar para la fianza, el primer mes de renta, muebles, toallas, sábanas, cortinas, alfombras. Si economizábamos, si a Alma le aumentaban el sueldo y si yo conseguía un buen trabajo, podríamos alquilar el apartamento seis meses después de mi graduación.
"Nos mudamos para Navidad," dije "y hacemos una fiesta para estrenar el apartamento." Imaginaba un apartamento, no muy diferente al de Misis Kormendi, lleno de gente cuyas caras ahora mismo aparecían borrosas porque todavía no las conocía.
Gozábamos tanto planificando nuestras vidas como chicas solteras en Manhattan, que no se nos ocurrió pedirle permiso a nuestras madres.
"De la única manera que tú sales de mi casa," sentenció Mami cuando le traje el tema, "es casada."
"Pero, yo no me quiero casar."
"Las muchachas decentes no viven solas en la ciudad."
"No vamos a estar viviendo solas. Alma y yo vamos a estar juntas en el mismo apartamento."
"Ni aun así," disparó, y cuando yo iba a proseguir con mis argumentos, levantó un dedo en dirección mía, "sólo porque estás en esa escuela para blanquitos," pero en ese punto yo me desconecté.
Alma tuvo la misma discusión con Titi Ana y tuvimos que aceptar que dos mujeres solas viviendo juntas seguían estando solas si no tenían un hombre que las velara.
Charlie nació en febrero y Don Carlos aprovechó el nacimiento de su hijo para colarse de nuevo en nuestras vidas. Se aparecía un día con una pañoleta para Mami, con un regalo de cumpleaños para Frankie, con un trajecito para Charlie. Jugaba _gin rummy_ con nosotros o se quedaba hablando con Tata y Don Julio hasta tarde y después, cuando creía que estábamos dormidos, subía al cuarto de Mami. Tan pronto se iba a trabajar, Tata empezaba a sermonear a Mami por haberlo aceptado de nuevo. Cuando regresaba por la noche, le servía la comida acompañándosela de insultos en voz baja. "Sinvergüenza," le decía cuando le ponía el plato de arroz y habichuelas en la mesa; "desgraciado," cuando le servía el café.
Mami se abochornaba por la frescura de Tata, pero a Don Carlos parecía no importarle. Con los ojos atrapados detrás de los cristales oscuros y una media sonrisa en los labios, como si lo que ella dijera fuera divertido más que ofensivo, Don Carlos, ni caso le hacía. Trataba de ganársela con regalos —un galón de vino Gallo, una caja de cigarrillos, un cuadro en terciopelo de John F. Kennedy y de Martin Luther King, cara a cara, frente al sangrante Corazón de Jesús. Tata le cogía las ofrendas, pero no cedía, y hasta llegué a preguntarme si Don Carlos no disfrutaría del abuso que ella le propinaba; y si parte de su razón para vivir con nosotros no sería poder escuchar un resumen honesto del tipo de persona que era. Suponía que tenía que querer a Mami para tolerarle sus ocho hijos y una mamá cantaletera; y ella lo quería también porque, muy pronto, los trajes oscuros de Don Carlos estaban colgando en el _closet_ como murciélagos gigantes y los dedos de Mami acariciaban los puños y el cuello de sus camisas blancas de algodón mientras se las planchaba todas las mañanas.
Luigi y La Muda no podían vivir juntos, pero tampoco podían vivir separados. Se separaban, se juntaban, se separaban otra vez, y volvían a unirse en un encuentro lloroso cuando él aparecía, como por casualidad, en alguna reunión de familia. La Muda, Tata y Mami tenían largas conversaciones que no nos permitían presenciar, pero, yo me las agenciaba para necesitar algo de la cocina siempre que La Muda venía de visita, y como ya era casi una mujer, Mami no me espantaba de allí como le hacía a mis hermanas menores. En una de sus visitas, La Muda tenía un golpe en la cara, lo que sugería que Luigi le había pegado. Era diffícil de creer que un hombre tan tranquilo y tan suave como él pudiera pegarle a nadie, pero era aún más difícil creer que La Muda mintiera sobre algo así.
Unos días más tarde Luigi vino a vernos, su triste figura encorvada dentro de su traje, como si se le hubiera encogido y la ropa le hubiera crecido alrededor. Había decidido regresar a Puerto Rico. Le rogamos que no se fuera, pero dijo que ya no aguantaba más el frío.
"Mírenme las manos," se quejaba. Los chichones en los nudillos eran enormes, y tenía los dedos doblados uno sobre el otro, como en puños sueltos. Miré para otro lado.
Caminó hasta la estación del tren arrastrando dolorosamente los pies por la acera. Había envejecido tanto en los últimos cinco años que era difícil creer que había sido joven y había hecho trucos de magia y había sido el amante de La ardiente Muda. Sentí que no lo volveríamos a ver; y menos de un mes después, supimos que había muerto. No estaba claro de qué. Alguien dijo que de artritis. Murió sufriendo unos espantosos dolores bajo el calor del sol de Puerto Rico. Alguien comentó por lo bajo que Luigi amaba tanto a La Muda que no pudo tolerar vivir sin ella y se suicidó. Murió de una borrachera, era la tercera teoría, la menos creíble, porque jamás lo vimos borracho. Nunca se supo. Simplemente, desapareció de nuestras vidas; consumido por el dolor, la pena o el licor; un recuerdo de ágiles dedos pálidos esparciendo magia al aire.
La primavera de mi año _senior_ trajo consigo informes diarios de las universidades donde mis compañeras habían solicitado, pero yo no. Misis Provet, la Dra. Dycke y el consejero escolar me animaban a seguir estudiando.
"No tengo dinero para ir a la universidad," les decía. "Necesito un trabajo para ayudar a mi mamá."
"A lo mejor puedes trabajar a tiempo parcial," sugirió la Dra. Dycke.
"La mayor parte de las universidades tienen programas de estudio y trabajo," añadió Misis Provet.
Pero no tenía interés en ir a la universidad, todavía no. Quería salir al mundo, ganarme la vida, ayudar a Mami, sí; pero también, dejar de depender de ella para cada necesidad.
El Sr. Murphy me ofreció trabajo a tiempo completo en su laboratorio de Brooklyn, pero había decidido buscar trabajo en Manhattan, en una de las nuevas y relucientes torres de oficinas que brotaban de la tierra como desafiantes y austeros fortines. El único problema era que yo no tenía destrezas que ofrecerle a un negocio.
Performing Arts ofrecía un curso de mecanografía diseñado para enseñarnos una destreza práctica por si acaso nuestros talentos no eran reconocidos enseguida cuando nos graduáramos. Me sentaba en la primera fila, los pies planos en el piso, la espalda recta, la cabeza erguida, los ojos fijos al frente, los dedos posados sobre el teclado, según Misis Barnes iba cantando las letras que teníamos que apretar sin mirar.
" _T_ mayúscula, _r_ minúscula, _J_ mayúscula, _m_ minúscula, seguro, _H P S V_ , punto y coma."
Cada tecla era un martillo que clavaba mi futuro en un rodillo de goma. Si no era actriz, secretaria. Si no era bailarina, secretaria. Si no era secretaria, ¿qué?
El Show de los Seniors, fue el último programa de mi clase para toda la escuela. Suponía que me tocaría representar a otra Cleopatra de alguna oscura obra de autor desconocido, puesto que ya había interpretado a todas las Cleopatras famosas. Me sorprendió que me seleccionaran para representer a la Virgen María con Laura Rama interpretando a Bernardette de Lourdes.
Cuando le conté a Mami que iba a hacer de la Virgen María, quedó en éxtasis. A diferencia de Cleopatra, Reina del Nilo, Mami sabía que la Virgen era un personaje respetable que no usaba un vestuario extravagante ni maquillaje recargado. Se ofreció a hacerme el vestido, pero la escuela se encargó de proveérmelo.
El primar día de ensayo me enteré de que mi parte no requería que actuara en el sentido tradicional. Sería una María danzante, sin diálogo. Mientras mis compañeros marcaban sus escenas, en un rincón del sótano, Miss Cahan y yo trabajábamos con el baile de la aparición.
La mía no era una famila religiosa, así es que mi idea de la Virgen María estaba basada en lo que había recogido de Abuela, la mamá de Papi, católica devota, de misa todas las mañanas y rosario todas las noches. "Santa María, Madre de Dios," me había enseñado, y en las improvisaciones de mi baile, trataba de dar con unos movimientos decorosos y evocadores dignos de la Madre de Dios.
Miss Cahan, sin embargo, sugería una interpretación menos piadosa. Su visión de María estaba más acorde con la escuela de movimiento de Martha Graham: fuerte, cortante y abstracta. Con un profundo salto, caía en escena, plantaba firmemente mi pierna derecha, y la iba enderezando mientras la izquierda se elevaba paralela al piso. Los ojos fijos en el suelo, los brazos extendidos, la espalda recta; mantenía el balance con la pierna derecha, mientras con el resto del cuerpo formaba la tilde de la T. Mantenía esta posición hasta que Bernardette de Lourdes se fijaba en mí y caía en trance. Entonces, todavía en una pierna, enderezaba el cuerpo mientras mi pierna izquierda se elevaba hacia arriba, arriba, arriba para formar un _split_ de pie, que yo sostenía con mi mano derecha. No era demasiado modesta, esta Virgen María, con to' por fuera.
Laura Rama se arrodilló al pie del escenario, mientras yo daba vueltas por la parte de atrás como una tigresa hambrienta, mi larga túnica siseando, la terrible visión de Bernardette. Ni una sola vez en todo el baile junté las manos en la posición tradicional de orar, ni se abrieron mis brazos suavemente para abrazar a la humanidad. Era una Virgen guerrera, llorando a mi Hijo. Con el torso contraído, lo buscaba en mi vientre vacío. Con los brazos extendidos hacia atrás, arqueaba mi corazón hacia el cielo, retando a Dios a llevarme a mí en vez de a Él, que sufría en la cruz. Cuando salí de escena, con una salida tan impetuosa como mi entrada, hubo una pausa seguida de dos o tres palmadas y finalmente, un verdadero aplauso. Corrí por las escaleras del escenario, por la parte de atrás hasta el salón de maquillaje, donde caí, hecha un manojo de nervios. Northern entró corriendo de los lados.
"¡Estuviste estupenda!," sonrió. "¡Estupendo vestuario también!"
Me reí y le di las gracias, pensando que estaba bromeando porque finalmente había dejado atrás el mantel amarillo de los días de Cleopatra. Cuando el próximo grupo de actores salía a escena, me doblé desde la cadera para estirar la espalda, con las piernas separadas en segunda posición y noté que, con las luces del espejo de maquillaje que estaba detrás de mí, mi túnica virginal era transparente. Me levanté en pánico. Durante el baile me habían estado iluminando desde atrás para realzar el efecto dramático de la coreografía Grahamesca.
Caí de rodillas y me tapé la cara con las manos. A mi alrededor, mis compañeros corrían de un lado para el otro preparándose para su momento en escena, mientras yo hacía un esfuerzo por desaparecer.
Después de la función, todo el mundo se reunió para una recepción en el sótano. Cuando bajé, Mami, mis hermanas, mis hermanos y Don Carlos me rodearon.
"Ay, Santo Dios," Mami estaba sin aliento, "tiene que ser pecado faltarle el respeto así a la Virgen." Estaba roja y miraba entre la gente, como si el mismísimo Dios viniera caminando hacia nosotros para castigarme allí, en el acto.
"Se te vio to' a través del traje," anunció Delsa; y una gente que estaba alrededor de otro estudiante, se viró hacia nosotros y se rió.
Miss Cahan se acercó. "¡Estuvo magnífico!" Me besó. "Precioso."
"Mi vestido..." balbuceé al borde de las lágrimas, "la túnica..."
"No te preocupes," me tranquilizó. "Estuvo maravilloso."
El sótano parecía un gallinero con el chachareo de maestras y maestros orgullosos y de estudiantes excitados. Durante tres años habíamos sido los críticos unos de otros, pero esta noche a todo el mundo le encantó el trabajo de los demás. Según fuimos abrazándonos, besándonos y felicitándonos, mi familia se retiró. No fue una gran distancia; dos o tres pies a lo sumo, pero era como un continente. Sentía su halón desde la esquina del salón donde se habían agrupado y reían y hablaban, aislados del barullo de actores volubles y maestros joviales. No podía alejarme, pero tampoco quería quedarme con ellos y perderme la camaradería entre actores después de una función. Mami, Don Carlos y mis hermanas y hermanos me halaban en una dirección, mis pares y maestros tiraban en otra. Inmóvil, me mantuve a mitad de camino entre ambos, incapaz de escoger, deseando que la fiesta no se alejara ni una pulgada de mí; y que mi familia también se mantuviera firmemente donde estaba. Al final, quedé sola entre los dos, y cuando vi claro que nadie me extrañaba en la animada reunión de actores y maestros, volví donde Mami y al ratito, ya estábamos en el tren, camino a Brooklyn.
En casa tuve que aguantar una representación de mi baile que mis hermanas hicieron para beneficio de Tata, Don Julio y los nenes que no habían ido. Verlas recrear mi Virgen María fue tan cómico que me reí hasta que se me saltaron las lágrimas. Más tarde, cuando todos nos acostamos y la casa estaba tranquila, lloré de verdad. No sabía por qué; no quería saber. Dejé que brotaran las lágrimas con la esperanza de que por la mañana mis ojos hinchados no me delataran.
Le pedí a Papi que viniera a mi graduación. Me escribió diciéndome que ya veríamos. "Ya veremos," generalmente quería decir que no, así es que no insistí, pero me sentí desilusionada. En los cinco años que dejé de ver a Papi, había crecido por lo menos cinco pulgadas, había aprendido a maquillarme, había adquirido un segundo idioma, me había vuelto lo suficientemente independiente como para viajar sola por Brooklyn y Manhattan, había tenido dos trabajos, me había convertido en bailarina, había logrado evitar los "algos" que le pueden pasar a las muchachas en la ciudad. Y él no había estado allí.
Me preguntaba si tendría él alguna idea de cómo eran nuestras vidas en Nueva York. Mis cartas casi nunca le describían las condiciones en que vivíamos. Ni siquiera en nuestros peores momentos le pedí ayuda. Era su responsabilidad interesarse por las necesidades de sus hijos, no la nuestra rogarle que se ocupara de nosotros.
Muchas veces sentí tanto coraje con él, que deseaba encontrar una manera de decírselo, pero nunca me atrevía a ser irrespetuosa, a arriesgarme a que se enojara. Mis cartas contestaban las suyas llenas de noticias pero, yo esperaba más de él. Deseaba los pequeños gestos de ternura que marcaron nuestra vida en Puerto Rico. El tiempo que se tomaba en explicarnos cosas, las horas que pasamos, uno al lado del otro, martillando clavos en las paredes. Su paciencia cuando me enseñó a mezclar cemento, a colocar un bloque de concreto sobre el cemento blando, a raspar con una espátula triangular el fango que salía por debajo del bloque. Me hacían falta los poemas que escribía, los chistes bobos que contaba, las melodías que tarareaba mientras trabajaba. Ahora que era casi una mujer, extrañaba a mi papá más que nunca. Pero no me atrevía a decírselo, temerosa de que mi necesidad fuera a sonarle a exigencia o le pareciera una crítica a la capacidad de Mami para cuidarnos. Lo que hice fue ahogar el hambre de un padre que se convertía cada vez más en una abstracción tan ilusoria, como el verde destello de un atardecer tropical.
Justo antes de los exámenes finales, se aparecieron por Performing Arts unos hombres bien trajeados. En un momento, se regó el rumor de que eran productores de Hollywood buscando artistas para una película. Visitaron algunas clases, pero parecían estar más interesados en la arquitectura de la escuela, que en los estudiantes o los maestros. Sin embargo, unos días más tarde, a algunos se nos comunicó que habíamos sido seleccionados para audicionar para la versión fílmica de _Up the Down Staircase_ de Bel Karman.
"¡Mami, me descubrieron!" dije, echándomelas, tan pronto llegué a casa.
"¿Te descubrieron haciendo qué?" preguntó.
Le expliqué lo poco que sabía. Estaban haciendo una película de un libro famoso sobre una escuela. El autor había sido maestro en Performing Arts. Los productores habían venido a examinar los alrededores porque a lo mejor filmaban allí y escogían a algunos muchachos y algunas muchachas para salir de estudiantes en la película.
"¡Ay, qué bueno!"
"Ten mucho cuidado," metió Tata la cuchara. "Algunas veces esa gente del cine lo que quiere es conocer a muchachitas jóvenes."
Tata nunca había conocido a nadie del mundo del cine y hasta donde yo sabía, ni siquiera había ido al cine, así es que su advertencia me entró por un oído y me salió por el otro. Aun así, fue un alivio cuando llegué a la entrevista en Warner Brothers y no había "un caucho" en la oficina del director de reparto.
El Sr. Jeffers era un hombre sin edad, de quijada cuadrada, cuya sonrisa estudiada, provocaba, sin embargo, en los demás, una sonrisa de oreja a oreja. Estaba sentado detrás de un enorme escritorio cubierto de fotografías en blanco y negro de cuanto aspirante a actor había en Nueva York. Yo no tenía una foto _head shot_ , pero él me dijo que no era necesario.
"Estamos buscando gente de verdad," dijo, "no necesariamente actores profesionales."
No me dio a leer de un libreto sino que me hizo preguntas diseñadas para hacerme hablar. Me di cuenta de que quería asegurarse de que no tenía acento, así es que pronuncié cada palabra con la dicción estándar, modulando la voz como me habían enseñado. Satisfecho, el Sr. Jeffers se puso de pie para acompañarme a la puerta y me sorprendió ver lo bajito que era. Sacó una tarjeta de presentación del _wallet_ que tenía en el bolsillo y me la entregó. Cuando la fui a coger, tomó mi mano en las suyas y me dobló los dedos alrededor de le tarjeta.
"Esta es mi línea directa," dijo. Asentí pero no me atreví a mirarle a los ojos. ¿Estaba coqueteando o estaba siendo amable? Me llevó por un laberinto de pasillos y oficinas hasta el área de la recepción.
"Llámame mañana," me dijo con su sonrisa perfecta.
Esa noche me encontré con Alma para cenar y hablamos de la entrevista.
"Es raro. No hizo nada más que agarrarme la mano un poquito más de lo necesario.
"¿Tú crees que se estaba prospasando contigo?"
"Me parece que sí, pero no estoy segura."
"No podemos sospechar de cada hombre que conocemos," comentó Alma.
"Quizá le estoy dando demasiada importancia. Pero me parece estar oyendo, 'Los hombres sólo quieren una cosa,' dije, imitando la voz de Mami, pero Alma no se dio cuenta.
"Lo mismo me pasa a mí. Lo pensó por unos momentos. Quizá nuestras madres no han conocido hombres buenos."
"Son buenos al principio," le recordé a Alma. "Entonces, cuando consiguen lo que quieren..."
"¡Ahora, suenas igual que tu mamá!" Alma se rió y yo me abochorné, pero tuve que aceptar que era verdad.
Desde que Mami y Titi Ana habían aguado nuestros planes de mudarnos juntas, los hombres habían sustituido a los apartamentos como tema principal de conversación cuando comíamos fuera. En casa estaban Don Julio y Don Carlos y teníamos visitas frecuentes de los tíos y los primos de Mami, que lo mismo venían solos que con sus esposas y novias. Pero Titi Ana no fomentaba mucho que familiares varones pasaran por su casa, especialmente mientras ella estaba trabajando y Alma y Corazón estaban solas.
Cuando de hombres se trataba, yo tenía conocimiento de primera mano y muchísima experiencia, en comparación con Alma, cuyo contacto principal con los hombres era su jefe, el comerciante de medias. A veces hablábamos del tipo de hombre con quien nos gustaría casarnos.
"Rico," decía yo cuando me preguntaba.
"Pero, ¿además de eso, qué otras cualidades? ¿Sentido del humor, que sea bueno?"
"No," insistía, "sólo rico." Se rió porque pensó que estaba bromeando. "Vamos a suponer que nuestras madres tienen razón y que los hombres sólo quieran una cosa," continué, "¿de qué vale dársela a cualquiera? Es lo único que tenemos para ofrecer."
"No estoy de acuerdo contigo," los ojos oscuros de Alma se agrandaron. "No puedes seguir pensando así."
"¿Por qué no? Los hombres piensan así de nosotras."
"No Negi, eso no es así." Sacudió la cabeza como tratando de sacarse mis palabras de su cabeza.
"No es broma. Cuando yo esté lista para entregar mi virginidad, va a ser para el mejor postor."
"Ay, Dios mío, eso es terrible. No bromees con eso. Eso no da gracia."
Me encantaba verla tan turbada. Con Alma, yo podía darme el lujo de ser chocante y decir cosas que no me atrevía a decir delante de nadie. Lo más divertido era que ella me lo creía. Yo le decía las cosas más locas y ella las tomaba en serio. Hablaba sin pensar, por el puro placer de verle la cara o de discutir algún punto con ella, o de escucharme a mi misma expresar opiniones que ni siquiera sabía que tenía hasta que se me derramaban de la boca.
"Tú vas a ver," le decía. "El primer hombre con quien lo haga, va a ser un millonario."
"Procura estar enamorada de él," me advertía.
"Seguro," le decía. "Una vez que yo sepa que es rico, me enamoro de él." Entonces, nos reímos las dos.
Cuando llamé al Sr. Jeffers, me dijo que tenía interés en que hiciera una prueba para Carole Blanca, un rol estelar, el papel más prominente en el cine para una actriz puertorriqueña.
"Ven preparada para hacer un monólogo corto," me dijo. "Y me llamas después."
"¿Y usted no va a estar allí?"
"No, esa no es mi área," rió. "Buena suerte."
La audición fue en un estudio de ensayo en la West 49 Street. Había varias sillas alineadas contra la pared en un pasillo que daba a una puerta cerrada. Cuando llegué arriba, salió una mujer volando de adentro, me preguntó el nombre, lo cotejó en una lista que tenía en su _clipboard_ , me indicó la última silla desocupada y volvió a desaparecer por la puerta.
Había tres actores antes que yo. Me descartaron tan pronto se dieron cuenta de que yo no era competencia para ellos. Los fueron llamando uno a uno y yo seguía adelantando en la fila. Cuando llegó mi turno, la señora que me había recibido me llevó a un teatro pequeño y oscuro, con un escenario tan chiquito que ni siquiera estaba más elevado que el nivel del público.
Había varias personas apiñadas en la parte de atrás. La Srta. Silver me presentó a un señor muy elegante, el Sr. Pakula, el productor, y a otro señor desgreñado, con un bigote grueso, el Sr. Mulligan, el director. Hablamos un ratito y me fueron haciendo las preguntas de rutina en estos casos. Entonces, el Sr. Pakula señaló el escenario y me dijo que estaban listos para escucharme.
"Pudes usar cualquiera de las piezas de utilería que encuentres allí," sugirió el Sr. Mulligan, lo que significaba que quería verme manejando utilería. Había escogido una escena de _Member of the Wedding_ en la que Frankie le dice a John Henry que quiere irse de su pueblito. No había ensayado la escena con la utilería, pero encontré un banco pegado de la pared y lo incorporé al monólogo. Cuando me senté a mitad del monólogo, el banco se tambaleó y yo di un salto, pero aun así, me mantuve en papel y terminé la escena como si todo hubiera sido planificado. Era la mejor prueba que había hecho hasta ahora. El Sr. Pakula y el Sr. Mulligan me dieron la mano, me dijeron que lo había hecho muy bien y que el Sr. Jeffers me avisaría en un par de días, tan pronto como entrevistaran a todas las que estaban siendo consideradas para el papel.
Me sentía orgullosa de mi misma. Había reaccionado apropiadamente ante cada situación y tenía esperanzas de conseguir el papel. A pesar de que eran pasadas las cinco, llamé al Sr. Jeffers que sonaba simpático y excitado.
"Estuviste estupenda esta mañana," me dijo.
"Querrá decir esta tarde," corregí.
"Si fueran a tomar la decisión en este momento, tú serías la persona," me aseguró y casi podía ver su sonrisa luminosa a través el teléfono. "Ven mañana por la tarde. Con toda seguridad te tengo buenas noticias."
Esa noche no pude dormir. Veía imágenes de mí misma como Carole Blanca, mi primer papel importante en una película de Warner Brothers, una importante compañía de Hollywood. No sería teatro legítimo, pero mi formación de Performing Arts me permitiría superar "la indicación" tan obvia de los actores de cine. Estaría brillante. Sería la primera vez en mi corta carrera que haría de una puertorriqueña. No María, ni Anita ni ninguna de las novias de Shark. Sería un personaje con un nombre, una muchacha lista, de mi edad.
Al día siguiente, cuando llegué a la oficina del Sr. Jeffers, pareció sorpendido de verme.
"Usted me dijo que viniera hoy," le recordé.
"Sí, claro." Se veía desconcertado, confundido, como si yo fuese la última persona que esperara ver. "¿Tú eres Esmeralda Santiago?"
"Sí."
"Bien. Dame un segundo." Buscó entre sus papeles. Le tomó un rato y me dio la impresión de que estaba dándole largas al asunto. Por fin, me preguntó si había traído la foto de cara.
"Usted me dijo que no era necesario."
"Es verdad, lo dije." Movió los papeles otra vez.
"Si éste no es un buen momento para usted, puedo regresar más tarde," le ofrecí.
"Sí. No. Está bien, está bien." Respiró profundo dos o tres veces, aguantó la respiración, se colocó las manos frente a la nariz, en gesto de oración, se me quedó mirando dos o tres segundos hasta que desvié la mirada.
"La verdad es," exhaló, "que te tenía confundida con otra muchacha."
"Ah."
"La otra muchacha," dijo, echándose hacia delante como si fuera a susurrar, aunque su voz permaneció igual. "La verdad es," repitió, "que tú no das la talla para el papel."
"Pero, usted dijo..."
"La otra muchacha, se ve más, ¿cómo te digo? Bueno, la verdad es," dijo por tercera vez, y yo deseé que me mintiera porque la tirantez en su voz me decía que yo no quería oír lo que iba a decir.
"La otra muchacha se ve más puertorriqueña."
"¿Qué?"
"No tienes el tipo. Eres una muchacha bonita. Así es el cine. Tiene que ver más que nada con el tipo."
"¿Soy demasiado bonita para ser puertorriqueña? ¿Eso es lo que está diciendo?"
"No te ves lo suficientemente puertorriqueña. Pero vas a salir en la película, no te preocupes. Hay muchísimas otras partes... un salón lleno de jóvenes..."
Sentí que me salía del cuerpo y me elevaba hasta la esquina del salón. Allí estaba el Sr. Jeffers, desdichado y pequeño, y yo sentada frente a él, las manos agarradas de los brazos de la butaca, como si al soltarlas fuera a salir disparada al espacio. Seguía hablando o por lo menos, me lo parecía, y mientras más se sonreía, más se desvanecía su sonrisa luminosa. Escribió algo en un papelito, me lo pasó por encima del escritorio y yo, que estaba sentada en la butaca, lo cogí, lo leí, lo doblé y lo metí en la carterita que tenía en la falda. Se levantó, extendió su mano. Yo no flotaba ya sobre mi cabeza. Le estaba dando la mano como si me hubiera hecho un favor. Salí del edificio en una nube; fui directamente a la biblioteca y encontré un retrato de Rita Moreno, otro de Chita Rivera y un tercero de José Ferrer. No era gente fea. Eran puertorriqueños hermosos. Pero, me pregunté, ¿parecen ellos puertorriqueños? De no haber sabido que lo eran, ¿hubiera dicho ahí va un compatriota? Al saber quiénes eran, no podía saber qué hubiera hecho de no haberlo sabido. Sólo sabía que, según el Sr. Jeffers, mi única conexión con el mundo de la cinematografía, la gente puertorriqueña no era bonita.
Cuando llegué a casa no mencioné para nada la humillación sufrida. Anuncié en voz alegre que me habían contratado para trabajar en la película, que me pagarían y que, obviamente, la regla de los diez años, no se aplicaba a mí. Una semana antes de la gradución, ya tenía trabajo como actriz. Me convencí de que era mucho más de lo que me había atrevido a soñar. Como tantas veces nos habían dicho, el rechazo es parte del trabajo. No se puede coger personal.
El auditorio de Performing Arts estaba lleno a capacidad. Cuando entramos con nuestras togas y birretes, el público se levantó a aplaudir. Era nuestra última actuación, el último día que apareceríamos en el auditorio siendo estudiantes. Papi no estaba. Con el rabo del ojo veía las sonrisas orgullosas de Mami. A su lado Don Carlos, engabanado y con sus gafas oscuras, lucía alto y digno, orgulloso también a pesar de que yo no era su hija. Sólo me habían permitido llevar dos invitados. Les había explicado que yo era la primera de nueve hijos en graduarme de Escuela Superior y que mi mamá me quería usar de ejemplo y traer a todos los niños para que me vieran. Pero, el auditorio de Performing Arts era pequeño. Sólo se daban dos invitaciones por estudiante.
Durante el programa, a muchos de mis compañeros los llamaron para recibir premios y honores. A mí no me llamaron pero no importaba. Yo sabía lo que había logrado. Ni mi mamá ni mi papá habían estudiado más allá de la escuela elemental. Y allí estaba yo, en un país extranjero, en un idioma extranjero, graduándome de una escuela para soñadores. De haberme detenido a pensar en mi futuro, me habría dado miedo. Pero lo que sentí ese brillante día de junio, fue la emoción del logro. Había logrado terminar la escuela superior sin salir encinta, sin salirme de la escuela, sin que "algo" me pasara. Tenía trabajo de actriz en una película, no sería un papel protagónico, pero por los menos me pagarían y quién sabe, a lo mejor hasta me descubrían.
Pero primero tenía que regresar a casa en Brooklyn, con mi mamá y mi padrastro a celebrar con mi hermana, la cajera de Woolworth's, y con mi hermano, el cocinero de pizza, mis otras seis hermanas y hermanos, mi abuela y su novio, mis primos, la sordomuda, el luchador y las hermanas americanizadas, con mi tío alcohólico. Ese mundo en Brooklyn, de donde yo derivaba tanto consuelo como ansiedad, era mi hogar; como lo era también el otro mundo cruzando el mar, donde mi padre todavía escribía poemas; como lo era también el otro mundo, al cruzar el río, donde yo tenía la intención de hacer mi vida. Tendría que aprender a montarlos todos, una jinete en tres caballos, cada uno tirando en una dirección diferente.
# "¿Quién tú te crees que eres?"
#
Una semana después de graduarme de Performing Arts, me encontraba en medio del patio de una escuela elemental en el Barrio, rodeada de otros aspirantes, para el primer día de filmación de _Up the Down Staircase_. El asistente del Sr. Mulligan nos dijo que contáramos con que tendríamos trabajo todos los días durante las próximas dos semanas. El patio sería nuestra base de operaciones, mientras el _crew_ estuviera filmando los exteriores frente a la escuela y al cruzar la calle.
La mayoría de los muchachos que salían en la película eran de las escuelas locales, pero unos cuantos eran profesionales con experiencia haciendo comerciales y películas. Habían estado antes en _sets_ y vinieron preparados con libros, el tejido, barajas y juegos de mesa para pasar el rato entre tomas.
La trama de _Up the Down Staircase_ giraba en torno a una joven maestra, Miss Barrett, en su primer trabajo en una escuela llena de estudiantes poco motivados en la ciudad de Nueva York. Entre los personajes estaba el payaso de la clase (judío), la chica gorda y fea (blanca), su mejor amiga (el papel que no conseguí porque no me veía lo suficientemente puertorriqueña), el joven Republicano del futuro (también blanco y también gordo), el joven sensible y destinado al fracaso (puertorriqueño negro), el rebelde sin causa italiano, la puta. Sandy Dennis hacía de maestra idealista. El reparto incluía también otros tipos: la "Maestra Dedicada," la "Solterona Frustrada," el "Principal Fascista," el "Poeta Alcohólico."
Cada vez que averiguaba el nombre de alguien asociado con la película, iba a la biblioteca y buscaba información. El director, Robert Mulligan, había ganado un Premio de la Academia por _To Kill a Mockingbird_ , producida por Alan Pakula. Ted Mosel, un dramaturgo muy respetado, escribió el libreto. Y los actores dramáticos, Roy Poole, Elizabeth Heckart, Maureen Stapleton, Ruth White y Vinnette Carroll (quien era maestra en Performing Arts) eran genuinos actores de teatro, como también lo era la estrella Sandy Dennis, quien acababa de protagonizar, junto a Elizabeth Taylor y Richard Burton, la película _Who's Afraid of Virginia Woolf?_ Me sentía orgullosa de estar en medio de tanto talento.
Muchos de los exteriores se filmaron en El Barrio, en los alrededores de la escuela elemental y las calles aledañas. Nos citaban al lugar de rodaje temprano en la mañana y a veces estábamos allí hasta tarde en la tarde. Hacíamos una misma escena dos o tres veces, y entonces esperábamos hasta que las luces, la cámara y el sonido estuvieran listos para hacer otra toma desde otra dirección. El trabajo era tedioso, pero, me dejaba mucho tiempo para leer, para aprender a jugar Monopolio y _Scrabble_ y para charlar con los demás _extras_. Como yo, esperaban dejar una impresión tal en la película, que serían descubiertos, se irían a Hollywood y se convertirían en estrellas. Hacíamos todo lo posible por llamar la atención del Director, el Productor y el equipo de producción. Dábamos masajes en hombros adoloridos, repartíamos café, coqueteábamos, oíamos con atención el montón de chistes zánganos. Dio resultado. Después que se filmaron los exteriores, algunos de nosotros fuimos seleccionados para aparecer en el salón de clases, lo que quería decir que tendríamos más trabajo y mejor paga.
La producción se movió a una escuela superior cerca de Lincoln Center, donde fueron filmadas las escenas del pasillo, la escalera y algunas de las del salón y de la oficina. Entonces, nos llamaron para ir al estudio de sonido en la West 20, donde se recreó el salón con todo, hasta con la vista exterior de la escuela que acabábamos de dejar, que se logró a través de una transparencia gigante. Las paredes se movían para dejar pasar las cámaras, las luces y los técnicos iban y venían entre tomas para ajustar las luces y los micrófonos, para empolvarle la cara a Sandy Dennis o echarle fijador de pelo.
Cuando no se le requería en el _set_ , Miss Dennis se iba a su camerino o se sentaba en una silla de director hasta donde se acercaban los muchachos y las muchachas lo suficientemente valientes como para arriesgrarse a que alguien los espantara de allí. Era simpática y parecía disfrutar cuando uno de nosotros le hablaba como si fuera una persona normal y no una estrella de cine. A veces hacía cosas que no sabíamos cómo interpretar. Una vez regresamos del almuerzo a filmar una escena que requería profunda emoción y concentración de parte de todo el mundo. Ensayamos la escena un montón de veces y se nos había advertido antes del receso que era una escena dificil de filmar, que teníamos que escuchar bien, concentrarnos y seguir las instrucciones cuidadosamente para facilitarle el trabajo a los actores principales. Estábamos nerviosos y cuando entró Miss Dennis nos enfocamos y nos preparamos para el momento en que en voz baja, el Sr. Mulligan dijera "¡Acción!" La cara de Sandy Dennis se contrajo, abrió la boca y de adentro salió un largo, redondo y estruendoso eructo. Todo el mundo quedó tieso. El Sr. Mulligan gritó "¡Corten!"
Miss Dennis se echó a reír. "No debí haber tomado cerveza en el almuerzo." Nosotros explotamos de la risa. Nos tomó un rato tranquilizarnos porque cada vez que el Sr. Mulligan decía acción, alguien se reía y al ratito, el elenco y el equipo de producción completo, estaban muertos de la risa.
Generalmente, una suplente del alto y del tipo de Sandy Dennis tomaba su lugar en lo que se ajustaban las luces y las cámaras. Pero a veces lo hacía ella misma. Un día se sentó en su escritorio mientras los técnicos trabajaban a su alrededor. Mi escritorio quedaba directamente frente al suyo y de vez en cuando ella levantaba la vista y me sonreía. De pronto, como salido del aire, me preguntó si tenía hermanas o hermanos.
"Sí, soy la mayor de nueve hijos."
"¡Nueve!"
"Cinco niñas y cuatro varones."
"¿Tu mamá nunca ha oído hablar del control de la natalidad?"
Detrás de mí alguien rió con sorna. "Ella no cree en eso," murmuré porque no sabía qué más decirle. Miss Dennis asintió, perdió el interés en mí y se puso a conversar con Liz, que se sentaba al lado mío.
El control de la natalidad había estado apareciendo en las noticias por el invento reciente de la píldora para evitar embarazos. Cada vez que se discutía el tema en casa, los adultos sentados alrededor de la mesa de la cocina, coincidían en que "La Pastilla" no era más que una licencia para que las mujeres jóvenes tuvieran relaciones sexuales sin casarse. El hecho de que mi mamá, mi abuela y casi todas nuestras parientas tuvieran relaciones sexuales sin casarse, ni se mencionaba. Si yo se los hacía notar, me regañaban por irrespetuosa. En todo caso, tampoco se me hubiera ocurrido sugerirle a Mami que evitara los bebés. A pesar de que ser parte de una familia grande era difícil para todo el mundo, no había ni una sola hermana, ni un solo hermano que no hubiera querido tener.
En mi caso, tenía decidido ya que había cambiado todos los pañales que quería cambiar en mi vida y había planificado apuntarme en la pastilla tan pronto se presentara la necesidad.
A veces, cuando nos despachaban temprano del _set_ de _Up the Down Staircase_ , me iba a ver las vitrinas por la Quinta Avenida o me pasaba las horas muertas en la Biblioteca del Lincoln Center escuchando musicales de Broadway. De vez en cuando se me acercaba algún hombre.
"Con el permiso, ¿hay alguien en este asiento?" Señalaba el asiento vacío al lado del mío y a mí me daban ganas de decirle, "Sí, mi primo invisible está ahí," pero nunca me atreví. Cuando venía a darme cuenta, estaba conversando con Dan o Fred, o Matt o Kevin. A veces me invitaban a tomar un café. Nos sentábamos frente a frente a hablar de teatro, porque todos los hombres que conocía en Manhattan, de día, durante la semana, eran actores desempleados. Mientras los oía discursar sobre si El Método estaba _paseé_ o si los actores de teatro genuinos estaban vendiéndose cuando hacían comerciales, trataba de decidir si aquello contaba como una cita o no. No sabía bien cuáles eran las reglas del _dating_ porque nunca lo había hecho. Y me hubiera sentido bastante estúpida preguntándoselo a quienes sí lo habían hecho, como la chica que hacía de puta en _Up the Down Staircase_ , o como Liz, mi compañera de asiento en el _set_. Leí _Sex and the Single Girl, The Group_ de Mary MacCarthy, algo de Harold Robbin, tratando de descifrar qué era lo que se hacía en una cita por si acaso alguna vez tenía una. Pero mis encuentros en la Biblioteca nunca llegaban más allá de un café y no había más candidatos.
Delsa tenía novio, Norma tenía novio, Héctor tenía novia. Pero ninguno de ellos tenía citas. Los novios de mis hermanas venían a casa los domingos, comían con nosotros, veían televisión con los nenes y se iban a una hora decente. La novia de Héctor venía con su mamá. Héctor iba a su casa y hacía lo que hacían los novios de Delsa y Norma en casa. No se les permitía ir a ningún sitio sin chaperona, que frecuentemente era alguno de los nenes más pequeños porque había que cuidarlos, una no se les podía escapar y choteaban si pasaba algo impropio.
Por ser yo la que tenía la mayor libertad, hubiera podido tener una cita clandestina en la ciudad, sin que nadie se enterara. No lo había hecho porque nadie me había invitado, pero empecé a planificar para cuando me tocara el momento. Todos los días, hacia el final del verano, según fueron concluyendo los trabajos de _Up the Down Staircase_ , me fui requedando más tiempo, e inventaba excusa tras excusa para llegar a casa mucho más tarde de la hora en que me esperaban. Mami fruncía el ceño, achicaba los ojos, apretaba los labios, en todas sus muecas habituales que yo sabía que querían decir que tenía sus dudas pero no iba a hacer nada todavía. Para no levantar sospechas sobre lo que en realidad no estaba haciendo, yo no abusaba. Poco a poco iba estirándole los límites de su tolerancia y cuando se quejaba de que había llegado tarde demasiados días corridos, al día siguiente, yo no salía si no había filmación. Jugaba con mis hermanas y hermanos, salía de tiendas con Mami o me quedaba en casa leyendo, con el pelo en rolos probando un nuevo peinado, y trataba de actuar como si no hubiera tenido nada que esconder, que de hecho no tenía, pero segura de que algún día sí lo tendría.
A pesar de que me había faja'o con la esperanza de ser descubierta durante la filmación de _Up the Down Staircase_ , al terminar la película no hubo ofertas de Hollywood, así es que tuve que decidir cuál sería mi próximo paso. Me di un mes para conseguir una oferta para un papel en otra película o como bailarina en alguna compañía. Compraba _Backstage_ y _Variety_ todas las semanas, hacía una lista de las audiciones para las que podía cualificar, llamaba a agentes de reparto, me aparecía por los estudios de ensayo cuando se anunciaban pruebas, pero no había un papel para una Cleopatra/bailarina clásica india/dama joven/puertorriqueña.
El final del verano y el comienzo del otoño fueron fuertes para Mami debido a los gastos tan grandes que tuvo para preparar a los nenes para la escuela y para otro invierno. Yo había ganado mucho dinero en el verano, pero había gastado la mayor parte en clases de baile y en comprar ropa apropiada para una actriz/bailarina que tenía que causar una buena impresión en las audiciones. Le daba a Mami parte de cada cheque y Don Carlos también aportaba, especialmente ahora que Mami estaba encinta de nuevo. Pero con todo y eso, no daba, así es que nos mudamos de la casa a un apartamento más pequeño en un tercer piso, que incluía agua y luz.
La dueña de la casa, Doña Lila, vivía en el segundo piso con sus dos hijos, uno de ellos, un par de años mayor que yo. Neftalí era esbelto, de tez color café con leche oscuro y unos impresionantes ojos verdes. Era el hombre más guapo que había visto y sus modales tan respectuosos, su voz suave y su sonrisa tierna me producían maripositas en el estómago y un cosquilleo cuando me miraba.
Mis hermanas y hermanos se dieron cuenta de que me gustaba Neftalí.
"¿Por qué es que ya tú no andas por la casa en rolos?" preguntaba Héctor.
"Sí, y subes y bajas por esas escaleras veinte veces al día," añadió Alicia.
"Antes tú nos pagabas pa' que sacáramos la basura cuando te tocaba a ti," se quejaba Raymond.
"Chica, Negi," rogaba Norma. "No me estoy ganando ni un chavo contigo."
Neftalí subía con frecuencia y se unía a los juegos de dominó y de _gin rummy_ que desde la mesa de la cocina, competían con la risa que venía del televisor en el otro cuarto. Era un jugador malísimo, lo que le hacía aún más divertido porque apostábamos en cada juego. Venía los domingos, como los novios de Delsa y de Norma. Le encantaba leer, cosa que me gustaba, pero sus libros favoritos eran los de ciencia ficción, que yo no entendía. Se había graduado de escuela superior y hablaba de ir a la universidad algún día. Mientras tanto, trabajaba en el _garment center_ , empujando carritos cargados de ropa recién hecha, desde la fábrica donde la cosían, hasta los almacenes, donde la embarcaban. Decía que era como levantar pesas y dejaba que mis hermanos se le colgaran del brazo doblado para mostrar sus bíceps.
"¡Estás enamorá'! Yo sabía que tú no lo decías en serio na' cuando decías... tú sabes," Alma se ruborizó, "lo de entregarle tu virginidad al mejor postor."
"Yo no voy a tener sexo con él ni na'," protesté, pero me ruboricé también, aunque por una razón diferente. Durante días había estado fantaseando con los besos de los labios café con leche de Neftalí. Y más de una vez, había dejado correr mis manos por mi cuerpo, imaginando que eran las suyas. "Además," continué, "no hemos esta'o solos todavía. Mami no me quita los ojos de encima cuando él está por ahí."
Era cierto. Pero también era cierto que Neftalí no mostraba ningún interés en estar a solas conmigo. Oportunidades había de más. Podía haberme acompañado cuando iba a alguno de los muchos mandados que me ofrecía a hacerle a Mami. Me podía haber esperado en la estación del tren cuando yo regresaba de mi trabajo. Podía haber subido mientras Mami estaba trabajando y Tata y los nenes veían televisión. Pero, no hacía nada. Se contentaba con estar con mi familia, contemplarme con sus inquietantes ojos verdes de vez en cuando y apostar generosamente cuando jugaba contra mí en _gin rummy_.
"Él va a entrar en acción cuando llegue el momento," especulaba Alma. "Él sabe que tu mamá espera que las cosas se hagan como Dios manda."
"Si por lo menos yo supiera que le gusto."
"No visitaría tanto si no le gustaras."
No estaba tan segura. Si le gustaba debería dejármelo saber. Debería mandarme flores, contratar un Mariachi para que me trajera una serenata, traerme chocolates, escribirme poemas. Debería hacer algo romántico que probara que yo le gustaba de una manera que no le gustaba nadie más. Como no hacía nada seguí los consejos de _Sex and the Single Girl_ y me hice la difícil. Cuando oía sus pasos en la escalera, me escondía en el cuarto. Empecé a pagarles otra vez a mis hermanos y hermanas para que hicieran las tareas que me tocaban, para no tener que pasar por su puerta cuatro o cinco veces al día. Dejé de anunciar cuándo estaría en casa.
Un día regresé y encontré a Doña Lila sentada en la mesa de la cocina. "Ese muchacho no es violento," murmuraba llorando. "En su vida ha mata'o una mosca." Mami y Tata estaban al lado de ella, pasándole la mano por los hombros, murmurándole ruiditos para consolarla. Pensé que a uno de sus hijos lo habían acusado de matar a alguien y rogué en silencio que no fuera Neftalí.
"A Neftalí lo llamaron del servicio," contestó Mami mi pregunta silenciosa, pero no tenía idea de lo que quería decir con eso de que "a Neftalí lo llamaron del servicio."
_"He's been drafted,"_ me interpretó Delsa.
Mami y Tata seguían sobándole los hombros a Doña Lila, tratando de convencerla de que porque a Neftalí lo llamara el Ejército, no quería decir que iría a Vietnam. Pero ninguna de nosotras se lo creía. Más de una vez le agradeció Mami a Dios y a las once mil vírgenes que Héctor tuviera sólo catorce años. Ella y Tata rezaban en voz alta porque la guerra terminara antes de que tuviera la edad para ser reclutado y lo mandaran a lo que todas temíamos sería una muerte segura.
Era difícil sacar en claro lo que estaba pasando en Vietnam. Las imágenes eran muy incongruentes. Veíamos en las noticias, reportajes que mostraban a los soldados riéndose y poniéndoles cuernitos a los compañeros mientras los locutores, con gran sobriedad, hablaban de los muertos. Veíamos un panorama exuberante y tropical y largas playas sembradas de palmas que tanto nos recordaban a Luquillo, en la costa norte de Puerto Rico. Soldados risueños, pintorescos arrozales; reporteros muy masculinos que se apoyaban contra los tanques del ejército, nos hablaban a través de las cámaras mientras que detrás de ellos, se veían hombres jóvenes jugueteando o llevándose unos a otros en camillas. No parecía de verdad.
Pero así era. Mi primer novio potencial, a punto de irse a la guerra. Se parecía demasiado a las radionovelas que escuchaba desde niña en las que el héroe guapo se iba a la guerra, mientras la bella heroína se quedaba en casa escribiéndole emotivas cartas y rechazando a los pretendientes que no le llegaban ni a los tobillos al amado. Me debatía entre sentir pena por Doña Lila y vivir el romance con un novio que luchaba por la democracia en un país lejano.
Esa noche Neftalí subió y yo no me escondí. Don Julio y Don Carlos, que habían peleado en Corea, le hablaron de lo que podía esperar durante el entrenamiento básico. "Te van a hacer un hombre," bromeaba Don Julio y Neftalí sonreía tímidamente y me miraba.
Al día siguiente me aseguré de sacar la basura y allí estaba Neftalí, al pie de la escalera con la de su casa.
"¿Me vas a esperar?" dijo tan suavecito que yo entendí, "¿Me lo vas a pesar?"
Lo debo haber mirado con una expresión bastante estúpida porque se me acercó y repitió la pregunta.
"Te voy a escribir," respondí.
"Voy a hablar con tu mamá," dijo, "para hacerlo oficial."
Había esperado con ansiedad por las promesas de amor de Neftalí y el mariposeo y el cosquilleo que me produjo pensar en él, se convirtieron en temblores y escalofríos. "¿Qué tu quieres decir con oficial?"
"Tú sabes," murmuró con una sonrisa tímida y se acercó para besarme.
Me eché para atrás. "No, no sé." Así no era que me lo había imaginado. Se suponía que él se hincara en una rodilla, me dijera que me quería, me ofreciera una sortija de brillantes y que por lo menos usara la palabra "matrimonio" en una oración completa. No estaba bien que él esperara que yo fuera a proponerme matrimonio a mí misma, allí parados en un pasillo oscuro, agarrando unas pesadas bolsas de basura.
"¿Y a ti que te pasa?" me dijo con un retintín tan familiar que pudo haber sido de Mami.
Me le escurrí por el lado y llegué hasta los zafacones en la acera. "¿Qué es lo que te pasa a ti?" quería preguntarle, pero no lo hice. Estaba segura de que él no entendía de esto más que yo. Tenía ganas de llorar. Se me acercó por detrás.
"Yo creía que yo te gustaba," gimió y su tono me irritó.
"Pues no," no pude evitar ser cruel. Unos minutos antes era mi sueño y ahora le estaba diciendo que no me gustaba. ¿Qué me pasaba? Corrí hacia el edificio, por las escaleras, hasta mi cuarto, metí la cara en la almohada y sollocé como si Neftalí me hubiera hecho algo terrible, cuando todo lo que había hecho era quererme. ¿O no? ¿Por qué no me lo decía? Estaba confundida, no entendía por qué había reaccionado como lo había hecho. Me sentí abochornada. Él se había quedado en la acera agarrando su bolsa de basura, mirándome como si hubiera perdido la mente. Y así era que me sentía. Desquiciada, demente, loca. ¿Quién iba a querer acercárseme?
"Hace un tiempito que Neftalí no sube a vernos," dijo Mami unos días más tarde, buscando con los ojos una respuesta. Yo levanté los hombros.
Camino al trabajo y de regreso, pasaba en puntillas por la puerta de Neftalí. Una parte de mí deseaba que nos encontráramos en el pasillo, que habláramos y que yo me excusara; pero no sabía qué le diría después de eso. Así es que fue un alivio cuando, a la semana, Doña Lila vino a decirnos que Neftalí se había ido a Puerto Rico a visitar a unos parientes, antes de presentarse para el entrenamiento básico. Me estaba velando cuando nos lo dijo y en sus ojos había resentimiento. Pero nunca dijo nada, ni Mami tampoco, ni yo tampoco. No había nada que decir. Cientos de veces repasé en mi mente la escena con la basura tratando de encontrar una explicación para mi comportamiento. Pero fue inútil. Me había portado mal y no me lo podía perdonar.
El término de un mes que me había dado para conseguir trabajo en actuación o en baile, llegó y pasó y muy pronto quedó claro que tenía que buscar trabajo por otra línea. Contesté un clasificado y la semana antes del Día del Trabajo, me recibió en la puerta de la oficina de Recursos Humanos de Fisher Scientific, el Sr. Kean, quien tenía la postura de hombros abiertos, y espalda erguida desde la cadera, típica de un ex-bailarín de ballet. Me pidió que llenara un formulario y me llevó entonces a un cuartito donde había una maquinilla sobre una mesita. De una tablilla que había junto a la puerta, tomó un cronómetro, una libreta de espiral y unos papeles que colocó junto a la maquinilla.
"Tenemos vacantes para mecanógrafas," me dijo, "así es que vamos a ver cuán rápida eres." El Sr. Kean me observó mientras yo ponía el papel en la maquinilla y lo alineaba para que las puntas quedaran parejas. Abrió la libreta encuadernada de espiral en una página al azar y la colocó junto a la maquinilla, prendió el cronómetro y dijo, "Empieza."
Escribí lo más rápidamente que pude, pero no había practicado desde que salí de Performing Arts y cometí tantos errores que cuando sonó el timbre me dio vergüenza enseñarle la página al Sr. Kean.
"Veo," marcó los errores en rojo. "No te sientas mal," me aseguró, "no todo el mundo nació para escribir maquinilla." Se rió y me hizo sentir mejor. "Deja ver qué otra cosa podemos encontrarte." Me llevó hasta su escritorio en la esquina de un salón lleno de escritorios que me recordaron la oficina del _welfare_. Rebuscó en un fichero de tarjetas tres por cinco, sacó un par de ellas, leyó unas notitas garabateadas y marcó un número de teléfono. "No te preocupes," me dijo. "Hay un trabajo en el cuarto de correspondencia."
Tomamos un ascensor hasta un salón del ancho y de la profundidad del edificio. Unos rectángulos de luces fosforescentes bañaban todo y a todos con una luz azulosa. El salón era un laberinto de filas de escritorios de metal grises. Amplios pasillos dividían el Departamento de Compras del de Ventas Internacionales y del rincón ruidoso donde se sentaban las mecanógrafas táquiti-táquiti, ocho horas al día, divididas por dos _coffee breaks_ de quince minutos y media hora para el almuerzo. En la esquina, al fondo, frente a una hilera de ventanas todas llenas de polvo, con vista a los techos, estaba el cuarto de la correspondencia. No era ningún cuarto nada, sino una sección dividida por una mesa larga, flanqueada por unos archivos puestos en forma de herradura, con suficiente espacio entre ellos para formar un pasadizo hasta el área de trabajo. Debajo de las ventanas había dos mesas más y al final un escritorio de madera y una butaca. El Sr. Kean tocó en la mesa como si hubiese sido una puerta. Una distinguida mujer rubia se levantó de detrás de uno de los archivos de donde había estado sacando cartapacios.
"Entra, querida," sonrió. Tenía un aire aristocrático que resultaba perfectamente apropiado a pesar del lugar donde estábamos. El Sr. Kean nos presentó e Ilsa Gold me entrevistó de pie, aunque había sillas junto a las mesas al lado de las ventanas. El Sr. Kean me llevó de vuelta a su oficina donde sonó el teléfono, como si hubiera estado sincronizado, justo cuando llegamos a su escritorio. "Estás contratada," anunció en un tono tan risueño que estoy segura de que estaba tan feliz por mí como yo misma.
Ilsa me explicó mis deberes. Por la mañana, tenía que abrir la correspondencia, sortearla, distribuirla; por la tarde recoger la que se iba a enviar, pasarla por el metro de la tarifa postal y tenérsela lista al cartero que venía tarde en el día. Entretanto, tenía que recoger y archivar cualquier documento en uno de los quince ficheros que formaba la herradura de nuestra oficina. Al finalizar el primer día tenía los dedos desbaratados de cortaduras de papel. Al día siguiente llegué a trabajar con curitas en cada dedo. Ilsa me miró intrigada pero no me dijo nada.
Había más trabajo del que podíamos manejar dos personas. Ilsa decía que tenía que contratar a otra persona para ayudarnos, pero que el candidato ideal no había aparecido aún.
"Soy muy exigente a la hora de escoger quien trabaja para mí," me aseguró. Hablaba con un acento que se volvía más pesado cuando se ponía nerviosa o tenía que hablar por teléfono. Le pregunté de dónde era.
"De muy lejos," me dijo con una misteriosa sonrisa. Me sentí mal por haber sido tan curiosa.
La mejor parte de mi trabajo era cuando recogía o repartía correspondencia. Me daba la oportunidad de visitar los departamentos, conversar con las secretarias o mecanógrafas, los oficiales de compra, los vendedores. Uno de ellos, Sidney, estaba siempre en su escritorio cuando yo pasaba.
"Es un buen muchacho," dijo Ilsa, lo que me dio risa. "¿Qué te dio gracia?"
"No parece que haya sido nunca un muchacho, es tan serio."
"Como debe ser," dijo y no elaboró y yo no pregunté porque ella era, con frecuencia, enigmática y cuando le pedía que me explicara, se enconchaba o enseguida buscaba algo que hacer.
Fisher Scientific tenía una cafetería para empleados, pero como tenía que haber siempre alguien disponible por si se necesitaba algún expediente, Ilsa y yo no podíamos tomar nuestros _breaks_ a la vez. De todos modos había una jerarquía que determinaba quién tomaba el _break_ con quién. Después de dos o tres incidentes incómodos, de sentarme con gente que era simpática conmigo cuando pasaba a recoger o a dejar la correspondencia, averigüé que mi lugar era con los oficinistas y otros empleados de mi nivel inferior. Los supervisores y las secretarias ejecutivas se sentaban con su propio grupo, como claques de escuela superior, pero en el nivel de adultos.
Había mucho chisme durante los _breaks_. Gus bebía demasiado, el matrimonio de Phil estaba mal, Loretta estaba encinta y sin esposo a la vista. Los problemas de la gente nos mantenían en suspenso desde el _coffee break_ de la mañana hasta el de la tarde, puesto que iban surgiendo detalles en el transcurso de las ocho horas de trabajo. Cuando no había nada más interesante, se cogía el tema de cómo se vestía la gente para venir a trabajar. Brenda era muy conservadora y qué pena, verdad, porque tenía un cuerpo tan bonito. Lucille, sin embargo, no tenía ni mínimamente el cuerpo necesario para lucir la ropa tan reveladora que insistía en ponerse. Penny se pintaba tanto el pelo que se quedó calva y por eso usaba peluca. Las piernas de Jean eran demasiado gordas para usar minifaldas y Roberto usaba demasiado perfume.
A mí me preocupaba que si yo no estaba presente, mis compañeros de trabajo fueran a hablar mal de mí, así es que yo volaba para la cafetería a la hora en punto del _break_ y me quedaba hasta que todo el mundo regresaba a su escritorio. En casa entretenía a la familia contándole los chismes y todos seguían las historias como si hubieran conocido a los protagonistas. Para el impacto dramático yo exageraba o añadía detalles que no eran parte del cuento original. Pronto empecé a creer que mis versiones eran la realidad y me sorprendía cuando los datos se desviaban de lo que debió haber pasado, dado el escenario que les había creado.
Un día, regresando a casa, al bajar la escalera del tren elevado, me sorprendió encontrarme a Neftalí esperándome. Durante las dos o tres semanas desde que se había ido a Puerto Rico, yo lo mandé a la guerra donde se destacó, recibí sus cartas de amor, le respondí con un interés reservado y distante, acepté su promesa de amor eterno, me casé en una catedral y mis hermanas y hermanos fueron mi séquito nupcial, me fui de luna de miel a Tahití y estaba a punto de tener gemelos, — todo eso en los quince minutos que me tomaba ir y venir de la estación del tren. Ahora, frente a frente, me daba cuenta que el Neftalí de mi imaginación era más alto y vestía mejor que el Neftalí de la vida real. También era más aplomado. El Neftalí de carne y hueso bajó la cabeza y murmuró hola, mientras yo me preguntaba qué era lo que veía en él hacía apenas tres semanas.
Caminamos uno al lado del otro por la acera congestionada. Era una tibia tarde de septiembre y las tiendas estaban abiertas. Cada puerta era la entrada de una cueva rica en tesoros: frutas y vegetales tropicales; periódicos y revistas; dulces de colores envueltos en papelitos brillosos; estantes de trajes, blusas y faldas cubiertas de plástico. Le gente entraba y salía arrastrando chillones carritos de compra. Bolsas de papel arrugadas a punto de explotar, llenas de abrigos apestosos a humedad comprados en la tienda de ropa usada. Las mujeres se sentaban en las entradas mientras sus hijos e hijas brincaban cuica, patinaban y tiraban tapitas de botellas contra una pared.
Neftalí y yo caminábamos por entre el gentío dejando entre nosotros el espacio suficiente para que cupiera un niño pequeño. Yo deseaba que tratara de tocarme, de robarme un beso, que hiciera un gesto que me indicara que éramos algo más que vecinos. Pero, todo lo que hacía era contarme de su viaje a Puerto Rico, lo que me hacía sentir celosa.
"Yo no había ido pa'llá desde que era un nene," me decía. "Esas quenepas, mano. Aquí ni se consiguen."
No le di importancia a que me hubiera llamado "mano," porque estaba saboreando la fruta redonda, de cáscara crujiente, resbalosa, dulce, de corazón duro, la quenepa de mi niñez.
Me tocó el hombro y regresé de un brinco. "Estabas eslembá," me explicó.
"Perdona."
"Bueno, estaba pensando si a ti te gustaría vivir en Puerto Rico."
"Algún día."
"Entonces nos podemos mudar pa'llá. Pa' Ponce, pa' que te puedas comer to'as las quenepas que quieras. Ya escogí el solar pa' la casa y to'."
"¿Tú piensas casarte conmigo?"
"Yo te gusto, ¿verdá'?" Y entonces añadió, en tono acusatorio, "Tú actúas como si yo te gustara."
"¿Tú me estás proponiendo matrimonio?"
"¿Qué quieres? ¿Que me arrodille?" Se arrodilló en la acera como si estuviera en la iglesia y me agarró la mano. "¿Esto es lo que quieres?"
La gente en la calle nos pasaba por el lado. "¡Dile que sí!" gritó alguien y escuché risas.
"¡Déjame!" Retiré la mano y salí corriendo por la acera.
"¿Quién tú te crees que eres?" me gritó. "Ahora eres una gran actriz de cine. ¿Verdá'? ¿No soy lo suficientemente bueno para ti, verdá'? ¿Es eso, ah? ¿Es eso?"
Su voz se perdió en el bullicio de la calle. Corrí tan rápido como me lo permitieron mis tacos altos; la cartera golpeándome al lado, como si alguien me estuviera siguiendo con un palo. ¿Que quién me creía que era? No estaba muy segura. Pero de lo que sí estaba segura era de que la esposa de Neftalí no iba a ser.
Había días en que dejaba el apartamento, cogía un tren, viajaba una hora, subía de la estación del _subway_ hacia las torcidas calles del Village, caminaba seis bloques hasta Fisher Scientific, subía por el ascensor y venía a darme cuenta de dónde estaba cuando se abrían las puertas al resplandor fluorescente y al traqueteo de las maquinillas del inmenso salón donde trabajaba. Era un enorme escenario, iluminado por todos lados, con un público que podía ver todos nuestros movimientos desde cualquier ángulo. Todo un teatro diario.
Ilsa afirmaba que me había contratado por mi actitud. "Eres positiva y entusiasta," decía, "si sigues así, vas a llegar lejos."
A veces me dolía la cara de tanto sonreír, de mantener el semblante alerta de la que está entusiasmada con lo que hace. Pero, la verdad es que mi trabajo era aburrido. Horas de archivar papeles que no podía leer porque Ilsa ponía cara cada vez que me miraba y notaba que el montón de papeles que tenía delante no bajaba. Esperaba con ansias la media hora que pasaba entregando y recogiendo la correspondencia que por lo menos me permitía charlar con los demás empleados. Pero hablar sin revelar demasiado de mí misma, me consumía una energía enorme. La gente quedaba espantada cuando se enteraba de que yo era la mayor de nueve hermanos y venía otro en camino. Su reacción me abochornaba, como si fuera mi culpa que Mami fuera tan fértil.
Cuando mis compañeros de trabajo me pedían detalles yo tomaba a broma nuestras condiciones de vida. "Nueve niños, tres adultos, en un apartamento de tres habitaciones," sonreía. "Suena peor de lo que es," insistía.
Si me apuraban mucho, admitía que Mami no estaba casada con el papá del bebé que iba a tener, y que tampoco se había casado con el papá de ninguno de sus hijos. Mis compañeros entrecerraban los ojos y apretaban los labios, mientras pasaban juicio sobre la clase de mujer que era Mami y por consiguiente, la clase de muchacha que sería yo.
"Pero no me dejan salir con muchachos," bromeaba, para hacerles saber que yo entendía la ironía, pero que aún así mi familia tenía valores que merecían respeto.
Más de una vez me dijeron que no "sonaba" puertorriqueña. "No tienes acento," comentó el Sr. Morton, uno de los supervisores, y yo le expliqué sobre Performing Arts y el hablar estándar. Cuando sugirió que yo no "actuaba" como puertorriqueña, me tragué el insulto. "Quizá, usted no ha conocido suficiente gente como nosotros," le sugerí, ofendida de que él se sorprendiera que las puertorriqueñas pudieran ser muchachas castas, competentes, que hablaran bien el inglés.
Yo sonreía, hacía mi trabajo, chismeaba. Al terminar el día, volvía sobre mis pasos hasta Brooklyn, a veces en la misma nube en que había salido, pero exhausta, después de una representación que había durado demasiado.
"¿Cómo te fue hoy?" me preguntaba Mami, todas las tardes según yo entraba por la puerta.
"Bien," sonreía risueña y me metía en el cuarto a cambiarme. Delsa, acostumbrada ya a la rutina, se bajaba de la litera y me dejaba sola.
Me removía el maquillaje y me desnudaba. Esmeralda Santiago permanecía entre los pliegues de cada pieza de ropa que me quitaba y guardaba. Desnuda, sin nombre, me acostaba en mi cama y dormía. Media hora después, aparecía Negi, vestida con la ropa cómoda que usaba en casa. Otra función estaba a punto de empezar, ésta, en español.
# "Las perlas traen lágrimas."
#
Después de semanas de estar entrevistando gente, Ilsa contrató a otra oficinista, Regina.
"Es preciosa, ¿verdad?" comentó Ilsa un día, mientras Regina se alejaba de nosotras.
"Les está causando tortícolis," me reí.
Los hombres en la oficina —todos— estiraban el cuello cuando Regina pasaba. Sus ojos la iban siguiendo según ella se movía de escritorio en escritorio, ondulando las caderas y las nalgas, de una manera muy poco americana. Algunos de nuestros compañeros, empezaban a sudar literalmente cuando Regina se les acercaba. Cuando hablaba en su voz ronquita, con acento brasileño, sus susurros y murmullos producían escalofríos evidentes en los cuerpos de los hombres.
Regina no parecía estar consciente de su belleza. Vestía en faldas largas, blusas de cuellitos modestos y mangas, zapatos de tacón bajo. Prefería los colores opacos, se recogía la melena que le llegaba a los hombros, en un moño suelto en la nuca, usaba poco maquillaje, un poquito de lápiz labial y máscara.
Ilsa me asignó que la adiestrara. Regina se pasaba detrás de mí y pronto aprendió las tareas sencillas que conllevaban nuestros trabajos de oficinistas de correspondencia y archivo. Al principio me molestaba que cada vez que me viraba, estuviera ella allí, hermosa y aturdida. Entonces, un día, mientras íbamos bajando para el _coffee break_ , me dio las gracias.
"¿Yo qué hice?"
"Yo estoy, como dice, choque cultural," me confió. "En Brasil no es así." Abrió los brazos como para abrazar al mundo.
"En Puerto Rico," le dije, "tampoco es así."
Ninguna de las dos tenía que decir nada más para entender lo que la otra quería decir; pero todavía no entendía por qué me daba las gracias.
"Yo no tengo amigas aquí," me dijo. "Sólo tú."
Me conmovió tanto que la abracé.
Durante el receso, no nos sentábamos con los demás oficianistas, sino que cogíamos una mesa para nosotras y hablábamos de nuestras vidas. Ella era hija única y había cuidado a su mamá durante una batalla de tres años contra un cáncer de seno. Cuando murió su mamá, su papá la mandó a Nueva York.
"Yo lloro todos días en tres meses," dijo. "Es horrible ver tu mamá morir, poquiño." Vivía con su tía paterna que tenía un puesto importante en las oficinas de Fisher Scientific en New Jersey. "Ella dice a mí: tres meses suficientes lágrimas. Tengo encontrar trabajo. Y pronto tengo que casar."
"¿Con quién te vas a casar?"
"Yo no sé."
Me imaginaba a la tía como la malvada madrastra de los cuentos de hadas, y pronto añadí las tribulaciones de Regina a las historias que yo componía para beneficio de mi familia. Mami y Tata estaban a punto de adoptarla.
"Pobrecita," decía Mami, "sin mamá y sola en esta ciudad."
"Y esa mujer," añadía Tata, "no tiene corazón."
Mami asentía. "Pobrecita," repetía y sacudía la cabeza pensando en la pobre Regina.
Como Ilsa se ponía tan nerviosa por teléfono y el inglés de Regina no era muy bueno, yo estaba a cargo de contestar las llamadas. La mayoría de las veces la gente llamaba para pedir un expediente o para avisarnos que había mucho correo que procesar ese día para que programáramos más tiempo para la recogida. Un día, cuando Ilsa acababa de irse a un _break_ , sonó el teléfono. Era Sidney, que siempre era bien buena gente conmigo. Su oficina quedaba como a veinte pies de la nuestra así es que, por lo general, se acercaba a pedirnos lo que necesitara.
Me viré para asegurarme de que estaba en su escritorio y me saludó con la mano. "¿En qué puedo servirte?" Lo saludé también.
"Sal conmigo el viernes por la noche." Sonrió.
"¿Una cita?" Me viré porque no quería que notara mi agitación.
"Si. A comer, al cine, lo que tú quieras."
"A comer suena bien," le dije suavecito al teléfono porque Regina se había dado cuenta de lo que estaba pasando y nos miraba divertida. "Gracias," colgué y me sentí estúpida por haberle dado las gracias. Tenía miedo de mirar hacia su escritorio no fuera a ser que me viera ruborizándome.
"Es bien bueno," dijo, espontáneamente, Regina.
"¿Cómo se lo digo a mi mamá?" me dije en voz alta y Regina sonrió.
Mami quería saber quién era Sidney, qué hacía, a dónde íbamos, cuánto tiempo íbamos a tardar. "Tráelo a casa pa' conocerlo."
"Es sólo a comer que voy, Mami. No me voy a casar con él."
"Es una señal de respeto," dijo. Tenía razón, pero yo no podía imaginarme a Sidney en nuestro apartamento, tan lleno de gente y de muebles. ¿Qué pensaría él si por casualidad Tata estuviera borracha cuando él llegara? ¿O si Mami estuviera en bata y en rolos, como se pasaba generalmente cuando estábamos en casa? ¿O si Don Carlos estuviera allí en su traje y gafas negras, sentado silenciosamente en la mesa de la cocina, con su sonrisa absorta en los labios? ¿O si Don Julio con la cara esbaratá como la de los boxeadores que han recibido demasiados golpes en la cabeza, estuviera tirado con los muchachos frente al programa de Lawrence Welk? ¿Y si mis hermanas y hermanos se burlaban del aspecto de Sidney? Era bajito, usaba espejuelos rectangulares, gruesos, que se le escurrían y le dejaban unos surcos rojos en el puente de la nariz. Tenía una voz suave, plañidera que lo hacía sonar como si se estuviera quejando aunque no lo estuviera haciendo realmente. Las manos le brotaban pequeñas y aniñadas de las mangas de la chaqueta y no reposaban en ningún sitio por más de un segundo. Yo las encontraba gráciles, pero Mami, sin duda, las imaginaría desabrochándome el brasier con mucha maestría.
Ilsa quedó atónita cuando se enteró de que iba a salir con Sidney. "¿Te invitó?" me preguntó incrédula.
"Claro," contesté, molesta de que pensara que había sido yo quien lo había invitado.
Miró hacia el escritorio de Sidney con una expresión sombría en el rostro. "Interesante," dijo pensativa.
"¿Hay algo que yo deba saber sobre él?"
"No, cariño," respondió Ilsa, "es que... me sorprende, eso es todo. Es un buen muchacho. Diviértete."
El día antes de la cita, Regina me acompaño a Gimbel's. Me gustaba ir de compras sola, pero quería causar una buena impresión y ese día necesitaba ayuda para escoger algo apropiado. Regina era la persona perfecta para controlar mis impulsos por la ropa teatral, colorida o dramática. Cuando llegué a casa con el conjunto azul marino, zapatos de tacón bajito y una carterita recatada, Mami no pudo disimular su sonrisa.
"¿Qué tiene de malo?" pregunté.
"Nada," dijo. "Está bien," y se viró tratando de aguantar la risa.
"Esa es ropa de vieja," fue la opinión de Tata.
"Lo compré en la sección de jóvenes," expliqué, pero al volverlo a mirar se me pareció al estilo de Regina. "Es elegante," añadí, repitiendo las palabras de Regina. "Se ve mejor puesto." No pude convencer a nadie.
Mientras me vestía al día siguiente, me decía que era mejor verme conservadora y aviejá' que como una cualquiera sacada de _West Side Story_. Cuando entré a la oficina, la gente se me quedó mirando y algunos comentaron lo mona que me veía, lo que me hizo sentir mejor.
Sidney no estuvo en su oficina en toda la mañana y yo estaba preocupada que fuera a cambiar de idea y no apareciera. Durante el almuerzo, Regina sacó una bolsita de la cartera.
"Ponte esto," me dijo. Dentro de la bolsita había un collar de perlas. "Eran de mi mamá," me explicó. "Se te van a ver lindas en tu noche especial."
Las perlas me pesaban en la mano, lánguidas como una tarde tropical. Las deseé. Mi deseo me avergonzó. "Regina, yo no me puedo poner esto," y se las devolví sin muchas ganas, "¿y si se me pierden?"
"Tú las cuidas bien, yo sé," me dijo. "Por favor, acepta usar."
Me coloqué las perlas alrededor del cuello y ella las abrochó. Cuando se echó para atrás para admirarlas, Regina me arregló suavemente el cuello de la blusa. Me sonrió con dulzura y se le humedecieron los ojos. "Me recuerdas a mi mamá," me dijo. Tuve que tragar gordo para no echarme a llorar.
Sidney llegó a la oficina a las cinco menos cinco.
"Lo siento, había mucho tráfico de New Jersey en el túnel."
Le aseguré que estaba bien, aliviada de que hubiera aparecido.
"Ve," dijo Ilsa, "píntate los labios. Nosotras acabamos aquí esto." Corrí hasta el baño, me arreglé la cara y el pelo, arreglé las perlas que tenía en el cuello. Relucían pálidas contra mi piel canela.
"Pensé que comamos cerca de donde estoy estacionado," sugirió Sidney mientras íbamos por la calle en una dirección que nunca antes había tomado. El aire estaba húmedo y una brisa fría que soplaba del Hudson, traspasó mi abrigo de paño hasta que empecé a temblar. Caminamos por una calle de adoquines, alrededor de unos camiones de entrega enormes que estaban estacionados en los muelles de carga.
"¿Vamos lejos?" pregunté después de caminar dos o tres bloques.
"Hasta ahí, a la vuelta de la esquina," respondió Sidney.
El restaurante era en un sótano. Un toldo aleteaba sobre la puerta que tenía un nombre escrito en unas letras tan oscuras que era imposible descifrarlas. Adentro, dos paredes de ladrillos estaban alineadas con mesas, cada una iluminada por una llama temblorosa dentro de un vaso rojo. El mantel y las servilletas resplandecían con un blanco fluorescente, flotaban en la oscuridad, cada uno con su rojo círculo de luz. Me recordó el altar de mi abuela en Puerto Rico, el misterio del rosario que rezaba todas las tardes.
Éramos los únicos clientes. El _bartender_ nos miró cuando entramos y nos indicó una mesa. En el salón en penumbras, las facciones de Sidney se suavizaron. Sus ojos, enormes detrás de sus lentes, eran dulces y había una tristeza en ellos que me movía a querer ser buena con él.
De atrás salió una mesera acomodándose unas horquillas en un vaporoso moño, estilo panal de abejas. "Yo tomaré la orden de bebidas," nos informó.
Nunca había tomado una bebida alcohólica expecto con mi familia en Navidad, cuando Mami preparaba varias botellas de coquito con leche de coco fresca y ron puertorriqueño. Cuando pedí una Coca-Cola, Sidney y la mesera se decepcionaron. Él pidió un _whiskey sour_.
"¿Tú no bebes?" preguntó.
"En casa solamente," contesté y él se rió. Me tomó un rato entender por qué. "No quise decir eso. Quiero decir..."
"Yo sé lo que quieres decir, no te preocupes."
Charlamos un ratito sobre la vida en Fisher Scientific, donde él trabajaba como vendedor de microscopios. Le gustaba su trabajo porque visitaba sus clientes en los diferentes estados en vez de tener que estar metido en una oficina todo el día. Recientemente, se había mudado de la casa de su madre viuda a su propio apartamento.
"No es gran cosa," me confesó. "Odio vivir solo, pero me gustaba menos vivir con mi mamá."
Ordenamos la cena del menú de los especiales, discutimos películas que habíamos visto, lugares que nos gustaría visitar, libros que habíamos leído. Hablamos de nuestros compañeros de trabajo y me dijo algo que yo no sabía. Ilsa, mi supervisora, era húngara y había sobrevivido los campos de concentración Nazis.
"No le gusta hablar de esa parte de su vida," me dijo Sidney.
"No la culpo." Eso explicaba muchas cosas. Su acento, para empezar. Esa mirada lejana que la sobrecogía, como si escuchara voces.
"Mírale el brazo izquierdo un día," me sugirió Sidney. "Tiene unos números tatuados ahí mismo." Me tocó cerca de la parte interior del codo.
Era fácil hablar con él, me escuchaba con interés. Nos quedamos en la mesa mucho rato después de haber terminado de comer, tomando café, hablando de baile y de música. Tocaba el violín y yo le admití que no sabía nada de música clásica, excepto la que había escuchado en las asambleas de Performing Arts.
"Mote-zart," corrigió mis intentos de nombrar a los compositores. Saqué un papel de mi cartera y escribí otros nombres. "¿Cómo se pronuncian éstos?" pregunté. "BEET-jóven." Repetía después de él. "Rack-MANNI-nov, Puu-CHII-nii."
Estaba lloviznando cuando salimos.
"¿Damos un paseo? Tengo una sombrilla en el carro."
Al principio, él sostenía la sombrilla de tal modo que yo quedaba protegida y él se mojaba. Cuando protesté porque él se estaba mojando, me atrajo hacia él, me tomó la mano y me besó en el cachete. Temblé de placer; era tan romántico pasear por la calle adoquinada, bajo la lluvia, con un hombre cariñoso que tocaba violín.
"Si mi mamá se entera de que salí con una _shiksa_ , me mata," soltó abruptamente Sidney.
"¿Una qué?" Me detuve tan bruscamente que caminó unos pasos antes de darse cuenta de que no estaba a su lado.
"Una _shiksa_. Una muchacha que no es judía."
No sabía si me estaba insultando o si debía sentirme halagada de que hubiera ido en contra de su madre para estar conmigo. Entendí entonces por qué a Ilsa le había sorprendido tanto que Sidney me hubiera invitado a salir. No se suponía que lo hiciera. "¿Eso va en contra de tu religión?"
"Más o menos," dijo, pero yo escuché un "Sí."
"Entonces, más vale que no me lleves a tu casa a conocerla." Me miró atónito como si la sola idea lo asustara. "Es una broma," le aseguré y me sonrió no muy convencido. "Se está haciendo tarde," decidí.
Corrimos hasta donde se había estacionado, huyéndole a lo que fuera que se había interpuesto entre los dos. La lluvia aumentó tan pronto nos metimos dentro del carro, protector y tranquilo. Lo dirigí hasta Brooklyn. Entrecerrando los ojos para protegerse de la luz de los demás carros, Sidney iba fijándose bien en los letreros y en los sitios donde tenía que doblar, para volver a salir. Traté de conversar, pero me detuvo. "Espérate un segundo, tengo que concentrarme. En la pizzería," continuó, hablando consigo mismo, "cojo a la derecha, después a la izquierda. Ya está." Se volvió hacia mí. "Ésta es tu calle," sonrió, "¿cuál es tu edificio?"
Nuestras persianas estaban cerradas, pero Mami estaba asomada por una rendija, mirando hacia la calle desde el tercer piso. Esperé a que Sidney se bajara del carro, diera la vuelta y me buscara con la sombrilla para no mojarme. Subimos por las escaleras despacio, porque yo oía pasos corriendo, cosas moviéndose y puertas cerrándose. En el último descanso me puse a buscar una llave que no tenía porque siempre había alguien en casa, hice como que se me había olvidado y toqué a la puerta. Mami abrió. Tenía puesta una bata de maternidad y unos pantalones; se había acomodado el pelo en unos rizos coquetos y se habá puesto un poco de color en los labios. Me pregunté si llevaría horas vestida así o si el correteo que había escuchado era el de la familia preparándose para recibir a Sidney. Mis hermanas y hermanos estaban sentados en el "caucho" y en las butacas, tiesos como el almidón, las caras lavadas, el pelo aplastado. Una sábana floreada dividía la cocina y el área de estar del cuarto de al frente, donde escuchaba a Tata acallando a Charlie. La cocina olía a café recién colado. Le presenté a Sidney a Mami y después a cada uno de los muchachos. Los más chiquitos se reían pachosos y se escondían detrás de los mayores. "Ofrécele café y bizocho," me sugirió Mami. En la mesa había un _coffee cake_ de supermercado, todavía en su caja.
"No," le contesté, "tiene que irse." Sidney me miró y luego la miró a ella esperando que yo tradujera. "Le dije que te esperaba un viaje largo a New Jersey."
"Ah, sí, es verdad." Pareció sorprenderse de que le recordara su lugar de residencia. Lo acompañé hasta la puerta.
"Te veo el lunes," prometí, y abrí para que saliera. Nueve pares de ojos siguieron todos nuestros movimientos. Fue un alivio cuando Sidney finalmente se despidió desde el umbral y bajó las escaleras. Tata salió del cuarto arrastrando los pies, con Charlie en los brazos. "¿Se fue?" preguntó con risa entrecortada. Cerré la puerta y me viré para enfrentar a mi familia que enseguida opinó.
"¡Qué bajito!"
"Tiene una narizota."
"El abrigo le apestaba."
"Los espejuelos son tan gordos."
"Por eso fue que invitó a Negi. Es ciego y no la puede ver."
Hice un débil intento de defender a Sidney. "Es un hombre bien bueno," pero fue inútil. Me di por vencida y contribuí a su risa al revelarles que su pasatiempo era el violín. Les pareció comiquísimo.
Mami miró el reloj de pared de la cocina. "Por lo menos se portó como un caballero y te trajo temprano. No son ni las diez," apuntó.
"A lo mejor ya Negi no soportaba estar con él más tiempo," dijo Delsa, muerta de la risa.
"¿De dónde sacaste esas perlas?" preguntó Tata, poniéndose seria de pronto.
"Regina me las prestó. Eran de su mamá."
"Quítatelas," chilló. Se me tiró encima a punto de arrancármelas del cuello. Me las protegí con la mano. "Las perlas traen lágrimas," me advirtió.
"Ay, Tata, déjate de supersticiones." Las perlas se sentían tibias contra mi cuello.
"Traen lágrimas," repitió, "especialmente, si son de otra persona. ¡Y de una difunta!" Se me tiró encima otra vez, pero yo la esquivé. Me metí en mi cuarto y cerré la puerta. Las perlas se sentían exquisitas. No había manera de que yo creyera que traían lágrimas. Lo único que tenía que hacer era escuchar la risa al otro lado de la puerta.
Pensé que me daría pachó ver a Sidney la semana siguiente, pero él estuvo fuera tres semanas y para cuando volvió, yo estaba enamorada de Otto.
Otto era un hombre grande, de piel y pelo dorado y una voz profunda que retumbaba desde su pecho de barril. Nuestros ojos se conectaron cuando fui a entregar el correo en el Departamento Internacional. Durante el resto de la mañana seguimos intercambiando miradas a través de la fluorescencia azul gris del salón. Desapareció a la hora de almuerzo, pero estaba allí cuando vine a su escritorio a recoger el correo.
"¿Tú eres Esmeralda, sí?" preguntó. El modo en que pronunció mi nombre, el sí al final de la oración, fue como una canción que se me quedó en la mente por horas. "Yo soy Otto," dijo. Le extendi la mano para saludarlo y él la retuvo y la apretó suavemente antes de soltarla. Casi me derrito ahí mismo. Me entregó un paquete de cartas con destino a Alemania. Le di las gracias y continué mi ronda, consciente de que me estaba mirando. A pesar de que siempre había resentido que los hombres examinaran mi cuerpo descaradamente, recibí con agrado la mirada de Otto y procuré mantenerme donde pudiera verme todo el tiempo que me tomó recoger la correspondencia. Esa noche fantaseé con cómo se sentiría estar entre sus brazos y continué soñando con él los días en que no volvió por la oficina.
La Navidad parpadeaba intermitentemente en rojo y verde en los vecindarios de Nueva York. En casa doblamos papel de libreta en forma de triángulos y los recortamos dándole formas artísticas para crear copos de nieve. Héctor cargaba a Raymond sobre sus hombros en lo que pegaba los copos en la esquina del techo. Don Carlos alzó a Franky para que encasquetara un ángel rubio en la punta de árbol de Navidad. Lágrimas plateadas chorreaban sobre las ramas plásticas cargadas de frágiles bolas de colores brillantes.
Fue una Navidad abundante. En casa todos los que teníamos edad suficiente, teníamos un empleo. Los domingos vibraban al son del tun tun tun de los parientes subiendo los tres pisos hasta nuestro apartamento. La mayoría traía paquetes envueltos que se ponían debajo del árbol, bajo el ojo vigilante de los chiquitines que los velaban como si el botin fuera a desaparecer si lo perdían de vista un momento.
La Muda y Gury vinieron un día con una bolsa de ropa que Delsa, Norma y yo, nos dividimos entre las tres porque Mami estaba encinta y no cabía en ninguna. Del fondo saqué un vestido en tafeta y chifón rosado con los puños de las mangas y el cuello recatado, bordado en perlitas.
"¿Es mala suerte usar estas perlas?" le sonreí a Tata.
"No, las de embuste no," rió y La Muda hizo el gesto de recortarle las mangas y el cuello al vestido, como para indicar que si las perlas fueran auténticas, yo tendría un traje sin mangas y muy escotado.
"Te lo puedes poner para el baile de la Armería," me sugirió Mami y mis hermanas y yo gritamos de gozo porque hacía meses que no salíamos a bailar.
A veces me encontraba con Alma y nos pasábamos horas en la Quinta Avenida, entre los turistas que se apretujaban y se empujaban frente a los muestrarios que las tiendas exhibían para atraernos. Cuando se trataba de gastar nuestro dinero, Alma y yo íbamos a Herald Square, donde los sueldos nos rendían más. Un día, mientras rebuscábamos en el cajón de zapatos de Orbach's, levanté la vista y me encontré con una cara familiar. Quedé fría, atónita, ante la imagen de Greta Garbo, doblada sobre un montón de zapatillas al 30 por ciento de descuento. Tenía puesto un _turtle neck_ y un abrigo negro; su rostro anguloso, pálido y luminoso, bajo el ala de un suave sombrero negro. Cuando me sintió observándola, se viró y desapareció entre la gente. Para cuando le hice un gesto a Alma, Garbo era un recuerdo.
Esa semana fui al salón de belleza y me corté la melena que me llegaba hasta los hombros, en un recorte recto hasta la barbilla, con partidura al medio, como el de Garbo. Compré un sombrerito en fieltro negro, que me bajaba hasta las orejas, tratando de reproducir el efecto del suave sombrero de Garbo. Fue inútil, no me parecía en nada a ella y el sombrero, lo único que hacía era aplastarme el pelo. Cuando me lo quitaba parecía que tenía una dita en la cabeza.
Guardé los regalos que compré en casa de Alma para que mi familia no fuera a descubrir lo que Santa Clause-Negi les iba a traer. En el apartamento de Titi Ana, la Navidad se celebraba muy sobriamente, con un par de guirnaldas de luces en las ventanas, un arbolito al lado del televisor, una modesta pila de regalos envueltos en papel brillante. Me quedé una noche en el cuartito al lado de la cocina a unas treinta yardas de las vías del tren elevado. Después que Titi Ana, Alma y Corazón se fueron a acostar, me paré en la ventana a mirar pasar los trenes. La gente que iba adentro eran fantasmas, espectros grises enmarcados en la oscuridad. Su anonimato me hizo sentir nostalgia de la tibieza de nuestro apartamento ruidoso. Me meti en la cama sintiéndome sola e invisible detrás de las cortinas de encaje de la ventana de Titi Ana.
El baile en la Armería fue un domingo por la noche. Nos quedamos hasta que la banda tocó la última nota y después desayunamos temprano en un _diner_. En casa, sólo tuve tiempo para bañarme, cambiarme a ropa de calle y regresar a la ciudad, a mi trabajo en Fisher Scientific. Medio despierta, me pasé dando tumbos toda la mañana hasta que Ilsa me sugirió que me fuera a casa a descansar. Ya estaba oscuro cuando salí para la estación y había una quietud extraña para ser media tarde. El aire frío me revivió lo suficiente para mantenerme erguida. Los pies adoloridos de tantas horas de salsa y merengue en tacos, me latían con cada paso.
Estaba a punto de cruzar Hudson Street cuando alguien me agarrópor detrás y me haló hacia la acera. Di un codazo y se lo metí en la cara a mi asaltante y salí corriendo en dirección contraria, pero me detuve cuando un camión me pasó por el lado volando bajito. Entonces me di cuenta de que el hombre detrás de mí estaba tratando de impedir que me pasara por encima. Cuando me viré, ahí estaba Otto apretándose el labio con los dedos.
"¡Ay, Dios mío, lo siento tanto!"
"Yo que pensé que sería un héroe." Trató de sonreír pero le dolía la herida en el labio.
"Tienes un poquito de sangre en un lado." Le ofrecí un pañuelito desechable, pero inclinó su cara hacia mí. Me daba vergüenza mirarlo a los ojos mientras le limpiaba la sangre del labio que se estaba hinchando rápidamente. "Necesitas hielo."
"Allí hay una cafetería," me dijo guiándome en esa dirección.
Según íbamos caminando, él con su mano en mi codo, deseé no haber gozado tanto la noche anterior. Tanía los ojos hinchados por la falta de sueño, el pelo, con su corte a lo Garbo, estaba todo para'o y escrespa'o porque no me había dado tiempo de lavármelo y estirármelo. No me había puesto maquillaje y había agarrado lo primero que encontré en el _closet_ —el conjunto que me había puesto para mi cita con Sidney que me hacía ver, me daba cuenta ahora, como una monja en ropa de calle.
Pero a Otto no le importaba. Nos sentamos uno frente al otro en la mesa de la ventana. "Encantadora," seguía diciendo y yo no sabía cómo contestar, excepto tartamudeando "gracias," lo que a él le parecía aún más enternecedor.
A diferencia de Sidney, con Otto no era muy fácil hablar porque su acento era pesado, su gramática confusa y con la bolsita de hielo en el labio no se le entendía nada. Le gustaban los restaurantes o restaurar, la cocina o _küchen_ , Audubon o autobahn. Después de mucho tratar, finalmente entendí que quería que lo acompañara a una fiesta de Navidad en casa de su hermana.
"Tengo que pedirle permiso a mi mamá," le dije, avergonzada de que a los dieciochos años, todavía tuviera que pedir permiso para ir a una fiesta.
"Encantadora," repitió.
Me acompañó hasta el tren y camino a Brooklyn, recordé sus manos fuertes en mi hombro. Me había salvado de ser aplastada por un camión. Era lo más romántico que me había pasado.
"¡Sola no!" dijo Mami cuando le pregunté si podía ir a Long Island con Otto.
"Es a casa de la hermana."
"A mi no me importa que vayas a ver al Papa. Puedes llevarte a uno de tus hermanos contigo o a una de tus hermanas. Pero tú no vas a ir tan lejos, sola con un hombre que yo nunca he visto." No había argumento que la persuadiera de que yo era lo suficientemente grande para cuidarme.
Regina se compadeció de mi problema y me ofreció una solución perfecta.
"Yo voy," me surgirió. A pesar de que Mami no había conocido a Regina, estuvo de acuerdo con que una muchacha que había quedado huérfana hacía tan poco y que había escogido el vestido azul marino tan poco favorecedor como la ropa ideal para una cita, era la chaperona perfecta. A Otto le pareció una idea espléndida que Regina viniera con nosotros. Su primo, Gilbert, necesitaba una pareja para la fiesta.
"Le va a gustar tu amiga," me aseguró Otto y quedó concertada la cita.
Se ofreció a recogerme en Brooklyn, cosa que impresionaría a Mami. La noche de la fiesta, Don Carlos y Don Julio decidieron quedarse en casa, sin duda a sugerencia de Mami. Vestido en su traje negro, Don Carlos estaba en la mesa de la cocina, sentado frente a Don Julio, también de lo más emperifolla'o en una camisa planchada y un pantalón nuevo. Con ellos estaban Héctor y Raymond, los varones mayores de la familia, con sus caras aseadas, el pelo recién lavado y repelado hacia atrás. Me aterraba el momento en que Otto se enfrentara a este patético intento de proteger mi virtud.
Estuve lista veinte minutos antes de la hora en que había quedado en llegar Otto. Mi intención era presentárselo a todo el mundo y salir de allí lo más pronto posible.
Sin embargo, cuando Otto y Gilbert se aparecieron en nuestra puerta, quedó claro que nos tomaría más tiempo salir de lo que había planificado. Dominaban el cuarto, dos enormes hombres teutónicos que hablaban muy poco inglés. Vestían trajes que lejos de hacerles parecer respectables, acentuaban su corpulencia, su masculinidad. Mami frunció el ceño e intercambió una mirada con Tata que sonrió vagamente y salió del cuarto para atender a Charlie que estaba gritando.
Otto me entregó una orquídea en un envase plástico que yo misma me prendí porque ni hablar que yo lo fuera a dejar acercárseme tanto delante de Mami. Su gesto consternado me preocupó. Deseé que Otto y Gilbert hubieran recogido a Regina en el camino a buscarme, para que Mami no tuviera que imaginarme ni un solo minuto a solas con dos hombres en un carro. Pero, ya era muy tarde. Don Carlos, que hablaba buen inglés, ya le había sacado el número de teléfono y la dirección de donde íbamos a estar. Le entregó a Otto su tarjeta de presentación y le hizo apuntar nuestro teléfono —como si yo no lo supiera— mientras Mami se aseguraba que yo llevaba una identificación encima.
"Por favor, Mami," rogué, "me estás abochornando."
"¿Qué tu quieres decir con abochornándote?" preguntó, levantando la voz lo suficiente como para que Otto y Gilbert desviaran la vista de los lentes verdes de Don Carlos y nos miraran. Mami les sonrió, se viró y a mí me frunció el ceño.
"Mejor nos vamos," sugerí, evitando su mirada, "o Regina va a pensar que nos perdimos." Tenía la esperanza de que la mención de Regina le recordara a Mami que yo tendría una chaperona y eso la hiciera relajarse un poco.
"Llámame cuando llegues," me dijo Mami mientras nos miraba bajar las escaleras en silencio.
Otto y Gilbert hablaron alemán entre sí y se rieron. Esperé una traducción, pero no hubo ninguna. Antes de montarme en el carro de Gilbert, miré hacia arriba. Mi familia entera estaba en la ventana rodeada de parpadeantes luces de Navidad.
Quizá esto era un error. Estos dos hombres que apenas conocía podían llevarme a cualquier sitio, violarme, tirarme por un puente. No pude relajarme en todo el trayecto hasta Lefrak City, donde íbamos a recoger a Regina. No registré cuando Otto mencionó que Gilbert y Regina ya habían salido antes, hasta que nos estacionamos frente al edificio y ella salió corriendo. Se veía espectacular con un traje ajustado que le marcaba las formas, un abrigo de piel, tacos altísimos y las perlas de su mamá, relucientes en su garganta. Su perfume invadió el carro. Un aroma floral que se quedó flotando en el aire.
"Wow," comenté y ella se rió.
"No son todos los días que una va de fiesta," dijo y hasta los hombres quedaron encantados con la alegría en su voz.
La hermana de Otto vivía en una calle con casas idénticas ubicadas en amplios jardines de grama. Santa Clauses, venados, duendes y cantantes de villancicos en miniatura competían entre sí junto a miles de lucecitas en los techos, aleros y persianas de las ventanas de casi todas las casas. Algo del vecindario me resultaba familiar. Entonces recordé que Archie y Verónica, Betty, Reggie y Jughead paseaban por una calle idéntica a ésta, sin las decoraciones, en los paquines que devoraba el primer año que estuve en Brooklyn.
Fue fácil darse cuenta dónde era la fiesta porque había muchísimos carros estacionados al frente y las cortinas de las ventanas estaban abiertas. Entramos por un camino iluminado por guirnaldas de lucecitas de Navidad colocadas en el suelo. Adentro, la casa estaba tibia y olía a canela, a clavos y a la madera que ardía en la chimenea. Una mujer rubia de huesos grandes nos recibió en la puerta y Otto la besó en ambas mejillas. Era Minna, su hermana mayor. Se parecían mucho, solo que Minna hablaba mucho mejor inglés.
"Estoy tan contenta de que hayas venido," me dijo apretándome la mano. "Otto me ha hablado tanto de ti." Regina y yo nos miramos pensando qué podría haberle dicho, puesto que apenas nos conocíamos.
Minna nos trató como a huéspedes de honor, nos presentó a todo el mundo, nos ofreció algo de beber y unas salchichitas en miniatura de una bandeja. Su esposo, Jim, era americano, pero tan rubio, de ojos azules y de pinta alemana como el resto de la gente en la sala. Tenía puesto un _leaderhosen_ y yo no estaba segura si realmente era una vestimenta típica o un relajo. Su trabajo principal era mantenerle lleno el vaso a todo el mundo, lo cual hacía con gusto. De vez en cuando rompía a cantar y la visita lo seguía en lo que yo supuse que serían villancicos alemanes.
Habíamos llegado cuando estaban a punto de servir la cena. La mesa del comedor estaba repleta de comida, organizada por clase. Un pavo, un jamón, un platón de albondiguitas y uno de fiambres, estaban al lado de una variedad de quesos, crema batida y mantequilla; al lado, había coloridas fuentes de vegetales: trozos de calabaza amarilla, cremosas papas majadas, habichuelas verdes salpicadas de minúsculas cebollitas blancas y remolacha rojo-sangre. Varías bandejas contenían panes crujientes, panecitos y bollitos de pan sazonados con diferentes semillitas. Una mesa lateral estaba dedicada a los bizcochos, pudines, galletitas y nueces cubiertas de chocolate. Era el despliegue de comida más abundante que había visto, cada grupo de alimento separado del otro por cintas y ramas de pino.
Otto y Gilbert nos guiaron por el _buffet_ , y nos animaban a que probásemos de todo. Se rieron de lo diligentemente que yo mantenía todo separado en mi plato para que los sabores no se contaminaran entre sí y de la cara de Regina cuando probó la crema batida y resultó no ser dulce sino picante por el rábano.
Después de cenar, bajamos a un sótano decorado, donde había sillas junto a las paredes, una barra, un _HiFi_ con una estiba de discos que caían uno a uno en un plato que giraba lentamente. Nancy Sinatra insistía en que sus botas eran para caminar, los Monkees eran ilusos creyentes de sueños y los Young Rascals prometían amor del bueno. Otto y Gilbert agitaban los brazos y las piernas en un estilo que yo había llegado a asociar con el baile americano, pero que ahora parecía ser una técnica internacional. Acostumbrada a los movimientos gráciles y seductores de la salsa, el merengue y el chachachá, me frustraba la distancia entre nuestros cuerpos, la sensación de que no estábamos bailando juntos sino cerca. Eso cambió cuando Percy Sledge empezó a berrear sobre los infortunios del hombre que ama a una mujer. Alguien bajó las luces. Otto se quitó la chaqueta, me atrajo hacia él y por fin estuve en sus brazos, mi cabeza recostada en su pecho ancho. Cada vez que subía la voz de Percy Sledge, Otto me apretaba más y yo me dejaba ir. Cuando terminó la canción, Otto me cogió de la mano y subimos por la escalera. Regina nos miró, se sonrió y acurrucó la cabeza en el hombro de Gilbert.
"¿A dónde vamos?" pregunté, pero Otto no me contestó. Fuimos por un pasillo. Él abrió la puerta de un cuarto que estaba oscuro, pero yo me negué a entrar. "Vamos a bajar de nuevo," sugerí. Me apretó contra la pared y me besó.
Fue divino su beso. Los labios, tibios y suaves. El calor entre nuestros cuerpos. La lenta insinuación de su lengua en mi boca. Irresistible. Cada vez que cogíamos aire, él me empujaba un poquito más hacia la puerta. Una pareja, camino a otro cuarto, nos pasó por el lado y sentí el olor del perfume de Regina. Otto me murmuró unas palabras al oído que no entendí. "Por favor," rogaba y yo me di cuenta que era mejor que me saliera de entre la pared y él. Sus besos eran insistentes, sus manos exploraban. Estaba abrumada, segura de que si esperaba un minuto más, no iba a poder resistirme a sus dedos curiosos, a su lengua caliente, al deseo de arrancarme la ropa y presentarme desnuda ante él. Era un hombre grande, pero yo era una bailarina musculosa. Con esfuerzo, me separé de él y corrí hasta el sótano lleno de gente donde los Trogg cantaban sobre su cosa loca.
Me senté en una de las sillas junto a la pared. Otto no me había seguido. Agradecía no tener que enfrentarme a él en ese momento. Minna se acercó y se sentó a mi lado. "¿Lo estás pasando bien?" preguntó.
"Muy bien," respondí, mi voz tirante. No se dio cuenta.
"Tú le gusta mucho a mi hermano," me confió. "Nunca nos ha traído una muchacha para que la conozcamos."
"A mí también me gusta," admití, con la esperanza de que si ella le llevaba ese mensaje, me perdonaría lo que acaba de hacer.
Dos ventanitas en la parte superior de la pared daban hacia la parte alta de la casa. Enormes copos de nieve titilaban entre las luces de Navidad. Minna siguió mi mirada. "¡Qué lindo!" exclamó. "Miren," gritó, "está nevando."
Algunas parejas dejaron de bailar para admirar la vista. Otto bajó las escaleras, yo pensaba que estaría enojado, pero traía un gesto medio cortado, me sonrió con dulzura, se me sentó al lado y me apretó la mano. Se volvió hacia la ventana que todos estaban mirando y yo miré también. Para mi horror, allí venían Mami y Don Carlos marchando hacia la puerta de entrada, gruesos copos de nieve golpeando sus rostros resueltos.
"Ay Dios mío," me levanté tan ligero que resbalé y caí de rodillas. Otto me ayudó a parar y corrí por las escaleras antes de que pudieran tocar. "¿Qué hacen ustedes aquí?" chillé. Mami tenía los labios apretados. Miró detrás de mí, la casa festiva, las sobras de comida en la mesa de Navidad, las caras de los curiosos que nos siguieron hasta la puerta.
"No llamaste," respondió Don Carlos. "Estábamos preocupados por ti."
Una enorme ola de humillación, alivio y vergüenza me cubrió. Diez minutos antes casi me entrego a Otto. ¿Y si Mami me hubiese encontrado en la cama con él?
Minna se me apareció al lado y me pasó el brazo por los hombros, los invitó a entrar y les ofreció algo de tomar. Pero Mami declinó con una sonrisa forzada y señaló el taxi al fondo del camino.
"¿Y tu amiga?" preguntó.
"Está en el baño," respondió Minna demasiado a prisa.
"¿Dónde está mi abrigo?" pregunté, con voz entrecortada. Jim lo sacó del _closet_ al lado de la puerta de entrada. Mami se le quedó mirando —un hombre hecho y derecho en pantaloncitos cortos en piel verde, con tirantes y medias rojas hasta la rodilla con dos bolitas de fieltro guindando. Otto me ayudó con el abrigo, inclinó la cabeza comprensivo cuando me lo ajusté y aplasté lo que quedaba del corsage que me había regalado. "Gracias," le dije a nadie en particular.
Me quería morir, deseaba que camino a casa el taxi chocara y nos matara a todos para no tener que volver a verle la cara a Otto. Pero, el taxista era cuidadoso e iba despacio, lo que me dio tiempo suficiente para gritarle a Mami y a Don Carlos.
"¿Cómo pudieron hacerme esto? Yo soy lo suficientemente grande para cuidarme sola."
"Baja la voz si no quieres un tapaboca."
El motivo de la ira de Mami era un enigma. Le discutí que le había pedido permiso, que había llevado a Otto a casa para que lo conociera, que había buscado una chaperona. Ella sabía dónde iba a estar, con quién iba a estar, cuándo regresaría. Don Carlos volvió a repetir que se me había olvidado llamarlos cuando llegué, pero yo les recordé que ellos tenían el número de teléfono de la hermana de Otto. ¿Por qué no llamaron para ver si había llegado bien? Habían pasado muchísimo trabajo y habían gastado un dineral para ir a buscarme, para humillarme delante de mis amigos y para darme una lección que no necesitaba. Estuve histérica todo el camino hasta Brooklyn. Tan pronto llegamos a casa, me arranqué el traje rosado con perlas falsas y lo dejé hecho trizas. Las perlitas se desprendieron de la tela, cayeron al piso tintineando y rodaron por debajo de las grietas de los zócalos, donde sabía que acechaban las cucarachas.
Regina no vino a trabajar el lunes pero Otto estaba allí. Ilsa y yo estábamos ajetreadas tratando de abrir y clasificar los paquetes de correspondencia entre las dos, cuando se acercó a la larga mesa que separaba nuestro departamento del de Compras. Era la misma persona, pero ahora lo veía a través de los ojos de Mami. A diferencia de Neftalí y de Sidney, Otto era un hombre, no un muchacho. No que eso lo hiciera menos atractivo. Al tenerlo de pie frente a mí, no pude evitar ruborizarme. Vergüenza y deseo se alternaban, se fundieron hasta que fueron una sola cosa.
"¿Podemos tomarnos un café, sí?" preguntó. Ilsa puso cara seria desde su escritorio.
"Mi _break_ es a las 10:30." Estaba feliz de que me hablara después del fiasco del sábado por la noche. Ilsa tosió con discreción para recordarme que tenía que volver a mi trabajo. Antes de retirarse, Otto inclinó la cabeza en dirección de Ilsa, gesto que yo encontré galante, pero a ella le enfureció. Murmuró dos o tres palabras en su idioma que sonaron bastante hostiles.
"¿Por qué ya no sales con Sidney?" me preguntó más tarde.
"Soy una _shiksa_." El tono defensivo en mi voz me sorprendió a mí, tanto como a Ilsa, que parpadeó con nerviosismo por unos segundos y después cambió la mirada.
Todo el mundo en la cafetería se quedó mirándonos a Otto y a mí, sentados solos en una mesa retirada de las demás. Me tuvo las manos cogidas los quince minutos que me permitían de _break_. En su inglés entrecortado, se disculpó por haberse propasado, lo que me sorprendió, pues yo había tenido tanto que ver con lo que pasó, como él.
"Tu mamá y tu papá es muy bueno," me aseguró. "Ellos cuidan bien."
"Me tratan como a una nena."
"Es bueno," me consoló. "Tú no eres muchacha americana. Ellas son muy libres."
"Yo quiero ser libre," le insinué, pero no lo cogió.
"Tú eres perfecta," me sonrió. "Mi novia," murmuró, y si llego a estar parada, las rodillas se me hubieran colapsado.
Más tarde, almorzamos en la cafetería donde una vez se había curado la herida que le causé. Tenía que ir a Suiza la semana entrante, me informó.
"Nosotros escribirnos," me ofreció.
Ilsa puso cara porque llegué tarde. Con la mirada atravesó las estibas de trabajo en las mesas. Mi excusa no le mejoró el humor. Más tarde, mientras estábamos archivando un montón de documentos en archivos contiguos, volví a excusarme.
"No estuvo bien," aceptó, "que me enojara tanto. No es contigo que tengo coraje, es con él." Con la cabeza hizo un gesto en dirección del Departamento Internacional. "Y tampoco es con él," corrigió. "Es con ellos." No tenía idea de qué estaba hablando. Me miró fijamente, con sus ojos azules. "Yo tuve muy mala experiencia con los alemanes," me explicó. Y entonces entendí.
"Pero Ilsa," le argumenté, "no pueden ser todos malos."
"Para mí, son todos iguales."
"Pero, no es justo."
"¿Justo? ¿La muerte de seis millones de judíos fue justa?" Subió la voz, pero no lo suficiente para que alguien la oyera. Tartamundeé que no, que no lo era, pero que era también injusto juzgar a toda una nación por los actos de unos pocos.
"¡Unos pocos!" Estaba consternada. "El país entero estuvo allí mientras los judíos eran asesinados. Mi mamá, mi papá, mis hermanas y mi hermano." La pasión de su voz era hipnótica y yo permanecí callada con la esperanza de que continuara, pero se mordió el labio y no dijo nada más.
"Lo siento tanto Ilsa." Le toqué el brazo, ella me apretó los dedos y sonrió con tristeza.
"Espero que nunca tengas que odiar," murmuró.
Regina regresó dos días después, todavía débil de un catarro. Ya había oído que Mami y Don Carlos me habían ido a rescatar a Long Island.
"¡Qué horrible! Minna dijo que tú estabas tan avergonzada... Todo el mundo se sintió mal."
"No importa. Otto quedó impresionado," me reí.
"Gilbert y yo nos vemos más." Regina se ruborizó.
"No se lo digas a Ilsa," le advertí.
Incapaz de convencer a Mami de que me dejara llegar a casa más tarde, sólo podía ver a Otto en el trabajo. Durante los días siguientes, almorzamos y tomamos el _break_ juntos. Ilsa refunfuñaba cada vez que me veía salir sin Regina, pero a mi no me importaba. Cualquiera que fuera su sentir contra los alemanes, era de ella y no mío. Acostumbrada a ser juzgada porque algunos puertorriqueños hacían cosas malas, yo no iba a hacerle lo mismo a Otto.
Supuse que Otto querría algo mío de recuerdo. Me corté un mechón de pelo y lo amarré con una cinta finita. Pero no me pidió nada y a mí me daba demasiado pachó admitir que se me hubiera ocurrido esa cursilería. Se fue justo después de Año Nuevo. Más allá de cogerme la mano y de darme, ocasionalmente, un besito en el cachete, nunca me tocó como me había tocado la noche de la fiesta de Navidad. Su trato caballeroso probaba que Mami tenía razón. "Un hombre que te quiere de verdá', te respecta." Yo se lo agradecí, pero no podía olvidar aquellas sensaciones de su lengua en mi boca, sus manos en mis senos, sus dedos inquisitivos tanteando. Era un hombre y su beso me había hecho sentir como una mujer.
# "Tenía la música por dentro..."
#
Fisher Scientific decidió mudar sus oficinas a New Jersey después del primero de enero. A Regina y a mí nos ofrecieron una promoción si nos transferíamos a las nuevas localidades. Con la promesa de que tenía un empleo en New Jersey, monté un caso para que me dejaran mudar más cerca del trabajo, como compañera de cuarto de Regina. Mami vetó el plan. "En Nueva York hay trabajos de más," alegó.
Antes de que la Compañía se mudara, aproveché un beneficio que ofrecían. Le pagaban parte de la matrícula a los empleados que quisieran continuar su educación. Don Carlos, que había estudiado contabilidad por la noche, me recomendó que averiguara en algunos de los Colegios Universitarios que ofrecían grados asociados y que eran más baratos que las famosas universidades de Nueva York. También ofrecían clases de noche y sabatinas, lo que me permitiría trabajar y estudiar.
Solicité al Manhattan Community College porque quedaba en la 51 con Sexta Avenida, cerca del área de los teatros y de los estudios de baile donde todavía tomaba clases. Las materias de los cursos eran administración de empresas, publicidad y mercadeo. Me matriculé en aquéllos en que podía salir antes de la una de la tarde. Después de clase, hacía trabajo temporero de recepcionista en algunas oficinas cerca de allí.
Un tiempito después de empezar las clases, fui a la librería del _college_ a comprar unos materiales que necesitaba. Al frente en la fila, había una muchacha como de mi edad, cuya presencia dominaba todo el pasillo hasta llegar a la librería. Tenía puestas unas botas marrón hasta las rodillas, una minifalda de cuero, una blusa marrón de chifón a través de la cual se le veía el brasier negro. Tenía el pelo arreglado en un montón de rizos dorados, agarrados con una pañoleta de chifón estampada con motivos de leopardo, cuyas puntas le caían sobre los hombros. Tenía un maquillaje elaborado que incluía hasta pestañas postizas.
Hartas de esperar en la fila largísima, las dos personas que estaban delante de mí, se marcharon molestas. La muchacha se viró, me sonrió de oreja a oreja y se presentó como Shoshana. "Estamos en la misma clase de Composición en Inglés," me informó.
Charlamos mientras esperábamos nuestro turno, y seguimos hablando durante el almuerzo en Horn & Hardart. Vivía en Queens con su papá y su mamá que eran tan anticuados como Mami.
"Es tan estúpido. Me paso la mitad del tiempo discutiendo con ellos," se quejó. Su mamá era especialmente crítica de la manera de vestir de Shoshana, lo que no me sorprendió. Si a mí se me hubiera ocurrido ponerme la mitad de lo que podría considerarse los conjuntos más conservadores de Shoshana, Mami me hubiera encerrado.
Shoshana nació en Israel y llegó a los Estados Unidos el mismo año que yo. Su papá y su mamá eran sobrevivientes del Holocausto, así es que me tomó un tiempo decirle de mi novio alemán.
"Entonces, es cierto," dijo pensativa, "las puertorriqueñas prefieren a los hombres rubios de ojos azules."
"¿De dónde tú te sacas eso?"
"De la escuela. Me lo dijo una compañera."
"Quizás estaba hablando de lo que le gusta a ella."
"Tú tienes un novio rubio de ojos azules," me señaló.
"Sí, pero es porque resultó así. El primer muchacho con el que yo salí era judío," añadí. "Pero no me podía llevar a conocer a su mamá."
"¡Muchacha! ¿Y causarle un ataque al corazón?"
Me sentía contenta cuando estaba con Shoshana, aunque de vez en cuando, hiciera conjeturas cómo que las puertorriqueñas preferíamos novios rubios. Si me ofendía y yo la corregía, movía la cabeza como si entendiera y pasaba a otro tema. Yo hacía lo mismo con ella.
Shoshana podía salir con muchachos, si eran judíos.
"Le estoy siendo fiel a Otto," le di como razón para no estar saliendo con nadie.
"¿Tú te crees que él se queda los fines de semana pensando en ti?" me preguntaba.
Las cartas de Otto no llegaban con la frecuencia que hubiera querido, pero traían noticias de noches en la ópera, la sinfónica y los museos. Describía sus caminatas por el bosque con tanto detalle que me hacía sentir que estaba allí. Mis noticias eran mucho menos interesantes, mayormente informes sobre mis cursos, el tiempo en Nueva York y la gente que conocía en mi trabajo de recepcionista a tiempo parcial. De vez en cuando fabricaba alguno que otro hombre exitoso que se interesaba en mi. Otto nunca reaccionaba a mis intentos de darle celos. También me inventaba amigas, felizmente casadas con hombres extranjeros, historias de matrimonios por _proxy_ en las que la novia y el novio estaban en ciudades diferentes (las novelas de Corín Tellado que todavía leía, tenían muchos de esos) y matrimonios en los que la novia hacía todos los arreglos, mientras el novio vivía en Europa. Tampoco respondía a esas indirectas.
Shoshana insistía en que por lo que ella veía, Otto y yo éramos amigos por correspondencia, y por lo tanto, yo debería salir con quien me diera la gana.
"Mi mamá no me quiere saliendo sola con hombres, hasta después que me case," le confesé.
"Mi mamá es lo mismo," rió Shoshana. "Es porque vienen del viejo continente."
Shoshana decía que la razón por la cual nuestras mamás nos decían que "no" tantas veces, era porque nosotras les hacíamos demasiadas preguntas. "¿Tiene ella que saber todo lo que tú haces?" preguntó Shoshana. Entonces, me sugirió que nos buscáramos un trabajo a tiempo parcial de noche, pero que les dijéramos a nuestras mamás que trabajábamos todas las noches. De ese modo, las noches que no tuviéramos trabajo podíamos salir.
Respondimos a un anuncio que solicitaba operadores telefónicos para de noche solamente y nos entrevistó el Sr. Vince, un hombre muy acicalado y perfumado, que usaba una sortija en el meñique, pantalones bien pegados, y una camisa desabotonada para exhibir su pecho velludo. Nos contrató en el acto y nos puso a trabajar esa misma noche.
Nuestro trabajo consistía en devolverle la llamada a la gente que había mostrado interés en saber cómo ganarse unas fabulosas vacaciones. La Compañía tenía anuncios de televisión de los diferentes lugares. Los televidentes llamaban a un número especial, que de hecho, era un servicio de contestadores, y ahí se les pedía el nombre, el número de teléfono, la hora más conveniente para recibir llamadas y qué anuncio habían visto.
El Sr. Vince nos dijo que no podíamos usar nuestros nombres verdaderos cuando devolviéramos las llamadas. Teníamos que escoger uno que fuera corto y fácil de recordar. Shoshana se convirtió en Miss Green y yo, en Miss Brown. El Sr. Vince nos dio un libreto. "¿Tú eres actriz; no debe darte mucho trabajo hacer esto." Sonrió.
Lo leímos en voz alta antes de que el Sr. Vince nos permitiera hacer la primera llamada.
"Buenas noches Sr. (o Sra.) _____. Le habla _____, devolviéndole la llamada. "¿Cómo está usted esta noche?" (Dale tiempo a que respondan. Si te preguntan como estás, dale las gracias.) "¿Usted llamó para saber cómo podía ganarse unas vacaciones en _______? ¿Alguna vez ha estado en _______?" (Sí: "Es un sitio fabuloso, ¿verdad?" No: "Ah, le va a encantar.")
Para ser elegible para el premio, el prospecto tenía que acceder a recibir una visita de venta. Si aceptaba, lo transferíamos al Sr. Vince, que fijaba entonces la fecha y la hora. Se nos pagaba por hora, pero si El Sr. Vince lograba vender un cierto número de vacaciones a los prospectos a quienes habíamos llamado, recibiríamos una comisión y la oportunidad de ir nosotras a una de esas vacaciones fabulosas.
"¿Cuántas tiene que vender?" le preguntó Shoshana.
"Yo te aviso cuando las venda," rió el Sr. Vince.
Trabajábamos en cubículos, cada una con su propio teléfono, un paquete de papelitos rosas de dejar mensajes, un par de lápices #2 y libretitas. Al principio el Sr. Vince monitoreaba nuestra parte de la conversación, parado detrás de nosotras mientras hablábamos con los prospectos o escuchando por una extensión. Pero, después que confirmó que estábamos hablando con sus clientes y no con nuestros amigos ("Hacen eso y las boto," amenazó), no era tan estricto. A veces, se iba y nos dejaba solas en la oficina porque en realidad no estaba demasiado ocupado. A pesar de todos nuestros esfuerzos, la mayoría de nuestros prospectos rehusaba la visita de venta, que los haría elegibles para ganarse unas vacaciones fabulosas, si le compraban otro viaje al Sr. Vince. Tan pronto salía él de la oficina, Shoshana llamaba a sus novios. Yo no tenía a quien llamar, así es que hablaba con sus novios también.
"¿Eres tan linda como Shoshana?" me preguntaban, y yo les contestaba que nadie era tan linda como Shoshana, y a ella le encantaba.
Muchas de las llamadas que devolvíamos eran de personas que no tenían ninguna intención de irse de vacaciones. "No dejen que les hagan perder el tiempo," nos regañaba el Sr. Vince. "Yo no les estoy pagando para que sean sus amigas."
Pero, a mí me gustaba escuchar. Si uno proveía un silencio interesado, la gente hablaba. Se quejaban de cónyuges poco atentos, de hijos malagradecidos, de sobrinas y sobrinos ingratos, de vecinos egoístas. A los difuntos se les recordaba con remordimiento.
_"No sabía lo mucho que dependía de él hasta que se fue."_
_"Era un ángel y yo no la supe apreciar."_
_"Él nunca supo cuánto lo quise."_
Más de una vez, me conmovieron hasta las lágrimas, las voces que flotaban desde la oscuridad hasta mis oídos. Nadie era feliz. Los dejaba hablar, les hacía preguntas, les señalaba las trampas en que habían caído y que los habían dejado tristes y solos. Si escuchaba con cuidado, podía oírme a mi misma, hablando dentro de veinte o treinta o cincuenta años. ¿Se resumiría mi vida en una serie de remordimientos y resentimientos? ¿Desearía darle marcha atrás al tiempo para revivir éste o aquel momento, para cambiar el desenlace como tanto lo deseaban las personas a quienes llamaba? ¿Cómo podría saber si la decisión que tomara hoy, vendría a atormentarme en los años venideros?
"Yo espero que nunca tengas que pasar por lo que yo pasé," empezó una mujer su historia y yo escuché con atención. Cada vida era un mensaje que yo tenía que decodificar, claves para lo que me esperaba. Ni un plano, sino más bien un mapa de donde tendría que escoger un camino.
Cuando no estábamos en clase o trabajando, Shoshana y yo íbamos a las grabaciones de los programas de televisión. Manhattan Community College quedaba a solo unos bloques de los estudios de CBS y de ABC y a dos bloques de NBC. Por el día nos sentábamos entre el público que iba a los programas de juegos, esperando a ver si nos seleccionaban para concursar. Nunca lo hicieron. Después de un tiempo, los ujieres de NBC nos reconocían y nos guardaban sitios en los estudios. Eran muchachos agradables, nítidos, vestidos en uniformes azules. Cada una de nosotras tenía su favorito. El mío era Andy, un gordito pelirrojo, con pecas en todas las partes visibles de su cuerpo, incluyendo los nudillos y los lóbulos de las orejas. La mayoría de las veces trabajaba el turno de noche y se aseguraba siempre de que yo pudiera entrar a ver las grabaciones del _Johnny Carson Show_. Andy me recordaba al Archie de mis paquines. Tenía la misma sonrisa boba y su sueño era escribir algún día, los chistes que Johnny Carson leía de las cartulinas.
"¿Tú me quieres decir que esos chistes no son de él?"
"Hay un ejército de escritores que hacen gracioso a Johnny Carson," me dijo.
"Pero, las improvisaciones..."
"Ah, esas son de él," contestó Andy. "El tipo es gracioso, pero los libretistas lo hacen más gracioso todavía."
Debido a que Andy trabajaba de noche, igual que yo, sólo podíamos salir de día, si yo no tenía clases. Íbamos a museos y a galerías de arte, almorzábamos en los carritos de _hot dog_ de la Quinta Avenida, nos pasábamos horas sentados en las cafeterías, cada cual metido en un libro diferente.
"¿Eso es lo que ustedes hacen cuando salen?" preguntó Shoshana.
"¿Leer una al lado del otro?"
Le expliqué que con Andy lo que tenía era una amistad, no un romance.
_"¡Oi!"_ se daba en la frente. "Eres incorregible."
"Él es lo único que tengo," me reí.
"Yo conozco algunos muchachos," me ofreció.
Sammy y Josh eran dos estudiantes de premédica, israelitas. Shoshana había salido con Josh un par de veces y él le había pedido que le presentara una muchacha a su mejor amigo. Así fue que una húmeda mañana de un domingo de junio, me encontré montada, tiesa y asustada, encima de un caballo en Van Cortland Park.
"Tienes que hacerle saber al caballo quién es el jefe," afirmó Sammy, su hablar todo enredado por el cigarrillo que tenía en la boca.
"Él es el jefe," le contesté.
"No, no, no, no." Sammy sacudió la cabeza y las cenizas volaron por todas partes. "Tú eres la jefa. ¡Tú!"
Se me hacía difícil creer que podría dominar la criatura temblorosa entre mis piernas. Viró sus ojos malévolos y húmedos para enfocarme, petrificada en su lomo hundido. Golpeó el cascajo con la pata como hacía Trigger cuando Roy Rogers le pedía que contara, sólo que este caballo no estaba contando. Estaba, lo podía asegurar, esperando el momento en que Sammy me entregara las riendas para salir disparado conmigo, bamboleándome, indefensa encima de él o colgando de un lado, enganchada todavía en el estribo. Les dije a Sammy, Shoshana y Josh que yo podía quedarme lo más feliz sentada en un banco, esperando a que volvieran de su paseo. Pero Shoshana insistió que ésta era una cita para divertirse. Según John, los caballos de Van Cortland Park eran viejos y dóciles y estaban a un paso en falso de parar en la fábrica de pega. Sammy me juró que era un jinete experto y que montaría a mi lado por si acaso lo necesitaba.
El caballo sabía para dónde iba. No importaba lo que hiciera yo con las riendas, él seguía pa' alante, detrás de los caballos de Josh y de Shoshana, como si hubiera estado amarrado a ellos. Aflojé mi agarre y miré a mi alrededor. A mi lado, Sammy iba hablándome en voz baja de su experiencia en los _kibbutz_ , donde había trabajado de electricista. Era muy delgado, de abundante pelo negro y ojos inquisitivos debajo de unas pobladas cejas. Fumaba cigarrillos sin filtro, uno detrás del otro. Tenía las puntas de los dedos y los dientes manchados de un tono mostaza opaco y de vez en cuando, un violento acceso de tos le enrojecía la cara y lo hacía doblegarse.
El camino cerca del establo estaba cubierto de árboles, pero al llegar a una curva se abría en un tramo largo, justo al lado de una avenida congestionada. Los carros y camiones retumbaban al pasar, pero los caballos acostumbrados a la congestión, no les hacían mucho caso. Seguían andando tranquilamente, el clic, cloc de sus patas, un contraste con el bullicio y las bocinas del tránsito. Josh y Sammy se hablaron en hebreo y Shoshana y Sammy cambiaron de sitio para que ella pudiera quedar a mi lado.
"Los muchachos quieren galopar," me explicó.
"¿Qué es eso?"
"Cuando los caballos van rápido."
Agarré las riendas de nuevo. Yo esperaba un _"Hi-yo Silver"_ o alguna exclamación parecida que animara a los caballos, pero Sammy y Josh simplemente hundieron sus talones en las barrigas de los animales y salieron corriendo. Los caballos de Shoshana y mío los siguieron aunque yo, por lo menos, no hice nada para incitar al mío. Halaba las riendas con todas las fuerzas de mis brazos, pero el caballo no me hacía ningún caso. El caballo de Shoshana era todavía más rápido que el mío y pronto me pasó por el lado como una flecha. De repente, bien alante, veo a Shoshana volar por el aire y caer de lado a sólo unas pulgadas de la avenida. En una maniobra digna de Annie Oakley, me escurrí del caballo todavía en movimiento y corrí hacia ella. Estaba inconsciente. Al instante, el tráfico de la avenida se detuvo, Josh y Sammy aparecieron y a la distancia los caballos podían verse galopando (si es eso lo que hacen los caballos cuando van ligero), hacia el establo con las riendas colgando, inútiles contra el suelo.
"Soy médico, soy médico," gritaban Sammy y Josh para mantener a la gente alejada de Shoshana.
"No se supone que uno mueva a alguien..." empecé a decir cuando Sammy la enderezó, pero me dio una miranda capaz de envenenar a la hiedra venenosa, así que hice mutis.
Se quejó, abrió los ojos y fue un alivio ver que estaba viva. Josh y Sammy la atendieron ansiosos hasta que la ambulancia llegó aullando hasta donde estábamos, yo me monté con ella y los muchachos nos siguieron en el carro de Sammy. Estaba pálida, pero consciente. Le tuve cogida la mano hasta que llegamos al hospital y cuando se la llevaron para examinarla, llamé a Mami.
"¿Alguien le avisó a su mamá?" preguntó Mami. Yo no lo había hecho y, probablemente, Sammy y Josh tampoco. Mami me dijo que era a la mamá de Shoshana a quien tenía que llamar, no a la mía.
Josh y Sammy entraron corriendo y mientras Josh se metió al cuarto donde se habían llevado a Shoshana ("Soy médico, soy médico") Sammy fue a llamar a la mamá. A Josh lo escoltaron hasta la sala de espera y allí nos quedamos en silencio hasta que salió un médico y nos llevó donde Shoshana. Estaba acostada en una cama alta, sus rizos dorados le enmarcaban el rostro como un halo. Las sábanas blancas acentuaban el efecto angelical. Se veía a la vez, vulnerable y _sexy_ y los tres hombres se hicieron un ocho. Los muchachos le hablaron en hebreo y entonces ella les pidió que la dejaran sola conmigo. Tan pronto salieron, me sonrió traviesa.
"Es mono, ¿verdad?"
"¿Cuál?"
"El médico."
"¿Cuál de ellos?"
"El de verdad, boba. Vamos a salir la semana que viene."
Shoshana estuvo en el hospital unos días. Le dieron de alta a tiempo para su cita con el Dr. Diamond, quien testificó cuando ella demandó a los que alquilaban los caballos. Transaron por el dinero suficiente para que Shoshana pudiera pasar el resto del verano en Israel. "Pero tú deberías salir con Sammy mientras yo esté fuera," me sugirió.
Con todo lo guapo que era Sammy, yo prefería mis tranquilas tardes con Andy. Shoshana viró los ojos. "¡Vas a morir jamona!" Nos reímos. Las dos teníamos diecinueve años y aunque estábamos desesperadas porque llegara el amor, sabíamos que teníamos tiempo todavía. Después de todo estábamos en América, no en el viejo continente.
No había vuelto a ver a Neftalí desde el día que trató de proponerme matrimonio en el medio de la calle. Su mamá, Doña Lila, todavía subía a visitarnos, pero casi nunca la veía. Entonces, poco después de nacer Cibi, Mami decidió que el apartamento en los altos del de Doña Lila era muy pequeño. Nos mudamos a una casa terrera, con un patio enorme, cuartos amplios y claros, de techos altos. En la parte de atrás, había un cuartito pegado a la cocina que yo pedí. Era lo suficientemente grande como para acomodar una camita, un escritorio, mi coqueta y la silla que hacía juego. Me pasé el "verano de amor" metida en ese cuarto, sin amor, escribiendo monografias sobre la historia de las relaciones públicas y el uso del humor en los carteles de publicidad.
Un día llegué a casa y encontré a Mami y a Tata en la mesa del comedor, con las caras tan tristes que supe que alguien había muerto.
"¿Quién?"
"Neftalí," contestó Mami.
Caí sentada en la silla, abrumada por imágenes de Neftalí, cosido a balas, en una trinchera en el lejano Vietnam. Pero, no fue así que ocurrió. Mami me explicó que el ejército había rechazado a Neftalí porque era adicto a la heroína.
"Lo arrestaron," me dijo, "y saltó por la ventana del cuartel de la policía."
"La verja tenía puyas."
Alcé las manos y con un gesto les pedí que pararan. Mi cerebro estaba todavía tratando de asimilar que a Neftalí lo había rechazado el servicio militar. Mami y Tata esperaron a que les indicara que estaba lista para saber más y volvieron a repetirme la información, como si la primera vez no hubiera estado lo suficientemente clara, y añadieron los detalles.
A Doña Lila le había dado un ataque de nervios tan fuerte cuando la llamaron de la policía, que la habían tenido que hospitalizar. Neftalí no le había dicho a nadie que el ejército no lo había aceptado. Según Doña Lila, se tiró por la ventana cuando lo arrestaron porque le daba vergüenza que todo el mundo se enterara de que estaba usando heroína.
"Por eso es que siempre usaba camisas de manga larga," reflexionó Tata, y yo me le quedé mirando.
"Yo nunca me di cuenta de eso," grité y me fui a mi cuarto seguida de las miradas preocupadas de Mami y de Tata.
Me tiré en la cama y cerré los ojos. Imágenes de Neftalí estallaban en mi mente en confusas secuencias. Neftalí levantando en brazos a mis hermanos para mostrar su musculatura. ¿Se estremeció porque le dolieron los pinchazos de las agujas en las venas? Las uñas chatas de Neftalí sobre las cartas españolas de Tata. ¿Perdía siempre porque no podía concentrarse? Los verdes ojos de Neftalí que me producían escalofríos. ¿Era esa mirada que a mí se me antojaba misteriosa, en realidad una mirada vacía? Era difícil reconciliar al héroe romántico que yo quería que fuera con quien había sido en realidad: un adicto que prefirió saltar por una ventana antes que enfrentar su problema.
Un domingo por la tarde, mi media hermana, Margie vino de visita. En los dos años que habían pasado desde que había visitado nuestro primer apartamento en la Avenida Pitkin con su mamá, nos habíamos mudado cuatro veces y Margie, por lo menos una. Se había casado recientemente con Néstor, un hombre cariñoso y sociable, algunos años mayor que ella. Estuvo parado detrás de ella, con su mano izquierda tocándole ligeramente la cintura, mientras ella nos lo presentaba y trataba de recordar nuestros nombres. Éramos ocho solamente cuando la vimos por última vez, y se sorprendió de que la familia hubiera crecido tan rápidamente en dos años.
Mami y Tata empezaron enseguida a preparar un arroz con pollo, con habichuelas guisadas. Néstor y Margie se quedaron en la mesa de la cocina contándonos de su apartamento nuevo en Yonkers.
"¿Por qué tan lejos?" preguntó Mami.
"Es casi en la colindancia con el Bronx," le dijo Néstor. Pero, para Mami, cualquier lugar fuera de los límites de Brooklyn o más arriba del distrito de ropa en Manhattan era territorio extranjero. Para ella era como si vivieran en otro país.
Margie y Néstor estaban interesados en conocer los más mínimos detalles de las noticias que podíamos darles. Querían saber en qué escuela estábamos, en qué trabajábamos, cuánto medíamos. Se excusó varias veces. "No quiero ser averiguá, pero es que hace tanto tiempo que no estamos juntos," decía. Me emocionaba su necesidad de estar en contacto con nosotros, de sentirse parte de nuestra familia. Néstor jugó con los nenes como si los hubiera conocido de toda la vida y Margie habló con sus hermanas, ayudó a Mami y a Tata en la cocina y le hizo caballito a Charlie y a Cibi en la falda. Se sentía cómoda, como si éstas fueran su casa, su mamá, su abuela, sus hermanos. Yo estaba encantada con lo franca que era, con lo dulce y sencilla. Antes de irse, Margie, le preguntó a Mami si, ahora que ella tenía su propio apartamento, podíamos vistarla de vez en cuando.
"¡Pues, claro que sí!" le dijo Mami y la abrazó.
Un par de semanas más tarde, Margie me esperó en la estación del tren de Yonkers. Caminamos un par de bloques hasta el edificio amarillo, ubicado en una loma donde ella y Néstor vivían en un apartamento alegre y soleado, decorado con el optimismo de los recién casados.
"Aquí es que vas a dormir." Abrió la puerta que daba a un cuartito cerca de la cocina. La cama estaba cubierta con un edredón mullido, con sus almohadones en combinación. Sobre una mesita de mimbre blanco con gavetas, había una lámpara. Al pie de la cama había un juego de toallas, una cesta con unos jabones chiquitos y una gorra de baño. De debajo de la cama sacó otra canastita. "Si te viene el periodo, aquí están los tampones." Una caja de tampones, acomodada en un nido de papel de seda rosado, perecía una ofrenda a la Diosa de la Menstruación.
Nunca había tocado un tampón ni por casualidad porque Mami me había advertido que podía perder la virginidad si los usaba. El mero hecho de que Margie pensara que los usaba, me hacía sentir grande, partícipe de los secretos de una mujer casada. Ella, ya no tenía que preocuparse por su virginidad y me pregunté si al ofrecerme el tampón, estaría ella probándome, a ver si yo me preocupaba por la mía.
Néstor estaba a punto de llegar del trabajo, así es que Margie me pidió que pusiera la mesa. En casa, poner la mesa consistía en poner las fuentes de comida en el centro de la mesa para que todo el mundo viniera a servirse su parte. Margie usaba _doilies_ , cuchillos, tenedores, un plato grande, un platillo de ensalada, un vaso para el agua, una taza para el café con su platillo. Tuvo que recordarme que pusiera cada cosa en la mesa. Había que llenar una jarra de agua con hielo, doblar las servilletas en forma de triángulo y ponerlas al lado izquierdo del plato. Hubo que sacar el juego de salero y pimentero del gabinete y alinerarlo junto al pote de ketchup y la azucarera. De tantas veces que me equivoqué me tomó el mismo tiempo poner la mesa que a ella preparar la comida para tres personas. Cerré los ojos y traté de recordar cómo era que ponían las mesas en los restaurantes, pero fue inútil. La mayor parte de mis experiencias comiendo fuera era en cafeterías y en el Automat, donde uno tenía suerte si le daban cubiertos.
"No, el vaso para el agua va a la derecha del plato," me corregía Margie. "El plato de ensalada va a la izquierda, así." Era amable, pero yo lo cogí personal y me pasé callada e incómoda toda la comida. Me ofrecí a fregar para compensar mi ineptitud en otras áreas. Margie se quedó en la cocina conmigo, lo que me hubiera alegrado en otras circunstancias. Pero, estaba tan consciente que era de esperarse que rompiera algo. Ella recogió el vaso del piso y me mandó a ver a Red Skelton con Néstor. Me alegré de que se fueran a acostar temprano y me acurruqué debajo de la elegante ropa de cama, ni reconfortada ni consolada.
A la mañana siguiente, me desperté con el olor de huevos fritos y café. Margie afanaba en la cocina mientras Néstor leía el periódico y tomaba su café con leche. Me metí al baño a lavarme. Encima del counter un Water Pik relucía blanco y antiséptico en la tablilla al lado del lavamanos. Me dio miedo tocarlo porque no sabía por qué parte del cuerpo de Margie o de Néstor se metía la manguerita. Se veía marital, tan íntimo como los tampones de algodón evueltos en papel blanco. Cuando salí, Néstor había acabado de desayunar.
"Más vale que termine de arreglarme," dijo yendo hacia el baño. Margie me sirvió un plato, cuidadosamente arreglado con dos huevos fritos, una lasca de jamón y tostadas cortadas en triángulos. Entonces, se sentó en la mesa a mordisquear un pedazo de pan y a charlar sobre lo que íbamos a hacer más tarde. Era difícil concentrarse en lo que decía por los sonidos que salían de detrás de la puerta del baño. La vibración de los aparatos eléctricos, las gárgaras, el agua corriendo, eran el contrapunto de los planes de Margie de caminar hasta el parque, almorzar en el _diner_ e ir de compras. Cuando Néstor salió, un olor limpio y fresco a menta y naranja saturó el cuarto. Margie lo acompañó hasta la puerta donde se besaron y se dijeron cositas. Después que se fue, Margie entró al baño y empezó de nuevo el zumbido y el gorgojeo.
"¿Tú no te lavas después de cada comida?" me preguntó y yo le contesté que sí entre dientes porque no era verdad, aunque yo sabía que debía hacerlo. "Puedes usar el Water Pik si quieres," me dijo. Todavía temerosa de tocar la manguerita, apreté un botón y salió un chorrito de agua, como el de un bebé haciendo pipi. "¿Sabes cómo usarlo?" me gritó desde la cocina y yo me emocioné de pensar que mi hermana mayor estaba a punto de impartirme conocimientos para adultos. Entró al baño, desenganchó la manguera y se echó agua en la boca como me hacía el dentista cuando me arreglaba las caries. Quedé absolutamente desencantada y se me debió haber notado porque a mitad de la demostración técnica, Margie se puso bizca, torció la boca en una mueca extraña y dejó que el agua se le chorreara por la barbilla. Nos miramos en el espejo y nos dio un ataque de risa que nos duró buena parte de la mañana porque cada vez que la miraba se metía el dedo en la boca, ronroneaba, gorgoteaba, se ponía bizca y hacía que se estaba lavando la boca.
Hablamos mucho de nuestro padre, a quien ella hacía años que no veía, pero con quien se escribía. Yo había vivido con él muchísimos más años que ella y le sorprendió saber que Papi cantara tan bien y escribiera poemas y décimas.
"Su letra es tan chiquita," dijo riéndose y me enseñó una hoja de papel con su letra inclinada, cada letra cuidadosamente trazada, los acentos sobre las íes casi horizontales. Fue Papi quien le dio nuestra dirección. "Le encantan tus cartas," me dijo, lo que me hizo sentir bien y culpable a la vez porque no le escribía con la frecuencia que debía hacerlo.
Durante las semanas siguientes mis hermanas y yo nos turnamos para pasarnos un tiempo con Margie y Néstor. Venían a visitarnos y nosotras regresábamos a su casa con ellos. O ella esperaba a alguna de nosotras en la estación y un par de días después, Néstor y ella nos regresaban a Brooklyn y se quedaban a disfrutar la comida rica de Mami. Una vez Margie abrazó a Mami y le murmuró que le gustaría que ella fuera su mamá. Mami nos repetía el comentario cada vez que uno de nosotros hacía algo particularmente irrespetuoso o impertinente, para recordarnos que había gente que sabía apreciarla aunque nosotros no supiéramos.
Un domingo por la tarde, Néstor nos informó que se mudaban para Miami. "Claro está, nos pueden ir a visitar allá," nos invitó Margie. Eso era poco probable. De haber tenido dinero para viajar, hubiéramos ido a Puerto Rico, donde hacía siete años que no íbamos. Cuando nos abrazamos, supe que pasaría mucho tiempo antes de que volviera a ver a Margie.
A mediados de agosto, recibí una invitación para la premiere de _Up the Down Staircase_ en el Radio City Music Hall, seguida de una recepción en el Hotel Warwick. Casi todos los actores que habían hecho de estudiantes estaban allí, de lo más emperifollados. Nos habían pedido que llegáramos temprano para podernos retratar en la hermosa escalera central. Como yo, muchos de los estudiantes nunca habían estado dentro de Radio City y todos estábamos tratando de no parecer demasiado deslumbrados. Pero una vez me acomodé en las butacas tapizadas del teatro, ya no pude aguantar más. Miraba boquiabierta el altísimo techo, las decoraciones doradas, las cientos de butacas inclinadas hacia el enorme escenario. Por primera vez vi la fila de puntapiés precisos de las Rockettes, las largas piernas que se movían al unísono, el taconeo que parecía venir de cada esquina de la sala.
Durante la presentación de la película, casi no pudimos concentrarnos porque mis compañeros actores y yo aplaudíamos o nos reíamos cada vez que aparecía alguno de nosotros en escena. En la fiesta, hablamos de cómo nos había ido después que terminó la filmación de la película. Sandy Dennis se había ganado un Oscar por _Who's Afraid of Virginia Woolf?_ , y los demás hicimos lo mejor que pudimos por colocar nuestros pobres logros en una luz igualmente espléndida.
Mi actuación no iba a ganar ningún premio, más bien, pasaría desapercibida. Pero, verme allí en la pantalla, renovó mi deseo de pararme otra vez frente a un público. Después de más de un año de trabajo en oficinas y de cursos poco motivadores, añoraba los nervios y la exitación antes de abrir el telón, los murmullos y el susurro de la expectación del público, el aplauso.
Volvi a revisar los avisos para audiciones en _Backstage, Show Business_ y en los tablones de edictos del International School of Dance, donde tomaba clases. Mi ilusión era bailar con algún grupo reconocido, como el de Matteo, pero pronto me descorazoné. Si bien en los últimos cuatro años había hecho grandes progresos como bailarina, mi competencia había empezado en la infancia. Podían tomar clases todos los días, podían dedicar sus vidas al baile. Muchos de los bailes tradicionales para principantes como _Allarippu_ , eran ya parte de ellos, que habían avanzado a coreografías más complicadas que requerían un alcance técnico y expresivo más amplio.
Iba a mis clases de baile, siempre que podía sacar el tiempo o el dinero, practicaba en casa, aunque mi familia se quejara de que el tintineo de las campanas de mi pulsera de tobillo y la música india atonal los volvía locos. Cada vez que consideraba dejar la universidad y usar el dinero de mis trabajos a tiempo parcial para dedicarme al arte, me reprochaba por ser tan ilusa e indulgente conmigo misma. Una artista tenía que sacrificarse por su arte, lo sabía bien. A una parte de mí le encantaba la idea romántica de ser una artista muerta de hambre. Pero, la voz que me hablaba con más fuerza me preguntaba qué posibilidades tendría una bailarina puertorriqueña subadiestrada en danza clásica india, de poder mantenerse a sí misma.
Nuestra casa en Glenmore tenía un sótano remodelado y habitable, además de un segundo piso donde Mami colocó las camitas de los nenes. Teníamos espacio de sobra, decía Mami. Quizás por eso fue que un día su prima Lólin se nos apareció en la puerta, del brazo del hombre con quien se había fugado. Lólin era flaquita, de unos ojos oscuros conmovedores y un carácter tranquilo. Era delicada y grácil, usaba el pelo largo y suelto como un ancho lazo negro entre sus hombros estrechos. Hablaba en un tono callado, con voz de gatita, y usaba con frecuencia el diminutivo, como si al hacerlo pudiera, a través de la palabra, hacerse aún más pequeña. Por eso no nos sorprendió que nos presentara al "esposo" como Toñito, en vez de Antonio.
Era tan delgado y callado como ella, de piel del color de la nuez moscada, pelo oscuro y rasgos taínos. Traían pocas pertenencias y ningún dinero, pero era obvio que estaban enamorados. Cada vez que él la miraba, ella se ruborizaba y bajaba los ojos. Cuando ella lo miraba a él, su mirada era como una caricia, suave, lenta, cargada de sentidos.
Mami no estaba muy contenta que digamos de tenerlos allí. Le caía bien Lólin, pero no se sentía muy cómoda con eso de tener un hombre guapo, joven, vigoroso —que no era nuestro hermano— en camiseta cerca de mí y de mis hermanas. Yo tenía diecinueve años. Delsa diecisiete, Norma dieciséis, Alicia catorce, Edna trece. Sabíamos lo que Lólin y Toñito hacían de noche en el sótano que Mami les había cedido, y aunque se esforzaban por ser discretos, era difícil hacer caso omiso de los quejidos y los suspiros que salían de su cuarto, del modo en que la mano de ella le acariciaba el muslo cuando se sentaban juntos y del modo en que él la mantenía abrazada cuando veían televisión.
Las tías y los tíos de Mami, las primas —Gury, La Muda y Margot— y otros parientes que casi nunca venían a vernos, se aparecieron por casa a ver a Lólin y a Toñito como si hubieran sido la atracción principal de un circo. En Puerto Rico, Tío Pedro no estaba muy contento con el marido que había escogido su hija mayor. Las muchas conversaciones telefónicas que escuché, eran súplicas para que fuera más flexible, para que aceptara a Toñito y, por lo menos, respetara el derecho que tenía Lólin de asumir las consecuencias de sus actos. Pero Tío Pedro era testarudo. Las tías, los tíos, las primas, Mami y Tata se pasaron horas en la mesa del comedor discutiendo lo que debían hacer. De vez en cuando, se oían en el sótano los románticos acordes de una guitarra con la que Toñito acompañaba sus canciones de amor, mientras Lólin lo escuchaba recostada. Los parientes se quejaban de que Toñito era un irresponsable porque, excepto por esa condená guitarra, no tenía en qué caerse muerto. Auguraban que la relación no duraría mucho. Lólin estaba acostumbrada a las comodidades, porque Tío Pedro era un comerciante, buen proveedor. Se había cegado con los encantos de Toñito, pera tan pronto se diera cuenta de lo vago que era, iba a regresar a Puerto Rico de rodillas, a pedirle perdón a su papá. Naturalmente, se sobrentendía, que Tío Pedro nunca la perdonaría ni aceptaría a Toñito, así es que el chisme estaba matizado siempre, con un dejo de compasión por la pobre, malaconsejada Lólin.
Mis hermanas y yo observábamos el drama. Durante años, Lólin y su hermana Tati, habían servido de modelos de niñas buenas y ahora resultaba que Lólin se había fugado con un tipo guapo que, a todas luces, carecía de medios para mantenerla. Y en Puerto Rico, Tati, que era más joven, ya se había casado, había tenido un nene y la habían abandonado. Tati, que era tan linda, vivaracha y alegre, ahora era una figura trágica. La desobediencia de Lólin no compaginaba con su naturaleza calmada, serena. La tías y las primas seguían usando a Tati y a Lólin de ejemplo —sólo que ahora eran ejemplos negativos.
De Tati decían, "¿Ven lo que pasa cuando las muchachas tienen mucha prisa de salir de sus casas y de la protección de sus papás?" El desafío de Lólin se lo achacaban a su docilidad. "Todo este tiempo fue la hija perfecta," decían en tono reflexivo. "Pero tenía la música por dentro." Cuando decían que Lólin tenía la música por dentro, nos miraban con intención, como dejándonos saber que si nos portábamos demasiado bien, empezarían a sospechar que nos traíamos algo entre manos.
Cuando Tío Pedro cedió finalmente y Lólin y Toñito regresaron a Puerto Rico, los parientes movían las cabezas y comentaban que era, justamente, la sobreprotección de Tío Pedro y Titi Sara lo que les había traído tantos problemas a sus hijas. Si no hubieran sido tan estrictos, Tati hubiese esperado un poco más antes de casarse y se hubiera evitado que la abandonaran tan joven, y Lólin hubiera conocido más muchachos y no se hubiera enredado con el primer manganzón que le hizo ojitos.
A mis hermanas y a mí nos aconsejaron que aprendiéramos de sus errores, que buscáramos un punto medio entre la impaciencia de Tati y el atrevimiento de Lólin. Era un camino sin precedente en mi familia. Cada día, cada tío, cada tía, cada prima adulta era un modelo de impulsividad y contradicción. Ni mencionar a Mami y a Tata que despepitaban reglas que no seguían, a diestra y siniestra, y eran el vivo ejemplo del aforismo "Haz lo que digo y no lo que hago." Tata nos advertía que no fumáramos ni bebiéramos, sentada en la mesa de la cocina con el cigarrillo en una mano y la cerveza en la otra. Mami hablaba de las bodas por la iglesia para nosotras y entonces, se usaba de ejemplo para mostrarnos lo frágiles que eran las uniones no santificadas.
"Pero Don Carlos estaba casado con aquella mujer y se divorció para estar contigo," empecé, y me mandó a callar.
"Ese matrimonio se había acaba'o mucho antes de que él me conociera," me dijo, lo que era cierto, pero ese no era el punto.
"Háganse de una educación para que puedan conseguir trabajo en oficinas y no en fábricas," nos aconsejaba Mami frecuentemente. Pero, al día siguiente nos mostraba un brasier cuidadosamente cosido. Con la cara resplandeciente de orgullo nos iba enseñando cada costura, nos señalaba lo difícil que era lograr que las dos agujas giraran hasta donde tenían que llegar, nos explicaba lo complicado que era trabajar con una tela tan delicada y nos enseñaba los broches novedosos. Hacía cosas útiles y bonitas con sus propias manos. Cuando quedaba cesante, se lamentaba de que sus destrezas no fueran suficientes para mantener a sus hijos.
"No sean como yo," insistía, "háganse de una profesión, no dependan de las fábricas para vivir."
Mientras más tiempo pasaba yo en casa, más confundida quedaba. No ibamos nunca a la iglesia, pero se suponía que me casara en la catedral. Tenía que ser buena, pero no demasiado buena o caería bajo sospecha. Si estaba demasiado ansiosa de irme de casa, mi vida podía convertirse en una tragedia. Si me quedaba mucho tiempo bajo el ala protectora de Mami, seguro que me engañarían los más experimentados en las reglas del mundo.
Había veces que salía para la escuela o el trabajo con la intención de no regresar, pero no tenía el valor de fugarme. A veces me quedaba mirando las lustrosas vías del tren y pensaba en lo fácil que sería tirarme en ellas, pero la idea de ser destrozada por toneladas de metal en movimiento, me hacía retroceder cuando el tren retumbaba cerca.
El hogar que había sido un refugio de los peligros de la ciudad se había convertido en una prisión de la cual quería escapar. La intensidad de mi vida familiar me tenía extenuada, el drama que no terminaba nunca, las crisis que surgían de la nada, se aplacaban, y abrían paso a otras que, también, eran sólo preludios de las próximas. Estaba hastiada de la lucha constante entre la vida que quería y la vida que tenía. Le tenía pavor a la soledad que se aferraba a mí en medio de la estridencia y el barullo de mi familia. No los quería culpar de mi infelicidad, pero tampoco quería contaminarlos con ella. Quería estar, como la Garbo, sola. Quería convertirme en La Sorda, sorda a las voces de mi familia, a sus mensajes contradictorios, a sus expectativas. Ansiaba poder ahuecar la mano cerca de mi boca, como hacen los cantantes, y escucharme a mí misma. Escuchar una voz, la mía, aunque estuviera llena de miedo e incertidumbre. Aunque me llevara a donde no debía ir.
# "¿Qué tamaño de brasier tú usas?"
#
El segundo semestre en Manhattan Community College, Shoshana y yo nos matriculamos en la clase Fundamentos de Matemáticas. El curso no era requisito para los estudiantes de concentración en Administración de Empresas pero ese otoño de 1967, el curso lo dictaba el chulísimo Profesor Grunwald. Shoshana estaba emocionadísima porque no solamente era el hombre más bello que jamás había visto, sino que también era judío. Según ella, como la clase se reunía tres veces a la semana y Míster Grunwald hacía horas de oficina para los que necesitaban ayuda, habría muchas oportunidades de que él se enamorara de una de nosotras.
"¿Pero, y si se enamora de ti y yo me pongo celosa?"
Shoshana lo consideró por un momento.
"No hagamos eso. Digamos que lo que es bueno para ti, es bueno para mí y viceversa. Así no hay celos."
Shoshana no tenía hermanas, yo sí. Su propuesta era noble, pero poco realista y así se lo dije.
"Está bien. Si me escoge a mí en vez de a ti, tienes que prometerme que te retirarás. Yo haré lo mismo."
"Así está mejor," acepté.
El primer día de clases, Shoshana y yo nos sentamos una al lado de la otra en la primera fila del salón que estaba lleno de mujeres vestidas, como nosotras, en sus mejores galas. Cuando entró Míster Grunwald, suspiramos a coro. No muy alto, ni muy bajo, perfectamente bien proporcionado de la cabeza a los pies. Míster Grunwald era tan bello como me lo había prometido Shoshana. Sus ojos azules, casi violetas, eran inteligentes y suaves. El pelo rubio-arena, se le rizaba por las orejas y por el cuello de la camisa. Bien afeitado, tenía la mandíbula cuadrada perfectamente formada, unos labios sensuales, una nariz perfecta. Tenía puesta una chaqueta en _corduroy_ marrón claro con parches de gamuza en los codos, mahones pegados, una camisa de botones en azul índigo, y una corbata con un diseño discreto. Cuando escribía sus fórmulas inescrutables en la pizarra, su letra era clara, precisa; los números bien formados; la _x_ enérgica y misteriosa. Él aseguraba que no nos estaba enseñando matemáticas, que lo que él enseñaba era lógica, pero a mí me tenía cara de matemática.
"Es guapo y eso," le dije a Shoshana, "pero esta clase está muy difícil. Yo mejor me doy de baja."
De ninguna manera quiso aceptar Shoshana esa decisión. "Lo que necesitas es una 'C' para pasar," me dijo. "Yo te ayudo."
Después de clase, Shoshana consultaba sus notas cuidadosamente tomadas y repetía al pie de la letra casi todo lo que acababa de decir Míster Grunwald. Dos veces por semana iba a su oficina y él corregía mis quizes deprimentes y mis exámenes delante de mí. Usaba colonia, una fragancia frutal que invadía mi nariz cuando se inclinaba a explicarme la relación del seno y el coseno con la tangente. Al hablar, arrastraba las palabras, las vocales largas y apacibles como una siesta. Quería vivir en sus diptongos, sumergida en sus _oes_ y _us_ , acariciada por sus _íes_. Pero la pasión de Míster Grunwald estaba en las regiones convexas y en el vértice de la parábola. Así como no se le ocurría a Shoshana que Míster Grunwald no se iba a enamorar de ninguna de nosotras, a él tampoco parecía ocurrírsele que yo, al igual que todas las demás muchachas de la clase, estaba enamorada de él.
Un día, mientras trataba de hacerme entender lo que para mí, nunca tendría sentido, Míster Grunwald se echó para atrás en su silla. "No trabajemos más en esto," sugirió.
Humillada de que se hubiera dado por vencido conmigo, me disculpé. "La matemática nunca ha sido mi fuerte."
Se frotó la cara con las dos manos. "¿Qué esperas lograr con tu educación universitaria?" preguntó desde detrás de sus dedos.
"Conseguir un buen trabajo," contesté.
Dejó caer las manos y me miró furioso. "¿Haciendo qué exactamente?"
"Publicidad, supongo." Me sudaba la frente, el bigote. "O mercadeo..."
"¿No tienes ni idea, verdad?" El tono de su voz, el registro bajo, la mirada suave con que acompañó sus palabras, hicieron que se me aguaran los ojos. Sacudí la cabeza. "¿Qué te gustaría hacer?" me preguntó y yo hubiera querido decirle, besarlo por todas partes, que era lo que estaba pensando, pero moví los hombros en un gesto de incertidumbre. "Shoshana me mencionó que eres bailarina," añadió. "¿Eres buena?"
Nadie me lo había preguntado y me tomó sólo unos segundos decidirme a contestar con honestidad, sin falsa modestia. "Soy muy buena," le dije. "Considerando lo tarde que empecé."
Me sonrió. "¿Danza moderna? ¿Ballet?"
Le devolví la sonrisa. "Soy, probablemente, la única bailarina puertorriqueña de danza clásica india que usted conoce."
El resto del tiempo de tutoría lo pasé describiéndole a Míster Grunwald las sutilezas de Bharata Natyam. Se mantuvo atento y me hizo comentarios que mostraban que estaba escuchándome.
"La música india progresa matemáticamente," intercaló en un momento y dejé de hablar para considerar esa posibilidad. Me observó mientras yo pensaba, como si fuese una experiencia nueva.
"Su... supongo que sí," dije por fin. Míster Grunwald se rió y me hizo sentir estúpida por haberle dado esa respuesta tan zángana.
Cuando le conté a Shoshana que me había pasado la sesión de tutoría hablando de baile con Míster Grunwald, quedó feliz. "¡Le gustas! Ahora, probablemente, te invite a un musical."
"Ese no es el tipo de baile que yo hago," protesté. A Shoshana se le había pasado el enamoramiento con Míster Grunwald ahora que estaba fascinada con el profesor de Fundamentos de la Publicidad. El profesor Delmar era mayor que la mayoría de los profesores del Manhattan Community College. Tenía el pelo canoso, los ojos grises, las facciones adornadas con unas arrugas deliberadamente formadas para resaltar su rostro bien parecido. Usaba trajes caros, bien entallados, que acentuaban su figura elegante, esbelta, de piernas largas. Míster Delmar caminaba por los pasillos del college como si fuese el dueño y atraía miradas de hombres y mujeres por igual, jóvenes y viejos. Me cayó mal en el acto, su aire tan compuesto me parecía engreído y calculado. Pero, Shoshana insistía en que eso me pasaba porque nunca había viajado a ningún sitio. "Es tan sofisticado, tan europeo," suspiraba.
Como cada una fantaseaba con un profesor distinto, no había celos. Nuestras conversaciones se iban en discutir hasta dónde llegaríamos si alguno de lo dos nos invitaba a salir. Las dos estábamos dispuestas a entregar la virginidad a la menor provocación, en el momento en que Míster Grunwald o Delmar insinuaran el más mínimo interés. Después de muchos intentos para que se fijara en ella, Shoshana decidió que Míster Delmar, nunca saldría con ella mientras fuera su estudiante. Lo dejó descansar durante el semestre de otoño y cifró sus esperanzas en la primavera. Por mi parte, la única manera de impresionar a Míster Grunwald era sumergiéndome en las transformaciones de figuras simétricas, y yo no estaba dispuesta a eso, ni siquiera con la promesa de una noche de pasión como premio. Continuaba escribiéndole a Otto mis largas cartas verborréicas, cuyas respuestas eran cada vez más cortas y menos frecuentes.
"Tú deberías dejarlo antes de que él te deje a ti," me aconsejó Shoshana. Dejé de escribirle y casi me pareció escuchar un enorme suspiro de alivio desde Suiza.
Al final del pasillo, donde estaban nuestros _lockers_ en Manhattan Community College, estaba el _lounge_ , un salón de recreo para estudiantes. Shoshana y yo nunca íbamos allí a estudiar, por la música tan alta que se oía detrás de las puertas cerradas. Nos gustaba la música, pero también nos gustaba oírnos cuando hablábamos. Entre clases, preferíamos ir a una cafetería o al Automat o nos encontrábamos con nuestros ujieres en la comisaría de la NBC. Pero un día, tenía urgencia de tomarme un café y me llegué hasta el _lounge_. El salón era grande, con un par de sillas maltrechas, un sofá pandea'o, una hilera de máquinas automáticas que vendían dulces, refrescos y bizcochitos envueltos en papel celofán. Debajo de la única ventana, había una mesita con una cafetera, sobrecitos de azúcar, un paquete de vasos de cartón y un pote de Cremora.
Al entrar me sentí como si me hubiera perdido en otro país. A mi izquierda, el salón vibraba con música Motown que salía de un tocadiscos portátil. Estudiantes negros en pequeños grupos, unos de pie, otros sentados, discutían de política mientras las Supremes cantaban _"The Happening."_ A mi derecha, en un volumen igual de alto, el ritmo de Eddie Palmieri destacaba los sonidos del espanglés. El medio del salón estaba casi vacío, excepto por unos pocos estudiantes blancos perdidos entre dos continentes vibrantes. La mayor parte de la gente me era familiar porque nos veíamos en clase o en los pasillos. Una de las muchachas, Gloria, me llamó hacia el lado del mambo.
"¿Tú eres puertorriqueña?" preguntó. Cuando le contesté que sí, se viró hacia el grupo. "¡Lo ven!" Se viró otra vez hacia mí. "Estos tipos aquí no me creían." Félix, uno de los muchachos, cogía una clase conmigo.
"Tú sabías que yo era puertorriqueña," le recordé.
"Yo se los dije," rió y chocó manos con otro muchacho que estaba cerca.
"¿Tú saliste en esa película sobre la escuela, verdad?" me preguntó otra muchacha.
Me encendí de placer porque me reconocieron. " _Up the Down Staircase_ , sí salí."
"¡Se los dije!" Otra vuelta de chocar manos. El timbre avisando el cambio de clases sonó y todos se apresuraron a recoger sus cosas.
"Nos vemos más tarde," dije. Nadie me contestó. Me fui sorprendida de que pudieran hablar de mí, pero que después de conocerme, a nadie le importara. Me pregunté si les habría causado una mala impresión y repasé la escena en mi mente muchas veces. ¡Había sido lo suficientemente simpática y abierta? ¿Les parecí demasiado echona por haber salido en una película? ¿Pude haber hecho algo más para caerles bien? Habían estado juntos, en un semicírculo mientras estuvimos hablando, como si me hubieran estado entrevistando. Pero entonces se dispersaron, me despacharon.
A lo mejor yo estaba hipersensible, me sugirió Shoshana más tarde, porque la mayoría de los estudiantes de Manhattan Community College eran negros o puertorriqueños y mi mejor amiga era judía.
"Quizás, en el fondo, dentro de ti, tú pienses que deberías ser amiga de ellos y no mía," dijo haciendo un puchero.
"Recuérdame no coger la clase de psicología el semestre que viene," le respondí. Desde que había empezado esa clase, todo lo que alguien dijera o hiciera estaba sujeto a interpretación. Pero sí, me molestaba que Shoshana pensara así y que una parte de mí —una partecita escondida— estuviera de acuerdo.
Dos o tres semanas después de empezar las clases, perdí mi trabajo de por las noches porque el Sr. Vince tuvo que cerrar el negocio. A pesar de los meses de publicidad y de las miles de llamadas a los prospectos clientes, no había podido vender suficientes vacaciones para mantenerme a mí y a los otros operadores de teléfonos empleados. Shoshana había dejado el trabajo hacía algunos meses, antes de irse para Israel en el verano. Varios hombres y mujeres llegaron y se fueron, pero yo trabajé con el Sr. Vince hasta el final, y el día que me dejó ir estaba a punto de llorar.
"Tan pronto mejora la cosa," me prometió, "te llamo." De bono, me dio el salario de una semana.
Fui a la oficina de empleo para estudiantes del _college_ , que tenía un programa a través del cual me daban crédito por trabajar en algo relacionado con mi concentración. La consejera me mandó al Advertising Checking Bureau. La Sra. Davis, mi supervisora, me prometió un horario flexible. "Tu educación es más importante que un empleo," me aseguró.
La Sra. Davis era una señora pequeñita, de pelo gris, que usaba faldas línea A y blusas vaporosas con puños y cuellos bien cerrados. Su escritorio quedaba cerca de la puerta de entrada, virado hacia el salón alineado con oficinas de paneles de cristal, para los supervisores y gerentes de rangos más altos. Los tres empleados de su departamento quedaban de frente a la Sra. Davis, pegados a la única pared de ventanas. Cada escritorio y cada tablillero que quedaba encima de los radiadores debajo de la ventana, estaba cubierto con estibas de periódicos, revistas, carteles doblados y anuncios para radio y televisión.
Mi trabajo era verificar que los anuncios para las cuentas que me habían asignado, salieran de acuerdo a los arreglos hechos entre el fabricante del producto y el detallista. El fabricante pagaba parte del anuncio. Mi trabajo era asegurarme de que si Amana pagaba el 30 por ciento del costo, el producto de Amana tomara por lo menos el 30 por ciento de las pulgadas por columnas del periódico o de la revista, o el 30 por ciento del anuncio de radio y televisión. Cada "verificador" estaba a cargo de varias cuentas de un área geográfica específica. Yo estaba a cargo de pequeños y grandes artefactos en el Upper Midwest. Los días que venía a la oficina, encontraba un paquete de recortes en mi escritorio y una lista de las cuentas, los acuerdos y los clientes. Con frecuencia, en vez de un recorte, estaba el periódico entero, que yo hojeaba hasta que encontraba el anuncio de mi cliente. Llegué a estar al tanto de menudencias tales como el tiempo en Ypsilanti, Michigan, el precio del trigo en Kankakee, Illinois y los resultados de las elecciones locales en Onalaska, Wisconsin. Por tercer año consecutivo, Tracey Dobbins en Rock Rapids, Iowa, ganó primer premio por su becerro en la exhibición de los Clúbes 4-H. El ruibarbo encurtido de la señora Sada Ulton fue el alimento de mayor venta en la feria del condado. Danny Finley había logrado la anotación ganadora en el partido de Retorno en Emmetsburg High School. Era un mundo tan distante de Brooklyn que me perdía en él, flotando entre comidas pro fondo de la iglesia, ferias agrícolas, nacimientos, muertes y funciones teatrales locales. De tiempo en tiempo, la Sra. Davis pasaba por nuestros escritorios para saber cómo iban las cosas o para preguntarnos si el logo de la RCA se destacaba lo suficiente en el anuncio de Sam's Appliance Mart. Pero al igual que a sus tres empleadas, le gustaba leer y muchas veces se reía de las payasadas de Lorenzo y Pepita o recortaba recetas de las páginas del _Philadelphia Inquirer_.
Sin la excusa de mi trabajo de noche, regresaba a casa todas las tardes, comía y me encerraba en mi cuarto a hacer mis asignaciones sobre figuras geométricas a las que Míster Grunwald nos hacía aplicarle la transformación de identidad.
Mientras más tiempo pasaba lejos de casa, más me sentía como una visita entre mi familia. El alboroto y el bullicio de mi casa, eran como una pausa entre las partes de mi verdadera vida en Manhattan, en los estudios de baile, en mis aventuras con Shoshana, en el _college_ , en el calendario social de Mishawaka, Indiana. Los fines de semana, cuando no tenía clases ni trabajo, me ponía al día con la vida de mis hermanas y hermanos. Delsa tenía un novio que se llamaba George, Héctor se destacaba en gimnasia en su Escuela Superior, Alicia cantaba en el coro de la escuela.
A los treinta y seis años y encinta de su hijo número once, Mami se veía gastada. Su andar era lento, su piel había perdido el lustre, su pelo arreglado en un recorte que le enmarcaba la cara, estaba seco y tenía horquetillas. Después de años de no poder pagar por cuidado dental, fue al dentista durante el verano y le sacó todos los dientes. La cara se le colapsó en la boca, desapareció su aire juvenil. La caja de dientes no le quedaba bien y sufrió cuatro meses de dolor antes de que el dentista accediera a arreglársela.
Tata se mudó unas semanas para irse a vivir con Don Julio, regresó, se mudó sola de nuevo. La fui a visitar al hospedaje donde vivía en una habitación apretada en la que había una cama, una butaca con el tapizado roto, una hornilla, unos vasos y platos descascarados. Al lado de la ventanilla había colocado su altar con las reliquias de familia y los santos que se suponía que le trajeran suerte cuando jugaba bolita. Ganó las veces suficientes para mantener su fe en ellos. El baño quedaba en el pasillo de abajo, pero ella tenía una escupidera debajo de la cama para no tener que hacer el viaje más veces de las necesarias. Después de unas cuantas semanas allí, regresó al sótano que una vez ocuparon Lólin y Toñito. Refunfuñaba y protestaba por toda la actividad que había en la casa. Ahora que éramos mayores, no le parecíamos tan simpáticos como cuando éramos nenes. El único al que todavía mimaba era a Franky, que a los cuatro años todavía era lindo y no se le alzaba cuando lo regañaba.
Yo entraba y salía en una nube de las actividades de la familia y notaba los cambios más obvios. Don Carlos vivía con nosotros. Norma se pintó el pelo de rojo. Don Carlos se mudó. El primo Paco dejó la lucha libre. Don Carlos volvió. Delsa sacó todas A'es en Matemáticas. Héctor le ayudó a Raymond a conseguir trabajo en la pizzería. Surgía una crisis, se tranquilizaba la cosa, volvía a surgir, y mantenía bien activas las tardes dominicales cuando las tías, los tíos, los primos y sus familiares se aparecían sin avisar para compartir la buena comida y el chisme que mantenía a todo el mundo entretenido de semana en semana. Yo me excusaba y me metía en el cuarto o me iba de la casa tan pronto podía zafarme, a veces con algunos de mis hermanos o hermanas, pero generalmente sola. Me iba a ver una doble tanda y me perdía en la versión de Hollywood de la vida, con sus mujeres elegantes, hombres masculinos, niños inexistentes, problemas resueltos por las armas o por el matrimonio, a veces, por los dos. De vez en cuando, iba a ver a Alma y Corazón y me sentaba en su apartamento tranquilo, a hablar de libros y a escuchar _rock and roll_ americano.
A Corazón le encantaban los Doors y los Bee Gees. "Óyete esto," decía, al poner el disco. Se acomodaba en el sofá y yo me tiraba al lado de ella, cerrábamos los ojos y escuchábamos. Ella entendía las letras de las canciones. Yo no. "¿Qué dice el coro?" preguntaba.
_"Come and maybe like my buyer?"_ adivinaba, y ella se estiraba de la risa.
Alma escribía poesía. Uno de sus poemas fue publicado en una antología. Colocó un pedacito de papel, como marcador de libro, en la página donde su nombre, en itálico, se veía autoritario y preciso. El poema, titulado "Ellos," era un soneto sobre la impotencia y la falta de poder. La última línea, "no me dejarán," me sorprendió tanto que levanté la mirada para preguntarle a quién se refería, pero lucía tan orgullosa y satisfecha consigo misma que no me atreví.
En el tablón de edictos de la International School of Dance alguien colocó un volante que decía: "Se solicitan Modelos, No necesita experiencia." Llamé al número y me dijeron que las modelos eran para una escuela de fotografía. A cambio de posar, las modelos recibían una foto 8×10 con brillo de cada estudiante que las retratara. No había desnudos. Las modelos traían su propia ropa y maquillaje. Se lo dije a Shoshana que enseguida estuvo de acuerdo con que lo hiciéramos.
La escuela era en el piso superior de un almacén en el West Side. Tenía un vestidor pequeño, con un espejo con luces y un closet donde las modelos colgaban su ropa. Nos pidieron que nos maquilláramos "al natural" y esperáramos en lo que nos llamaban. Había otras dos muchachas, Sharon y Beverly, que pensaban usar sus fotos para el portfolio que les exigían las agencias de modelaje. Eran más altas y tenían mejores pómulos que yo, así es que no me hicieron caso. A Shoshana la miraron con envidia. Era tan alta como ellas, pero mejor formada y sus facciones, bien proporcionadas y muy bonitas, estaban hechas para ser fotografiadas.
Nos sentamos en unas sillas que había en una fila fuera del vestidor y unos momentos después apareció el instructor seguido de un grupo de muchachos.
"Señoritas," dijo. "Esto funciona así: durante la primera media hora, más o menos, todo el mundo posa y todo el mundo puede tomar fotos. Pero, si alguno de los fotógrafos y las modelos desarrolla alguna afinidad, pueden irse a trabajar aparte al _seamless_ ," y nos señaló un enorme rollo de papel blanco que colgaba del techo hasta el piso, "o en el _set_." Sus manos aletearon en dirección de un telón de fondo gris oscuro con unos mancharones simulando nubes. "¿Estamos listos?" Dijimos que sí y nos condujo a otro _seamless_ , donde posamos en grupo, mientras los estudiantes se afanaban por tomar sus fotos, se movían de un lado a otro, buscando los mejores ángulos y trataban de no metérsele en el medio a los demás.
Me sentía tonta asumiendo poses de chica _"Wow,"_ pero Sharon y Beverly eran unas expertas. Cuando el instructor nos pidió a Shoshana y a mí que nos saliéramos para poderlas fotografiar a ellas solas, primero juntas y después individualmente, me di cuenta de la diferencia que había en tener aptitud para el modelaje. Sharon abrió los codos y se colocó las manos en la cintura como una mariposa en alerta y de algún modo logró parecer bidimensional. La especialidad de Beverly era el movimiento. Saltaba, y lograba mantenerse en el aire el tiempo suficiente para que los fotógrafos pudieran tomarle más fotos de las que me pudieron tomar a mí quieta. Era alucinante ver cómo Sharon y Beverly iban de una pose a otra, cada una diferente, cada una impresionante. Shoshana y yo nos miramos con desánimo. No había manera de que pudiéramos equiparar eso.
Nos sorprendimos cuando al terminar la primera sesión a Beverly y a Sharon les dijeron que se podían ir, pero tres muchachos pidieron fotografiarnos a Shoshana y a mí juntas. Posamos de perfil, primero mirándonos una a la otra, y después, mirando en la misma dirección. Los tres muchachos trabajaban en equipo; se arreglaban las luces uno a otro, hacían turno para usar la cámara colocada en un trípode, y el instructor desde el fondo, les daba sugerencias de cómo colocarnos o iluminarnos mejor. También nos fotografiaron individualmente, en diferentes cambios de ropa, maquillaje y peinados que Shoshana y yo nos hacíamos una a la otra. La sesión tomó la mañana entera. Al final estábamos exhaustas, pero el instructor nos pidió que regresáramos otro día para trabajar con otro grupo de estudiantes, y accedimos en el acto.
"¿Tú puedes creer esto?" deliraba Shoshana. "¡Despacharon a las profesionales y nos prefirieron a nosotras!" Se nos abría la posibilidad de una carrera que nunca habíamos considerado. "Si logramos unas fotos buenas," sugirió Shoshana, "podemos organizar un buen portfolio y llevarlo a las agencias." Imaginábamos que la mismísima Eileen Ford nos contrataría y nos pondría en la portada de _Vogue_.
"¡Competencia para Twiggy!" dije echándomelas.
"Aunque yo creo que tú eres más _Seventeen_ ," consideraba Shoshana.
"Nunca he visto a nadie con mi tez en esa portada," me quejé.
Hicimos dos o tres sesiones más en la escuela de fotografía. Las 8×10 que nos dieron los muchachos eran en blanco y negro, con brillo, de alto contraste, muy diferentes a como imaginamos que serían cuando posamos.
"¿Tú crees que éstas se pueden usar para el portfolio de modelo?" le pregunté un día a Shoshana, mientras revisábamos nuestras fotos. Sombras profundas nos distorsionaban las facciones, las yuxtaposiciones dramáticas nos hacían virar las fotos de lado a ver si nos podíamos reconocer.
"Son artísticas." Estaba tan poco convencida como yo. Aún así, nos compramos un portfolio negro y organizamos nuestras fotos con las más "artísticas" al final. Fue idea de Shoshana que después que organizáramos nuestros portfolios, consultáramos con nuestros profesores de publicidad. "Después de todo, ellos tienen agencias y ven miles de modelos," fue su argumento.
Su profesor era el elegantísimo, Míster Delmar. El mío era el Dr. Henning, largo como un jugador de baloncesto, de enormes pies y manos, y una cabeza imponente cubierta de rizos grises. Usaba trajes que le colgaban en pliegues y caídas como si hubieran sido togas de paño de lana.
Cuando le pedí que si podía examinar mi portfolio, me sugirió que fuera a su oficina que quedaba sobre Tad's Steaks en la Séptima Avenida. En un oscuro salón al final de un pasillo igual de oscuro que olía a carne asada, estaba el Dr. Henning sentado detrás de un enorme escritorio de roble, delante de una ventana. Una luz gris ceniza le daba por detrás e iluminaba el polvo que flotaba en el aire. Me señaló una silla de piel que había frente a su escritorio, y me hundí en sus chirridos mohosos.
"Están muy bien," dijo examinando las fotografías. Viró el portfolio como hacíamos Shoshana y yo, tratando de buscarles la vuelta a las fotos más artísticas. Me miró. "¿Cuánto tú mides?"
"Cinco pies, cuatro pulgadas."
"Las modelos de ropa son más altas," me dijo. "Por lo menos, cinco ocho. Pero, tú podrías hacer trabajo para catálogos. ¿Qué tamaño de brasier tú usas?"
"¿Perdón?"
"No estoy siendo fresco," me tranquilizó. "Hay un mercado para modelos que se especializan en ropa interior de mujer, brasieres y fajas, ese tipo de cosa." Estaba espantada y se me debe haber notado porque el Dr. Henning levantó las manos con las palmas hacia mí como para protegerse de cualquier cosa que yo le fuera a tirar. "Es para catálogos respetables como los de Sears y JC Penney." Se estiró en su silla hasta alcanzar un libro grueso que tenía detrás. "Déjame enseñarte."
Esperé hasta que pude hablar sin echarme a llorar. "Gracias, pero..."
Hojeó el catálogo hasta que llegó a las últimas páginas. "Generalmente, no fotografían la cara, así es que nadie te va a reconocer." Inclinó el catálogo hacia mí. Fotos en blanco y negro de torsos de mujeres con brasieres de algodón, aparecían impresas en los márgenes de la derecha y de la izquierda. Unos bloques de texto daban los detalles sobre el estilo, el precio, los tamaños disponibles. "Hay mucho dinero en esto," prometió.
"No es la clase de modelaje que tenía en mente," le dije, tratando de mantener la compostura. Temblaba de rabia y de humillación, pero no quería decir ni hacer nada estúpido. Después de todo, era mi profesor y el que me iba a dar una nota al final del semestre. "Gracias de todos modos."
"No lo decidas ahora mismo. Yo sé que no es el tipo de cosa que las muchachas agarran enseguida." Devolvió el catálogo a su sitio.
"Estoy segura de que eso no es lo mío." Cogí mi portfolio. Me acompañó hasta la puerta. "Le agradezco el tiempo que se ha tomado."
El largo pasillo se sentía aún más largo con él mirándome desde la puerta. Traté de caminar sin que las caderas se me movieran de lado a lado, sin que los senos se me bambolearan. Ninguna parte de mi cuerpo debería parecerle insinuante a quien acababa de recomendarme una carrera como modelo de brasieres. Esa idea, ¿se le habría ocurrido mientras miraba mis fotos? O me habría estado ligando desde diferentes ángulos mientras yo estaba sentada en clase apuntando sus famosas consignas publicitarias. ¿Cómo podría mirarlo a la cara otra vez? En ropa apretada, nunca. De eso estaba segura.
A Shoshana no le pareció que tenía porqué ofenderme. "Alguien tiene que modelar brasieres," razonó. "¿Por qué no tú?"
"A mí no me dejan ni ponerme bikinis. ¿Cómo le voy a decir a mi mamá que modelo brasieres?"
"Tú le dices demasiado," me dijo.
"Ese no es el punto, Shoshana."
No le hizo caso a mi molestia. "¿Ella no hace brasieres?"
"Sí, pero, eso no quiere decir que ella quiera que me las ponga sin camisa."
La cita de Shoshana con Míster Delmar le fue mucho mejor. A él le pareció que podía ser modelo, pero necesitaba otras fotos. "Consíguete unas que sean más realistas, no este arte fantasioso," fue su recomendación.
Como me faltaban cuatro pulgadas para ser modelo, decidí que no quería hacerme más fotografías.
"Puedes usarlas cuando vayas a alguna audición," me sugirió Shoshana. "¿Tú no me dijiste que siempre piden una foto de cara?"
Siempre me había sentido en desventaja en las audiciones por no tener un _composite_ ni una foto de cara porque costaban más de lo que yo estaba dispusta a invertir en mi carrera como actriz. Pero, estas fotos no se parecían en nada a las fotos de cara que otra gente traía a las audiciones. Hicimos otra cita para que nos fotografiaran, y esta vez, trajimos ropa sencilla y nos pusimos poco maquillaje. Otra vez nos retrataron juntas, pero al final de la sesión, diferentes estudiantes nos pidieron que posáramos para ellos individualmente.
El joven que me lo pidió a mí era de la India. Era huesudo, levemente más alto que yo, de hombros caídos. En una voz suave, deferente, hablaba un inglés musical que al principio no entendí porque me sonaba demasiado rápido. Una vez me acostumbré, me gustaron sus vocales fuertes, expresivas, enérgicas y el modo en que diferenciaba cada sílaba de la otra.
"Mi nombre es Shanti," me dijo, mientras montaba las luces para el retrato.
Trabajamos bien juntos. Era amable y considerado, me dejaba tomar un descanso entre tareas, hacía gestos leves con sus labios o su cabeza o sus dedos huesudos, para pedirme que me moviera o que mantuviera una pose. Fue como si nos hubiéramos conocido de mucho tiempo y al final de la sesión me pidió si era posible hacer algunos exteriores.
"¿La escuela lo permite?" le pregunté.
"Sí, claro," me dijo. "Se supone que aprendamos a hacerlos también."
Nos encontramos ese domingo por la tarde en Central Park. Caminamos un rato y cuando él veía algún fondo bonito, me hacía posar allí. Después de un par de horas, ya tenía suficientes fotos y nos fuimos pero no sin que antes me pidiera que nos volviéramos a encontrar el fin de semana siguiente, esta vez en el Village.
Ése domingo andaregueamos un rato por allí y me retrató cerca de un grupo de _hippies_ en ropas extravagantes y con el pelo todo desaliñado. Me sentía incómoda cerca de ellos y se reflejó en las fotos. Los _hippies_ le hacían muecas a la cámara, mientras yo posaba muy recatada frente a ellos, con mi cartera apretada contra el pecho. Después me retraté en medio de unos viejitos que observaban un juego de ajedrez. No tenía idea de cómo se jugaba, así es que presté mucha atención para ver si le cogía el golpe. Pero había tan poca acción, que era imposible descifrarlo y cuando levanté la vista para ver si Shanti había terminado con los retratos, mi mirada se encontró con los ojos de mi profesor de matemáticas, Míster Grunwald. Detrás de él, Shanti retrató mi gesto atónito.
"¡Hola!" Míster Grunwald parecía contento de verme, cosa rara, porque en clase no me hacía ningún caso para poder concentrarse en la gente que entendía la multiplicación escalar. Sonrió. "¿Siempre viajas con un fotógrafo?" preguntó. Se lo presenté a Shanti y los tres caminamos por entre las filas de la mesas de ajedrez rodeadas de espectadores, como si el juego realmente tuviera algo excitante. Se rió cuando comenté que no había nada que ver. Al lado de nosotros, Shanti ni hablaba, ni tomaba fotos y yo sentía crecer su mal humor como un globo cuando se llena de aire. Míster Grunwald lo sintió también porque después de un par de bloques, se excusó y siguió su camino en dirección contraria.
"¿Es tu novio?" preguntó Shanti tan pronto vio que Míster Grunwald no podía oírlo.
"No. Es mi maestro de matemática," me ruboricé.
"Veo." Sonaba molesto y a mí me dio coraje. ¿Qué le importaba a él si Míster Grunwald era mi novio o mi profesor de matemática? "Creo que terminamos por hoy," me dijo.
"Está bien." Mire en la dirección que había cogido Míster Grunwald pensando que tal vez podría alcanzarlo.
"Muy bien," dijo Shanti y salió disparado. Desapareció enseguida entre la multitud y yo me quedé allí, sorprendida ante su reacción. En las dos semanas que habíamos trabajado juntos, no se me había ocurrido que el interés de Shanti fuera más allá de lo profesional. Era difícil imaginar que cuando me miraba a través del lente, veía algo más que una modelo. Caminé un rato por el Village preguntándome si quería ser el objeto del afecto de Shanti, además del de su arte. Cuando me di cuenta de que seguía paseando con la esperanza de encontrarme de nuevo con Míster Grunwald, supe la respuesta.
La semana siguiente, Míster Grunwald me pidió que me quedara después de la clase. Mientras los demás salían, me entregó una copia de _Backstage_ , con un anuncio de una audición circulado en rojo.
"Vi esto," me dijo, "y me pareció perfecto para ti."
"¿Usted es actor?" Me debí haber imaginado que un hombre tan guapo como él, era del mundo del teatro.
"No," me dijo, "Pero mi _roommate_ sí."
Shoshana se había quedado afuera esperándome. Cuando le conté que Míster Grunwald me había dicho que su _roommate_ era de teatro, quedó sin habla.
Le pregunté por qué se veía tan desencantada.
"Es homosexual."
"¡Por favor!"
"Piénsalo. Es soltero, vive en el Village y tiene _roommate_."
Me pareció el razonamiento más absurdo que había oído nunca. Si alguien sabía de homosexuales era yo, que estaba rodeada de ellos en las clases de baile, le argumentaba. Pero, no había cómo disuadir a Shoshana. Decía que algunos homosexuales no lo parecen. "A que su _roommate_ es..." y viraba los ojos, "bien loca."
Había una sola manera de saberlo. Teníamos que ver el _roommate_. Si seguíamos a Míster Grunwald después de clases, veríamos donde vivía. Podíamos echar un vistazo a través de la ventana. O si esperábamos cerca de la puerta, a lo mejor los veríamos salir juntos.
En la excitación de planificar cómo íbamos a seguir a Míster Grunwald sin ser vistas, casi se me olvida para qué me había pedido que me quedara después de la clase. El anuncio para la audición que apareció en _Backstage_ pedía una damita joven para una compañía de teatro infantil que estaba en planes de montar una fábula india que se presentaría en Broadway. Era un sueño hecho realidad —un papel que podía representar y en el que podía aprovechar mi tipo y mi entrenamiento. Llamé para pedir cita y me dijeron que no tenía que llevar nada preparado porque leería del libreto.
Si Shanti volvía a hablarme a lo mejor podía coger un acento indio y hasta tirarle dos o tres palabritas en hindú si fuera necesario. Nos encontramos para almorzar y me enseñó las fotos que me había tomado la semana anterior. Me fijé bien en las inflexiones de su voz, traté de recoger el ritmo de su hablar. Se rió de mis intentos de imitar su acento.
"Esto no se aprende en un día," reía. "Toma la vida entera."
La audición era en el Michael's Rehearsal Studio en la Octava Avenida. Había otras actrices esperando antes que yo, pero dos de ellas eran muy viejas para ser damita joven y las demás no se veían tan indias como yo. Me arreglé para verme lo más india posible sin tener puesto un Sari: el pelo trenzado y dividido al medio, los ojos bordeados de kohl, una gotita de esmalte de uña rojo en la frente.
Cuando me pidieron que pasara, Bill, el director y Vera, la productora, se miraron. Me hicieron dos o tres preguntas sobre mi experiencia previa y me pidieron que leyera del libreto. En la escena, un personaje de nombre Soni, le explicaba al personaje llamado Babu, que estaba prisionera en una torre porque su tío tenía planificado casarla con un rajá. A Soni le permitían salir de la torre a rezar, pero tenía una cadena amarada a la cintura que su tío halaba cuando era hora de regresar.
Tan pronto escuchó mi mal logrado acento indio, Bill me interrumpió y me pidió que leyera con mi acento natural. Vera me pidió entonces, que improvisara una escena en la que un mono entraba por una ventana y le ofrecía a Soni ayudarla a escapar de la torre. Hice lo mejor que pude para parecer sorprendida, asustada, curiosa, agradecida. Al final, Vera me pidió mi número de teléfono y me dijo que se comunicaría conmigo.
Bill y Vera eran muy profesionales y reservados y no me dejaron ver muy claramente si les había o no gustado mi audición. Yo quería hacer el papel de Soni más que nada en el mundo y cuando salí del estudio, revisé en mi mente todo lo que había hecho y dicho, tratando de pensar si había algo más que podía haber hecho para asegurarme el papel. Era sábado, casi la hora de la función de matiné. Caminé por Broadway, pasé por las marquesinas con los nombres de las obras famosas y de los no menos famosos actores, entre los turistas que se quedaban boquiabiertos con los carteles vulgares que había frente a la tiendas de pornografía que competían con los teatros legítimos. Doblé en la esquina y me detuve frente a la fachada chocolate con puertas coloradas de Performing Arts. Apoyé la frente contra el cristal. Las cajas de madera tan familiares estaban amontonadas cerca de los _lockers_ en el sótano; los pupitres, acomodados en semicírculo, como si se fuera a montar una escena para la facultad y el estudiantado. Temblaba de la ansiedad y tuve que apoyarme contra las columnas del frente de la escuela cerrada, hasta que me pasó el mareo y se me quitó el temblor.
Colgué el teléfono y me desplomé contra la pared de la cocina. Mami entró en pánico. "¿Qué pasa?"
"Me dieron el papel," me dije a mí misma, incrédula. "Voy a salir en una obra." La miré. "En Broadway."
"¿Eso es bueno?" preguntó. Supo que lo era cuando saqué a Cibi de su corral en el medio de la cocina y la bailé por toda la casa. "Voy a estar en Broadway. Voy a ser una estrella," cantaba. Cibi se rió, me babeó y yo la devolví al corral.
Tata se arrastró desde su esquina en el sótano, Delsa y Norma dejaron el televisor prendido en la sala y corrieron a la cocina, Raymond y Franky llegaron del patio, el resto de mis hermanas y hermanos bajaron de sus habitaciones. La casa estaba llena de la gente que yo quería, ansiosa de escuchar la buena noticia. Sin saber de qué se trataba, mis hermanas y hermanos, mi mamá y mi abuela, podían darse cuenta de que era algo maravilloso, por lo contenta que yo estaba. Les repetí lo que me dijo Vera. La obra era para un público joven. Era una, de un repertorio de obras que se montaban en las escuelas y teatros alrededor del Northeast. La compañía Children's Theater International, había recibido múltiples distinciones y premios por su excelencia y profesionalismo.
Mi familia estaba impresionada. No me preguntaron si me pondría alguna de esa vestimenta extraña que me habían visto antes, o la campanas amarradas en los tobillos, o el punto de esmalte de uñas rojo que había que remover con acetona, en la frente. No me relajaron con los sonidos extraños que salían de mi cuarto cuando practicaba mis bailes indios, que aumentarían ahora que estaría trabajando regularmente. Estaban tan contentos como yo, lo que me alegraba aún más, porque era maravilloso hacer algo que no sólo me hacía sentir bien a mí, sino que hacía sonreír a los demás.
Llamé a Shoshana, a Shanti, a Alma, a Corazón, a todo el que conocía. Se lo hubiera dicho a los extraños también, si me hubiera atrevido.
Los ensayos empezaron esa semana en una buhardilla en Christopher Street en el Village. Algunos miembros del elenco habían actuado en _Petey and the Pogo Stick_ y _Hans Brinker_ , otras producciones del Children's Theater International. La obra en que saldría yo, había estado en repertorio el año anterior.
"A lo mejor conoces a la muchacha que hizo de Soni," me dijo Vera. "Estuvo en tu escuela. Priscilla López."
"Sí, se graduó un año antes que yo." Me emocionaba hacer un papel que había sido estrenado por Priscilla, una de las actrices más talentosas de Performing Arts cuando yo estudiaba allí. Vera también me dijo que aunque les complacía el hecho de que fuera una bailarina india clásica, mi papel no requería que bailara.
"Hay una escena de baile," añadió, "pero ya tenemos a quién la va a hacer." Me dio pena, pero se me pasó tan pronto leí el libreto completo y me di cuenta de que Soni, mi personaje, tenía un papel más importante en la historia que la bailarina que atendía al rajá.
El papel estelar de Babu estaba a cargo de Allan, un actor y cantante, cuya franqueza y simpatía me conquistaron instantáneamente. En la obra, él rescataba a Soni de su torre en la prisión y no requería demasiada actuación de mi parte, enamorarme de él en cada presentación y permanecer flechada entre funciones.
Allan y Bill se conocían hacía años, habían trabajado juntos y eran buenos amigos. Ambos tenían maravillosas voces entrenadas y con frecuencia les hacía alguna pregunta sólo por oírlos hablar.
El otro miembro del elenco que llegué a conocer bien fue Tom. En la primera escena de la obra, cuando Soni y Babu se conocen, Tom, en su papel de dios-mono permanecía sentado en la posición de loto en un nicho en el escenario. Después de que el tío malvado arrastraba a Soni fuera de escena, Babu rezaba pidiendo un modo de ayudarla a escapar. Cuando Tom abría sus ojos y hablaba, el público gritaba porque no esperaba que la estatua cobrara vida.
Después de los ensayos, con frecuencia me reunía con Bill, Allan o Tom para una cena tarde o un café. Llevaban en el teatro mucho más tiempo que yo y contaban historias simpatiquísimas sobre sus percances y malos ratos en y fuera de escena.
Vera vivía en Westchester County y viajaba para los ensayos. En unos momentos era como una mamá ansiosa y en otros, toda una mujer profesional. Si alguno de nosotros tosía, buscaba en su bolso grande alguna pastillita para la garganta y nos la daba, pero si alguno llegaba tarde, se aseguraba de que entendiéramos que la próxima vez teníamos que ser más responables. Con frecuencia nos recordaba nuestras obligaciones como actores. "El que hagamos teatro para la niñez, no quiere decir que seamos condescendientes con nuestro público."
Ensayábamos por las noches y los fines de semana y siempre salía del teatro sintiéndome afortunada de estar trabajando con personas tan talentosas y comprometidas. Era divertido improvisar con Allan y con Tom y entonces, trabajar del libreto con Bill, que nos exigía muchísimo, pero nos hacía sentir como si fuéramos la gente más brillante que había dirigido. Con el pasar de las semanas de ensayo, según fui comprendiendo lo que Bill esperaba de mi actuación, el personaje de Soni evolucionó.
"No tan estilizado," me regañaba, cuando trataba de incorporarle movimientos de danza india a la representación que hacía de Soni.
Con el College por la mañanas, el Advertising Checking Bureau por las tardes y el Children's Theater por las noches y los fines de semana, pasaba la mayor parte del tiempo fuera de casa. Veía a Shoshana en clase y con frecuencia almorzábamos juntas antes de irme a trabajar.
No nos olvidamos de Míster Grunwald. Un día lo seguimos en el _subway_ hasta su parada en Waverly Place. Era tan fácil seguirlo. Iba totalmente desentendido del mundo que lo rodeaba, parecía estar absorto en profundos pensamientos matemáticos y mantenía los ojos fijos en los obstáculos que pudieran hacerlo tropezar, pero en nada que le quedara más lejos. Una vez entró al _subway_ , se sumergió en un libro grueso con una parábola y unas fórmulas en la portada. En el vagón de al lado, Shoshana y yo esperamos hasta que se levantó y se detuvo junto a las puertas. Tan pronto se abrieron, se bajó. Esperamos un momento y entonces lo seguimos a la distancia, tratando de no llamar la atención a pesar de los ataques de risa que nos entraban cada vez que pensábamos en lo que estábamos haciendo. Míster Grunwald subió las escaleras de la estación que conducían hacia la calle y lo perdimos entre la multitud. Pero Shoshana pronto lo divisó en un kiosco comprando un periódico. Dejó caer unas monedas en la mano del vendedor, dobló en la esquina y desapareció.
"Qué raro," comentó Shoshana, mientras mirábamos a escondidas desde la esquina de un edificio hacia una calle de hermosa casas _brownstones_ de piedra rojiza, de portales muy cuidados y jardineras en las ventanas. "Tiene que haber entrado en una de esas casas."
"Tiene que haber sido la primera," le indiqué, "no tuvo tiempo suficiente para ir más lejos." No bien había terminado de hablar, de la planta baja que daba a la calle salió Míster Grunwald siguiendo a un perrito blanco, vaporoso y ñoño y con mucha prisa por llegar a una boca de incendio.
"¿Qué te dije?" dijo Shoshana triunfante. Fuera o no fuera Míster Grunwald homosexual, no cabía duda de que su preferencia en perros balanceaba la ecuación —como él mismo hubiera dicho— en dirección de la sospecha de Shoshana. "Un perro de loca," proclamó, virando los ojos, como si yo no me hubiera dado cuenta. "Ay, chica, lo siento tanto," me dijo apenada cuando vio la cara que yo puse.
"No parece la clase de hombre que tendría un perro así," fue todo lo que pude decir.
Shoshana me llevó hasta un restaurante que quedaba cerca y me pidió un tazón de sopa y un sándwich para compensar mi desilusión. Cuando me sentí más fortalecida, hablamos de lo difícil que era encontrar el hombre perfecto.
"Quizás es que somos demasiado exigentes," reflexionó.
"Hubiera podido tener un pastor alemán, por lo menos," dije, obsesionada con la imagen de este hombre magnífico pegado a un perrito ñoño.
Las ventanas del restaurante daban a una intersección congestionada cerca de la entrada del _subway_. Una mezcolanza de _hippies_ , gente de negocio, viejitas, ancianos, gente pidiendo, músicos callejeros, desfilaban de arriba a abajo, como para entretenernos. En medio de la descripción de su hombre ideal Shoshana gritó como si la hubieran pinchado. Por la acera, caminando hacia nosotras venía Míster Grunwald, todavía detrás del perrito ñoño y abrazado a la cintura de una pelirroja patilarga. Cada dos o tres pasos se detenían para besarse, lo que fatsidiaba a la gente que venía detrás, que con cara de disgusto tenían que dar la vuelta para no tropezar con ellos.
"El perro es de ella," decidí.
Bill y Allan se morían de la risa cuando les hice el cuento.
"¿Te gustan los pastores alemanes?"
"Por lo menos son perros de verdad, no plumeros andantes." Bill y Allan se miraron y volvieron a reírse. Pasó una semana antes de que yo entendiera dónde estaba la gracia. Un día, Allan tuvo que regresar de prisa a su apartamento en el Upper West Side. Vivía en el segundo piso al fondo de una casa _brownstone_ y según fuimos subiendo la escalera, un profundo ladrido que solo podría venir de un perro enorme, llenó el pasillo. "Espera aquí," me dijo Allan. Entró al apartamento en lo que yo lo esperaba en el pasillo. Unos segundos después, abrió la puerta y salió agarrando con la mano izquierda, al pastor alemán más grande que había visto en mi vida. "Este es Tristán," me dijo. El perro me metió el hocico ahí mismo y Allan tuvo que agarrarlo para que no me siguiera empujando contra la pared. "Le gustan las chicas," comentó sonriendo. Le puso la cadena y caminamos medio bloque hasta Central Park, donde Allan jugó con Tristán, mientras yo me quedé recostada de un árbol. Era enternecedor ver el afecto entre ellos, el modo en que el perro le seguía cada movimiento, se detenía si Allan se detenía, se movía cuando Allan lo hacía. Mirándolos me di cuenta de que Allan era el primer hombre, fuera de mi familia, que había querido. Me había enamorado de varios —Neftalí, Otto, Míster Grunwald— pero lo que sentía por Allan era diferente a la fantasías románticas que había tejido en torno a otros hombres. No fantaseaba con casarme con Allan, ni siquiera con besarlo. Quería estar con él para hablar y hacer tonterías y oír sus cuentos. Me encantaba su risa, el modo en que le brillaban los ojos cuando estaba orgulloso o satisfecho. Entre nosotros no había juegos sexuales. De haberlos habido, me habría desilusionado.
Shoshana no entendía mi relación con Allan. Muchísimas veces discutíamos si era posible la amistad entre un hombre y una mujer, sin que hubiera interés sexual. Ella decía que no, yo que sí. O mejor dicho, yo esperaba que fuera posible. No quería pensar que, durante el resto de mi vida, cada hombre que conociera tuviera que ser evaluado como un posible encuentro sexual. Como un ejemplo de mi habilidad para tener amigos varones que no fueran novios, le señalaba mis encuentros frecuentes con Shanti.
"Él está enamorado de ti," insistía Shoshana. "Lo que pasa es que tú te niegas a admitirlo."
La devoción de Shanti por mí me halagaba mucho. Trabajábamos bien en equipo y continuábamos colaborando una con el otro, a pesar de que con frecuencia nos poníamos quisquillosos cuando estábamos juntos. Siempre estaba criticando mi dieta que consistía principalmente de _hot dogs_ de carritos, arropados en col agria, acompañados de Yoo-Hoo; _pizza_ y uvita, o cremosos palitos de Jacob y café.
"Eres una bailarina," me recordaba Shanti. "Tienes que alimentarte mejor."
"Por los menos, yo no fumo," le replicaba mirando con desprecio su cigarrillo.
Un cálido y soleado día de invierno, nos sentamos en las escaleras de la Biblioteca en la Quinta Avenida, después que él me había tomado una serie de fotos encima de los leones que vigilaban la entrada. "Tú no eres la muchacha más bonita que yo he retratado," admitió, "pero cuando te miro a través del lente, me veo a mí mismo."
"No me austes así," le dije bruscamente.
Cuando a Shanti le daba con ponerse metafísico, yo me ponía bien pesada. Una vez me dijo que nuestras almas estaban conectadas y yo me le quedé mirando como si estuviera loco. "Yo no tengo alma," le solté finalmente.
Se quedó en silencio un rato largo, y entonces habló, casi en un suspiro. "Yo veo tu alma aunque tú no la veas."
Ahora me tocó a mí quedarme muda. Su fe en algo dentro de mí que yo no podía ver, me hacía sentir inadecuada e inmadura. Tuve que defenderme. "Ves lo que quieres ver, no lo que está ahí."
En otra ocasión, insistió en leerme la mano. "Aquí está la línea de la vida," me acarició la curva que está entre el pulgar y el índice que va hasta el pliegue de la mano. "Vas a tener una vida larga," me aseguró. "Pero éstas," dijo, y me señaló unas líneas irregulares, "indican enfermedades."
Retiré mi mano. "Esas son zanganerías," le dije. "No creo en nada de eso." La verdad era que me asustaba pensar que él pudiera saber algo de mí a través de mis manos. Si era así, ¿habría otras señales en la forma de mis labios o cejas, en la manera en que se me rizaba el pelo? Si las había, no quería saber qué significaban. ¿Qué más daba si me quedaban diez, veinte, cincuenta años de vida? ¿O si al día siguiente me iba a pasar un carro por encima? ¿Por qué querría saber lo que me deparaba el destino?
"No puedo predecir qué va a pasar," protestaba, "sólo puedo interpretar lo que ya pasó."
"Eso lo puedo hacer yo," le repliqué.
No importaba lo grosera que yo fuera con él o lo mucho que él me criticara, siempre sacábamos tiempo para estar juntos. Sabíamos que las fotos que tomábamos no eran comerciales y no aparecerían nunca en ninguna revista, ni se reproducirían por cientos como tiros de cara para audiciones. Los fines de semana nos encontrábamos en Central Park, en el Lincoln Center o en el Empire State Building, donde me tomaba fotos en las que yo salía tan remota e inaccesible como era para él. Todas las semanas me entregaba unas cuantas fotos ocho por diez, con brillo, que yo estudiaba como si fueran un rompecabezas, cada rasgo, sombra o línea, una parte de un todo más grande e indefinido. Me sentía protegida por su formalidad, su quietud solemne. Pero me asustaba cuando captaba otra parte de mí, mis ojos suplicándole al espectador algo que yo no podía definir.
Según se fue acercando el día de mi debut en Broadway, me fui acomodando en mi papel de Soni. Hacíamos presentaciones en la escuelas locales, lo que me daba la oportunidad de familiarizarme con la escenografía y de sentirme cómoda con mis dos vestuarios. Bill y Vera tenían planificadas varias presentaciones adicionales después de la temporada en Nueva York y una gira fuera de la ciudad. Preparé a Mami para la posibilidad de que estuviera fuera de casa dos semanas o más con el elenco de _Babu_. Aparte de alguna que otra noche en casa de mis primas Alma y Corazón, y las visitas a Margie en Yonkers, nunca había dormido fuera de casa. Esperaba que Mami formara un lío, pero sólo hizo unas cuantas preguntas sobre dónde iríamos y parecía estar tranquila con esa posibilidad.
A pesar de que el Sr. Grunwald me dio C en la clase, lo invité al estreno. Después de todo, él era el responsable de la audición con que había logrado el papel. Los dos sabíamos que la C era un acto de generosidad, considerando mi progreso negativo en matemáticas o, como él lo llamaba, en pensamiento analítico. Para el trabajo final tuvimos que esbozar una teoría y probarla utilizando la progresión lógica. Yo me enfrasqué en probar que la civilización comenzó en Puerto Rico.
"Pero usted nos dijo que la teoría no tenía que ser cierta," le argumenté cuando discutimos mi trabajo. "Usted lo que pidió fue que la probáramos lógicamente."
"Pero no lo hiciste," sostuvo.
La obra estrenaría y tendría una temporada corta en el Longacre Theater durante las fiestas navideñas. Unos pocos meses antes, Sandy Dennis había protagonizado _Daphne in Cottage D_ en ese mismo escenario. Para el primer ensayo general me asignaron su camerino, que ostentaba en la puerta una estrella. Cada vez que abría la puerta, la estrella frente a mis ojos me llenaba de un orgullo, que yo contenía para no parecer demasiado engreída. Quería compartir mi alegría con alguien, sin sonar demasiado vanidosa o echona, así es que le escribí a Papi en Puerto Rico. Le mandé un programa donde aparecíamos Allan y yo con nuestras "coronas reales" y los elaborados vestuarios con bordados y lentejuelas que nos había diseñando Robert de Mora para el final. Le describí las horas de trabajo que tomaba montar un espectáculo, la gente involucrada, la trama fantasiosa. Le conté del camerino, de los espejos del grande de la pared rodeados de luces, del baño privado, las alfombras, el sofá viejito pero cómodo donde tomaba siestas entre funciones. Como no estaba allí para verlo con sus propios ojos, Papi lo veía a través de los míos y por primera vez me alegré de que no viviera con nosotros porque ahora había alguien cuya visión de mi mundo dependía de mi perspectiva.
El escenario del Longacre era enorme. Unas horas antes del estreno, me paré en el centro y miré las filas de butacas vacías y vi, no un teatro vacío sino, un reto. Mi misión era transformar un salón lleno de adultos agotados de hacer compras de Navidad y de niñas y niños inquietos, en un público. Si podía creerme que yo, una muchacha puertorriqueña de Brooklyn era una princesa india, cautiva en una torre, que fue rescatada por un mono, y que se casó con un príncipe, mi público me creería. Si podía hacerlo, podía hacer cualquier cosa.
Mami, mis hermanas y hermanos vinieron a la primera función. Estaba tan nerviosa que lo hice todo volando y para los saludos estaba atontada y exhausta. Cuando regresé al camerino para cambiarme, me recibió un enorme arreglo de flores de parte de Vera y Bill, otro del Sr. Grunwald y un tercer arreglo de Shanti. En unos minutos, el camerino se llenó de gente. Cuando vino trabastidores, Mami también traía flores, aunque un poco marchitas de haber tenido que compartir sus brazos con Franky. Míster Grunwald pasó por el camerino, desde la puerta saludó el barullo y desapareció.
Bill y Vera fueron especialmente atentos con Mami y ella me comentó después, que se notaba que eran personas respetables, serias.
"Cuídeme bien a mi hija," le pidió Mami cuando Vera mencionó la gira.
"Yo soy mamá también," le respondió Vera. "No se preocupe." Mami la abrazó.
Don Carlos trajo a sus hijos. La Muda apareció por allí. Shoshana vino con Josh y Sammy. Shanti me retrató maquillándome y con el vestido de princesa cautiva en la torre.
"Esto no se parece en nada a lo que usan las muchachas indias," se quejó. "Esto es para el harén."
"El diseñador se tomó algunas libertades," le dije, "pero el vestuario sirve bien su propósito en escena y eso es lo que importa." Shanti sacudió los hombros.
Todos los días antes de la función, me bajaba del _subway_ que venía de Brooklyn, caminaba despacio por Broadway, y escuchaba la conmoción como si fuera una música maravillosa. La bocinas de los taxis formaban un estruendo. Los turistas chachareaban en una abundancia de dialectos, imposibles de entender, pero familiares. Los Hare Krishnas tintineaban sus cimbales de dedos, aporreaban sus tambores, cantaban su cántico gozoso. Doblaba la esquina y me sonreía con el intrépido Babu en la marquesina del Longacre. Al frente había unos enormes carteles de Allan y míos, de Tom como el Mono-dios, del rajá y de su bailarina. Entraba al teatro por la puerta de los artistas, flotaba como en un sueño hasta mi camerino de primera actriz, y captaba mi imagen en el enorme espejo. No era la muchacha más bonita que Shanti había retratado, ni la actriz más famosa que se había graduado de Performing Arts. Sola con mi reflejo me preguntaba qué me había llevado allí. Estaba agradecida pero no sabía a quién darle las gracias.
# "No se vería bien."
#
Como Vera vivía en Westchester County y estaba a cargo de presentar allí una serie de teatro infantil, organizó varias funciones de _Babu_ en las escuelas de su área. El elenco se reunía en el estudio de ensayos y Bill guiaba una guagua Volkswagen crema y marrón bordeando el Río Hudson hacia Scarsdale o Bronxville, Tarrytown o Elmsford, Mamaroneck o White Plains. No nos quedábamos mucho tiempo en las comunidades donde hacíamos las funciones porque el elenco estaba siempre loco por volver a la ciudad. Algunos alegaban que la alergia del polen se les agravaba de sólo mirar los árboles. Otros recordaban su niñez en comunidades similares y se pasaban malhumorados y pensativos todo el camino de ida y vuelta.
El público bien educado y mayormente blanco de Westchester County, contrastaba con la niñez expresiva y franca de las escuelas de la ciudad de Nueva York. Cuando las cortinas se abrían para mostrarme rezando frente a un dios de piedra en el escenario de un auditorio de una escuela de suburbio, se escuchaba un aplauso amable y respetuoso y se percibía una atención intensa. En Brooklyn Academy of Music, en Town Hall o en la escuela de la ciudad de Nueva York, el aplauso de los estudiantes de tercero, cuarto y quinto grado iba acompañado de pitos, gritería y comentarios. Requería mucha concentración esperar mientras las maestras trataban de controlar a los estudiantes que gritaban: "¡Qué buena estás!" o "¡Mami!" o "¡Oye, cosita linda!" Una vez el público estaba más o menos callado, Allan hacía su entrada y descubría a Soni, cautiva, con una cadena alrededor de la cintura. Mientras discutíamos el problema en que me encontraba, una halón me asustaba; el público no podía ver a Bill ni al supervisor de escena arrastrándome fuera del escenario mientras yo le suplicaba a Babu que me ayudara. En los suburbios, éste era el punto de mayor intensidad dramática. En la ciudad, la muchachería gritaba. "Síguela, mano," una recomendación obvia, pero poco práctica en términos dramáticos.
Según se acercaba la primavera, el itinerario de funciones se intensificó y se dio la gira prometida. Tomé un tiempo libre del college y del Advertising Checking Bureau para poder ir a Maine, New Hampshire, Massachusetts. El plan era guiar por el norte hacia Bangor y entonces, ir haciendo representaciones por toda la costa hasta Nueva York. Bill, Vera y el elenco viajaba en la guagua VW, mientras el supervisor de escenario nos seguía en un camión que llevaba la escenografía y el vestuario.
Un domingo por la mañana temprano, nos encontramos todos en la esquina de la 55 y Sexta Avenida. La consabida guagua VW estaba estacionada en la esquina y detrás de ella, el camión alquilado.
"¡Nanook del Norte!" bromeó Allan cuando me vio, emburujada como si nuestro destino fuera el Polo Norte y no Nueva Inglaterra. Estábamos a mediados de marzo y aunque en Nueva York empezaba a florecer, había consultado varios periódicos regionales en el Advertising Checking Bureau y sabía que había que prepararse para mal tiempo, desde nieve y agua-nieve hasta lluvias torrenciales.
El elenco negoció dónde sentarse, un proceso que Vera comparó con el de sus cuatro hijos porfiando sobre quién iba a ir al frente, quién necesitaba parar frecuentemente para ir al baño y quién se tenía que sentar junto a la ventana porque si no, vomitaba. Fuimos despojándonos de la ciudad según avanzábamos hacia el norte por la Interstate 95. Cada vez que aparecía el letrero de una salida conocida, alguien hacía un cuento de teatros y repertorios de verano, o de las peripecias de quedarse varado en New Haven en plena tormenta, o de las obras que se hacían en las afueras que nunca llegaron a la ciudad. Lee, que hacía el papel de la nodriza de Soni, empezó una ronda de canciones de campamento, que yo no sabía. Mientras todos cantaban, yo marcaba el ritmo dando palmas o silbaba.
Nos deteníamos a comer en _diners_ que quedaban a veces a cinco o diez millas de la autopista. Millie's Coffee Haus, Aunt Polly's Place, The Towne Line Diner, The Harbor View (sin agua a la vista) —todos, ofrecían enormes platos de una comida deliciosa y barata. Éramos un grupo de tomadores de café y generalmente entrábamos a los _diners_ con la agonía del síndrome de la abstinencia de la cafeína que nos ponía irritables e impacientes hasta que el líquido negro y oloroso nos llegaba al sistema. Las camareras veteranas reconocían nuestra mirada aturdida tan pronto entrábamos olfateando desesperadamente el aire. No nos preguntaban si queríamos café. No bien nos sentábamos nos servían la taza llena, entonces nos entregaban un tentador menú con una larga lista de opciones. En la parte de atrás del mostrador, había unas vitrinas refrigeradas llenas de doradas tartas de manzana, tartas de limón y merengue, crujientes _cobblers_ de frutas, pudines, tapioca cremosa. Con excepción de Lee, que era una vegetariana estricta, el resto éramos comensales indiscriminados, deseosos de probar las especialidades locales, como la leche con café de Rhode Island, la sopa de almejas y crema de Massachusetts, las almejas de New Hampshire, los mariscos al vapor de Maine.
Pasamos nuestra primera noche en Lewiston, Maine. Bill y Vera estaban nerviosos en cuanto a cómo serían los arreglos de alojamiento en los _bed and breakfast_ locales. Sus temores se disiparon cuando llegamos a una hermosa residencia Victoriana en los altos de una loma.
"Parece una casita de libro de cuentos," exclamé encantada con las cortinas de encaje en las ventanas, los aleros profusamente adornados, la puerta de cristal con un diseño grabado. La dueña de la casa, una mujer de tez rosada llamada Misis Hoch tenía el fuego encendido y unos panecillos en el horno. Lee, Allan y yo nos quedamos con ella; el resto del elenco y el _crew_ se quedaron cerca, en otras casas.
Fuimos a cenar a un restaurante local. Tan pronto entramos, me di cuenta de cuánto sobresalíamos del resto de los parroquianos, diez nuyorquinos en vestimenta urbana. Yo era la persona más oscura en aquel lugar, y las miradas que recibí fueron como dardos. La camarera arrimó un par de mesas, mientras nosotros esperábamos amontonados en la puerta. Me sentía avergonzada por la conmoción que habíamos causado, consciente de que éramos forasteros en esa pequeña localidad. Cada movimiento nuestro era observado por las personas de allí, que parpadeaban y cambiaban la vista cuando alguno de nosotros las miraba. Hice hincapié en sentarme entre Allan y Bill.
"Me siento tan oscura," murmuré. Bill sonrió y me pasó el brazo.
El color de mi piel era algo que yo notaba todo los días cuando me desnudaba para ducharme o bañarme. Cuando me probaba ropa nueva, el color de mi piel determinaba si podría usar ciertos verdes o amarillos. El negro me hacía ver más pálida, el blanco tenía el efecto contrario. El rosa subido me daba un rubor saludable, mientras que ciertos azules creaban sombras cenizas alrededor de mis ojos y labios. Era importante para mí saber estas cosas a la hora de escoger los vestuarios porque en Performing Arts nos habían enseñado que la selección de color que hacía un personaje decía mucho sobre él. Los principios que había aprendido allí, se colaban en la selección de mi ropa de diario. Prefería los colores tropicales, brillantes, pero evitaba los estampados llamativos. Más allá del color de mi ropa no había nada de su estilo que me hiciera sobresalir. Mis faldas nunca eran demasiado cortas, ni mis pantalones muy pegados, ni mis blusas demasiado escotadas. Así es que era el color de mi piel, pensaba, lo que llevaba a la gente en Lewiston, Bangor, y Portland, en New Hamphire, Massachusetts Rhode Island y Connecticut a quedárseme mirando. Dondequiera que parábamos a representar nuestra fábula india, yo era la persona de piel más oscura en el salón, en el _diner_ , en la escuela, en la tienda, en todo el pueblo.
"Debo ser la única puertorriqueña que ha estado jamás en Woonsocket," bromeé una vez, después de una visita particularmente tensa a un _diner_. Los otros se rieron pero nadie dijo nada más. El color de mi piel o mi origen puertorriqueño no eran tópicos de las conversaciones locuaces en la guagua VW o en las comidas. Así como los demás daban por sentada su blancura, yo tenía que hacer lo mismo con mi piel oscura. Sólo que a ellos no se les quedaban mirando como a mí.
Al principio me intimidaba tanta atención. Según fuimos avanzando en la gira, me volví desafiante, interpretaba las miradas como un reto y en los restaurantes me aseguraba de sentarme donde todo el mundo pudiera verme, una cara oscura entre las claras. Cuando eso no alteró el modo en que me sentía, decidí educar a la gente sobre Puerto Rico. Una mañana de fríos vientos en Salem, Massachuetts, me hacía recordar los tibios amaneceres del campo puertorriqueño. Caminando por la orilla del mar en Newport, Rhode Island, con mis compañeros actores, sentía el impulso de describir la Bahía de San Juan. Una porción de arroz _pilaf_ al lado de mi _meatloaf_ suscitaba recuerdos del arroz tierno y sueltecito de mi mamá. Aprovechaba cualquier oportunidad para hablar de Puerto Rico y de los puertorriqueños, aún cuando el tema de conversación no tuviera nada que ver ni con la etnicidad ni con la cultura. Las camareras, los guardias escolares, el portero de uno de los hoteles donde nos quedamos, el dependiente de la farmacia donde fui a comprar toallas sanitarias, el cajero de L. L. Bean — todos se enteraron de que yo era puertorriqueña, de que Puerto Rico estaba en el Caribe, de que los puertorriqueños somos ciudadanos americanos de nacimiento, de que hablamos español como primer idioma, de que el inglés es requisito en nuestras escuelas. Sí, había muchos puertorriqueños en Nueva York, pero también había muchos en otras ciudades como Chicago y Miami. Si lograba cambiar lo ignorante que eran sobre mí, tal vez podrían mirar al próximo puertorriqueño que pasara por allí con respeto en lugar de con sospecha.
Cuando terminó la gira y regresamos a Nueva York, me sentía como una mujer de mundo. Había atravesado el vasto horizonte de los Estados Unidos que no podía ver desde el suelo, pero el viaje me dejó cautelosa de aventurarme más lejos en el continente. ¿Cómo sería si, como habían planificado Vera y Bill, nos íbamos de gira por el sur? ¿Me podrían prohibir la entrada a los restaurantes? Yo sabía que había leyes que no lo permitían, gracias en parte a Martin Luther King, Jr., cuyo retrato colgaba en nuestra sala. Pero también sabía que las leyes no significaban nada para la gente que odia. Yo no era negra; yo no era blanca. Ese intermedio racial en el que existía hacía que la gente me evaluara en el acto. Sus ojos parpadeaban, mientras sus cerebros calibraban el nivel de pigmentación que estaban dispuestos a tolerar. ¿Es lo suficientemente clara para ser blanca? ¿Es tan oscura como para ser negra? En Nueva York yo era puertorriqueña, una identidad que cargaba en sí misma todo un cuadro de estereotipos negativos que yo batallaba por superar. En otros lugares donde había menos puertorriqueños, de dónde venía era lo de menos. Simplemente, era demasiado negra para ser blanca, demasiado blanca para ser negra.
Las semanas siguientes fueron un torbellino, tratando de ponerme al día. Había faltado al _college_ quince días y regresé a hacer montones de asignaciones que había perdido y a leer cientos de páginas para alcanzar a mis compañeros. En Advertising Checking Bureau, una intimidante estiba de recortes de periódicos esperaba encima de mi escritorio a que los examinara y los aprobara. La Sra. Davis sonreía mientras yo hojeaba rápidamente las páginas sin detenerme a ponerme al día de los últimos acontecimientos en Grand Rapids, Michigan, o Baraboo, Wisconsin.
Shanti llamó para ver si nos poníamos de acuerdo para hacer más sesiones de fotografía, pero no pude. Había dejado la escuela de fotografía para aceptar un trabajo en un laboratorio haciendo ampliaciones de las fotos de otra gente. "Ahora puedo revelar a color," me dijo orgulloso.
Hicimos unas cuantas presentaciones más de _Babu_ ; entonces, llorosos y tristones, nos despedimos por el verano y nos regamos en diferentes compañías de repertorio lejos de Nueva York. Bill y Vera nos prometieron trabajo para el otoño y una gira en el área de Washington, D.C. Pensaban añadir otra producción al programa, esta vez era una fábula japonesa con una parte para damita joven.
Planifiqué dedicarme a estudiar y a trabajar en el verano para poder tomarme libre el semestre para actuar en _Babu y_ —esperaba— que también en la nueva producción. Uno de los cursos que cogí, Historia del Arte, requería que visitara algún museo semanalmente. A veces arrastraba a alguna de mis hermanas para que fuera conmigo —generalmente Edna, y nos pasábamos la tarde del sábado o el domingo mirando obras que ninguna de las dos entendía. En casa, escribía un ensayo sobre la obra de arte asignada esa semana. Mis fines de semana se llenaron de estrés porque aunque me gustaba el arte, no podía explicarlo. De vez en cuando Mami y Tata me tocaban a la puerta del cuarto porque escuchaban quejidos. Era yo, frustrada por el reto de la pinturas ante las cuales reaccionaba emocionalmente, pero no encontraba nada que decir.
"Si el artista quería decir algo más que lo que está en el cuadro," discutía con la profesora, "debió haber sido escritor, no pintor." Ella insistía en que la pintura estaba llena de claves vitales y sutilezas que revelaban su significado, pero que cada detalle tenía que ser estudiado individualmente.
"Si te colocas frente a una pintura el tiempo suficiente," me aseguraba Miss Prince, "su sentido se te irá aclarando."
Un domingo por la tarde, mientras observaba cuidadosamente los puntos de Seurat, se me acercó una mujer. "Siento molestarla," sonrió dulcemente, "¿pero tiene usted idea de cómo llegar al restaurante desde aquí?" Tenía el pelo rubio peinado en "tisin" al estilo _bouffant_ que había hecho famoso Jacqueline Kennedy, ocho años antes.
Saqué de mi cartera el folleto con un mapa de las galerías del museo y le tracé la ruta que debía seguir hasta el primer piso. Mientras estábamos inclinadas sobre el mapa, se acercó un hombre. "Hola," sonrió. Obviamente, estaba emparentado con ella, tenía los mismos ojos alertas, el pelo color arena, la sonrisa simpática y el melifluo acento sureño. Ella me lo presentó como Avery Lee, su hermano, y se presentó a sí misma como Patsy. "Usted ha sido tan amaaable," dijo, estirando la palabra hasta que pareció interminable. "¿Nos acompaña a tomar un café?"
Bajamos la escaleras y ella me dijo que vivía en El Paso. "Pero me encanta venir a Nueva York," dijo, "a los museos, los restaurantes maravillosos... ¿Usted vive aquí?"
Para cuando llegamos al primer piso, Patsy ya me había sacado que vivía en Brooklyn, que era soltera, estudiante universitaria, bailarina y actriz. "¡Santo Dios!" exclamó efusivamente, "usted sí que tiene una vida muy interesante." Según llegamos a la fila de la cafetería, se acordó de que tenía que llamar a su esposo. Le indiqué dónde estaban los teléfonos y me quedé sola con Avery Lee, que nos había seguido atento y silencioso todo el tiempo que Patsy estuvo sacándome mi historia.
"¿No deberíamos pedirle algo?" sugerí, pero Avery Lee contestó que él no sabía lo que a ella le gustaba. Los minutos siguientes fueron medio incómodos mientras yo esperaba a que Patsy regresara.
"Tengo que ser honesto con usted," me confió Avery Lee. "Ella no va a regresar."
"¿Por qué no?"
"Porque así fue que lo planeamos."
Si no hubiésemos estado en un lugar público como la cafetería del Met, hubiera entrado en pánico. "¿Que quééé? ¿Qué quiere usted decir?"
"Usted llevaba mucho rato frente a esa pintura," me dijo. "Me paré al lado suyo pero usted siguió mirándola y mirándola."
"Estaba haciendo mi asignación," le admití.
"No la quise asustar, así es que le pedí a Patsy que me ayudara."
"¿Ustedes son familia?" le pregunté.
"Claro que sí. Ella es mi hermana." A pesar de su treta, había algo en Avery Lee, una franqueza que me gustaba, y su acento, tan lento y claro, me traían el recuerdo del adorable Sr. Grunwald. Avery Lee, sin embargo, no se parecía en nada a mi maestro de matemáticas. Físicamente, se parecía más a Otto —grande y musculoso, con labios finos y definidos y quijada cuadrada.
"Usted probablemente sepa qué espectáculos pueden verse en Nueva York, como usted es actriz." También estaba bajo la impresión de que yo podía recomendarle los mejores restaurantes. Tuve que admitirle que la vida cultural de la ciudad era algo sobre lo que yo leía, pero no participaba debido a su alto costo.
"Entonces, puede ser nuestra guía," sugirió. "Usted sabe a donde ir y estamos aquí para pasarla bien." Cuando yo vacilé, él insistió. "Anímese. Patsy tiene a su esposo y yo estoy aquí por mi cuenta. Usted puede ser mi acompañante," sonrió.
Le dije a Mami que había conseguido un trabajo como guía de un turista tejano. A la mañana siguiente aparecí en el edificio de apartamentos donde Avery Lee y Patsy se estaban quedando con unos amigos. El portero me anunció y al rato, Avery Lee bajó solo. Cuando pregunté por Patsy me contestó que tenía migraña. Visitamos el Empire State Building, el Museo de Arte Moderno, el Lincoln Center. Almorzamos en el Waldorf Astoria. Él quería cenar en el Plaza, pero yo no estaba vestida apropiadamente, así es que fuimos a Bloomingdales.
"Avery Lee, no me siento cómoda con que me compres ropa," protesté. "Voy hasta casa y me cambio." Pero de ninguna manera aceptó.
Escogí un vestido sencillo que estaba en especial, pero después tuvimos que comprar zapatos y una cartera en combinación. Me debatía entre el placer de compar lo que con mi propio dinero no podía, y la preocupación por el costo que yo sabía que tendría.
"Sé lo que estás pensando," me leyó la mente, "pero créeme, me encanta hacer esto por ti. Me encanta verte sonreír."
Así es que sonreí por todo el Departamento de Juniors donde Avery Lee me compró un ajuar completo para el día siguiente, cuando iríamos a ver _Man of La Mancha_ , y un bikini y una bata de playa que me pondría para la playa el día después.
Resultaba raro llegar a un restaurante de lujo con los paquetes, pero eso fue lo que hicimos. La luz de las velas, el vino que pidió para la cena, su hablar lento, todo era embriagante. Tuve que excusarme dos o tres veces para ir al baño de damas, donde apoyaba la cabeza contra las losetas frías hasta que me dejaba de dar vueltas y podía hablar sin arrastrar las palabras. Después de cenar, caminamos cogidos de la mano alrededor de la fuente frente al hotel. El último hombre que había besado había sido Otto, hacía ya año y medio. Avery Lee besaba con igual pasión y sus manos también se corrían, como las de Otto.
"Voy a pedir una habitación," ofreció Avery Lee y caminó hacia el Plaza.
Ahí se me quitó el mareo enseguidita. "No, más vale que me vaya a casa." Entrecerró los ojos como si las luces de la fuente no hubieran sido lo suficientemente brillantes para verme. Se viró, se metió las manos en los bolsillos, se alejó de mi dos o tres pasos y yo esperaba que en cualquier momento pateara el piso, bajara la cabeza y dijera, "¡Ay, contra!"
"¿A tu casa?" preguntó, como si fuera una palabra recién acuñada.
Balbuceé que el _subway_ se ponía peligroso si esperaba hasta más tarde. Mis paquetes estaban todavía en el hotel, pero yo no me atrevía a entrar con él de nuevo, no fuera a ser que flaqueara y accediera a subir con él. Esperé afuera en lo que los buscó. En mi mente oía la voz de Shoshana, "¡So idiota! ¡Es un millonario tejano!" Perder la virginidad en el Plaza hubiera sido el final perfecto para un día perfecto, pero no me atreví. Mientras Avery Lee me acompañaba hasta la estación del tren, abatido y callado, sentí que tenía que darle alguna explicación.
"Debes saber que yo nunca antes he estado tan cerca de entregarme a alguien," le confesé.
Sonrió, me besó en la frente y me entregó mis paquetes. "Te veo mañana," me dijo.
Durante los próximos tres días nos encontramos, comimos, vimos obras de teatro, caminamos por Central Park, nos besamos a la menor provocación. Llovió el día que se suponía que fuéramos a Jones Beach, así es que nos fuimos mejor al cine. "¿Eso es lo que le dicen _petting_ , verdad?" le pregunté después de una escena de besos y caricias particularmente caldeada.
"Sí," jadeó.
"¿Cuál es la diferencia entre _petting y necking?_ " le pregunté y él me hizo la demostración. Todas las noches trataba de que me fuera al hotel con él y todas las veces, me resistí. Una noche me acompañó todo el trayecto hasta Brooklyn. Nos besamos, nos hablamos, nos volvimos a besar.
"Vente conmigo a Texas," me ofreció cuando el tren se estaba acercando a mi estación. "Te consigo un apartamento, un carro, lo que tú necesites."
"¿Me estás pidiendo que sea tu amante?" le pregunté coqueta porque estaba segura de que estaba bromeando.
"¡Así mismo!" Sonrió, pero esta vez no sentí el encanto.
"Si vas a pasar todo ese trabajo, ¿por qué no te casas conmigo de una vez?"
"No se vería bien," confesó, "que yo tuviera una esposa española."
Quedé tan estupefacta que por poco se me pasa la parada. Las puertas traquetearon al abrirse y estaban a punto de cerrar cuando yo salté y salí demasiado rápido para que Avery Lee me alcanzara. El tren siguió y lo dejó todavía sentado en el banco de plástico, boquiabierto con mi agilidad.
Tengo que haber malinterpretado. Él no puede haber querido decir lo que dijo. Ni una sola vez en estos últimos días había sentido yo que la impresión de Avery Lee sobre mí estuviera matizada por el estereotipo de la Latina caliente. Yo era la virginal María de _West Side Story_ , pero él me veía como la promiscua Anita.
Caminé hasta casa por las oscuras calles de Brooklyn, entré, me puse la piyama y me acosté boca arriba a mirar la oscuridad. Estuvo mal aceptar la ropa que me compró, las cenas, el teatro, el paseo romántico por Central Park en el coche tirado por caballos. Besarlo y acariciarlo estaba bien, me explicó Shoshana, mientras no fuera sólo para incitarlo, sin llegar más lejos. Pero yo me sentía abochornada de lo cerca que había estado de enredarme con él en una cama.
Por la mañana temprano me llamó y me rogó que nos encontráramos. "Déjame explicarte," me dijo, y yo accedí a encontrarme con él en una cafetería cerca de mi trabajo.
"No sonó bien," gagueó tan pronto nos sentamos, "la manera en que lo dije."
"¿Puedes hacerlo sonar mejor?" Estaba decidida a hacerlo retorcerse, como lo había hecho yo toda la noche recordando sus besos y sintiéndome sucia y usada.
"¡Demonios!" exclamó, y se ruborizó cuando los clientes de la cafetería se viraron a mirarnos. Se inclinó hacia mí. "Mi papá ha tenido una mexicana veinte años," me dijo. "La quiere más que a su vida," añadió.
"¿Una qué mexicana?" le disparé.
"Estoy siendo honesto contigo," contestó resentido.
Me picaban los ojos y me estaba dando trabajo respirar. Debajo de la mesa las manos me temblaban con ganas de estrangularlo. Pero él era impenetrable a mis emociones. Suavemente giró mi cara hacia la suya. Murmurando me dijo que tenía ambiciones políticas, que tenía que casarse con una "niña bien de Texas," de familia prominente. Alguien que pudiera ayudarlo a salir electo. "Demonios" volvió a exclamar, "echándose para atrás de nuevo, "el mismo LBJ lo hizo. El matrimonio no significa nada."
Me levanté y recogí mis cosas. "Yo no quiero ser tu amante," le siseé. "Ahora mismo, ni siquiera quiero estar en el mismo sitio que tú."
"Siéntate," me ordenó Avery Lee. "Todo el mundo nos está mirando."
Me senté, vencida. Era demasiado tarde para hacer una salida dramática o para hacerme la santurrona. Avery Lee apuntó un número de teléfono en su tarjeta de presentación. "Esta es mi línea privaadaaa." ¡Ay, esas vocales tan largas! "Llámame cuando cambies de opinióoon." Me quedé mirando la tarjeta, su tonta cara esperanzada. Tenía ganas de escupirla. Se puso de pie y me ayudó a levantarme de mi asiento y caminó conmigo hasta el Advertising Checking Bureau que quedaba a medio bloque de distancia. En el ascensor trató de besarme, pero yo me retiré. Cuando se abrieron las puertas cogí aire, enderecé los hombros, me tragué el dolor que me apretaba la garganta y me hacía cosquillas en los ojos. Según fue subiendo el ascensor, absorbí el insulto de Avery Lee tan plenamente como absorbe la tinta el papel de periódico. Quizá era demasiado orgullosa y ambiciosa. Quizá los años en Performing Arts, el adiestramiento en baile exótico, el trabajo de cine, el espectáculo en Broadway, me habían hecho desarrollar una opinión de mi misma más elevada de la que merecía. Quizás Avery Lee vio lo que era realmente; una muchacha "española" lo suficientemente buena para acostarse con ella, pero no para casarse.
Esa noche saqué las fotografías que me había tomado Shanti y las estudié. En una, estaba sentada en la grama, mi cuerpo hacia la cámara, mi cara en perfil altivo, la mirada hacia el horizonte lejano. Tenía una pañoleta amarrada alrededor de la frente —la diadema de Cleopatra. Cuando Shanti tomó esa foto, me había hecho aguantar la pose mucho rato. "Quieta," me murmuró una y otra vez, hasta que tuve que dejar de respirar para complacerlo.
En otra foto aparecía reclinada hacia la pared de granito en la punta del Empire State Building. Detrás de mí, Brooklyn parecía flotar en una densa nube gris como un simulacro de ciudad, nada excepto unos pálidos rectángulos y unas manchas magulladas. Entre Brooklyn y yo, el East River era una plancha plana y helada. La foto fue tomada una fría mañana de mucho viento, justo cuando el sol atravesaba las nubes de modo que una parte de mí estaba sobrepuesta y la otra, medio oscura. Tenía una expresión desolada, como si acabara de escuchar una mala noticia.
Según fui hojeando el portafolio de mi natimuerta carrera de modelaje, no me vi a mí misma. Vi a la muchacha española de Avery Lee, seria pero triste, con ojos cautelosos y en cada retrato, sola, los bordes de la foto encasillando soledad.
Unos días depués, en la Quinta Avenida, mientras me inclinaba hacia la guagua que se acercaba, un hombre me tocó el hombro y me preguntó cómo llegar a Rockefeller Center. Me viré y me encontré con los ojos verdemar de Jurgen que no estaba, en realidad, perdido sino cautivado por mis pupilas color marrón.
"Yo sé donde queda," me admitió. "Hay un restaurante allí. ¿Me acompaña a tomar un té?"
Seguí a Jurgen hasta el restaurante en la planta baja de Rockefeller Center que en el invierno daba a una pista de patinaje pero en el verano tenía unas mesas cubiertas con sombrillas brillantes.
Jurgen hablaba un excelente inglés con un acento encantador. Cuando cometía algún error gramatical o de pronunciación se daba en los labios con el dedo índice y el del corazón, como si la culpa fuera de los labios y no del cerebro. Había nacido en Hamburgo, pero no vivía allí. "¿Dónde, entonces?" le pregunté.
"Por todas partes," rió.
Su piel era translúcida, la superficie suave y uniforme. Los labios con frecuencia se entreabrían en una sonrisa pícara que dejaba ver unos dientes pequeños y chatos como si le hubieran limado los bordes.
Como mucha gente en Nueva York, Jurgen estaba de paso. Me invitaría a un té en Rockefeller Center y de ahí regresaría a Alemania o a donde quiera que fuera su próxima parada.
"Los Ángeles," dijo. "Después Egipto."
"¡Qué divino!" suspiré, y él se rió.
Seguimos hablando y yo caí en las mismas respuestas ambiguas de siempre a las preguntas típicas. Pero Jurgen escuchaba con cuidado, pedía detalles que nadie más se molestaba en pedir. Antes de que el mozo volviera a llenarme el vaso de té, ya le había contado a Jurgen todo lo que había que saber de mí, incluyendo que era virgen, que no me dejaban salir con muchachos hasta que me casara y que recientemente, me habían ofrecido una posición como amante de un millonario tejano demasiado ambicioso para casarse con una "muchacha española." Me escuchó, rió, me miró con dulzura y preocupación. Cuando se me salieron las lágrimas, sacó un pañuelo del bolsillo y me secó los cachetes. Cuando me estaba secando las lágrimas, me dio vergüenza haberle dicho tanto y me excusé un momentito, con la intención de salir por la otra puerta y caer en el _subway_. Pero, primero tenía que ir al baño, lavarme la cara, peinarme, ponerme _lip gloss_. Cuando salí, Jurgen estaba en el pasillo que daba a los baños. "Pensé que te habías perdido," me dijo. Me guió hasta la calle y caminamos por la Quinta Avenida hacia Central Park. En la Calle 59 me cogió la mano y para cuando llegamos frente al Plaza en la 59 con Quinta, su brazo rodeaba mis hombros y el mío, su cintura. Paseando por Central Park le conté que el último alemán que me había cogido la mano me había salvado de morir aplastada por un camión. Él bromeó diciendo que los alemanes eran muy oportunos.
Le pregunté dónde había nacido y me contó de su niñez en Hamburgo. Su mamá y su papá todavía vivían allí, me dijo, pero hacía algunos años que no los veía. Me preguntó si extrañaba a mi papá y por poco rompo a llorar de nuevo.
"Tú debes pensar que soy una llorona," me disculpé.
"No," me acarició el pelo, "es tierno." Era fácil estar con Jurgen, hablarle sobre asuntos que nunca había compartido con nadie, excepto con Shoshana. De vez en cuando, me acordaba que lo acababa de conocer y me preguntaba qué sería lo que tenía que me hacía sentir como si lo hubiera conocido de años.
Caminamos hacia un restaurante frente a Lincoln Center y Jurgen me presentó a Donny, el _bartender_ , con quien se estaba quedando en su apartamento.
"¿De dónde eres?" le pregunté cuando oí su acento.
"De Irlanda," rió. Tenía el pelo negro y los ojos azules, era más bajito que Jurgen, llenito, un poco mayor aunque decía tener la misma edad, veintinueve. Él y Jurgen intercambiaron algunas palabras en alemán. Me di cuenta de que Donny había dicho algo de mí por la mirada cariñosa, orgullosa que me dio Jurgen.
"¿Dónde aprendiste alemán?" le pregunté a Donny. Los hombres se miraron.
"Éste habla un alemán terrible," rió Jurgen. "Como un niño de primaria." Donny se sonrojó. Charlamos otro ratito y entonces, Donny nos invitó a venir con él y la novia a Jones Beach al día siguiente. Cuando titubeé, Jurgen se ofreció a llamar a mi mamá y a sacarme permiso.
"No, está bien," le dije, segura de que se estaba burlando de mí.
Jurgen dijo que tenía un compromiso y me pidió que lo acompañara hasta casa de Donny en lo que se cambiaba.
"No puedo," le dije, "tengo que irme a casa."
"No me va a tomar mucho rato," insistió Jurgen. "Es aquí cerquita." Desde su puesto detrás del bar, Donny me animó a que fuera. "No te preocupes, él es un hombre decente. No te va a hacer nada. Te doy mi palabra."
"Te acompaño hasta la esquina," le ofrecí. Cuando llegamos allí, Jurgen me cogió la mano y me fue llevando por la acera. "De verdad me tengo que ir," protesté. "Mi mamá se va a preocupar."
"Me va a tomar sólo un minuto ponerme el traje," me dijo Jurgen.
El apartamento quedaba a dos bloques, en un edificio de ladrillos amarillos que no tenía portero, pero sí, dos puertas de seguridad. Adentro, el pasillo ancho estaba oscuro y fresco, las paredes y los pisos cubiertos con unas losetas color mostaza que le hacían eco al tac tac de nuestros pasos camino al ascensor. Subimos al quinto piso, uno al lado del otro. El corazón se me quería salir según iba revisando en mi mente cada patada y cada golpe que mi primo Paco, el luchador, nos había enseñado a mí y a mis hermanas por si acaso alguna vez teníamos que defendernos.
El apartamento quedaba al final de un pasillo largo que tenía una ventana que daba hacia el Hudson. Adentro había dos habitaciones muy recogidas, con pocos muebles y una cocina como las que tienen los aviones, sin platos, ollas o comida a la vista.
Tan pronto entramos, Jurgen trató de besarme. Me resistí pero después pensé que si me dejaba besar a lo mejor él se tranquilizaba y yo podía escapar. Fue delicado, no se me pegó, ni me puso las manos donde no debía. Dio un paso atrás, me cogió la mano y la besó reverentemente. "Estamos hechos el uno para el otro," dijo.
"¿Ah?"
Me miró a los ojos. "Cásate conmigo."
"¿Perdón?"
"Cásate _conmigo_ ," y se dio con la mano en el pecho como si hubiera etado diciendo, "Yo Tarzán, tú Jane."
"¿Tú tienes que estar bromeando?"
"Hablo en serio."
No pude controlar la pavera que me dio. Estaba delante de mí, con mi mano en la suya y la sonrisa pícara en los labios. Se me ocurrió entonces que podía ser un psicópata y que no era muy buena idea reírse de él.
"¿Sí?" apuntó.
"Nos conocimos hace tres horas," le recordé.
"¿Sí?"
Quería salir viva de allí. "Está bien, vamos a casarnos."
"Magnífico." Me abrazó, me besó los ojos, la frente. "Mi esposa." Ahora, pensé, cuando trate de llevarme al cuarto le doy una patá' y salgo corriendo. Jurgen me soltó y se echó para atrás. "Me visto ahora," dijo. "Espera un momento, por favor." Arrastró una silla desde donde estaba contra la pared y la sostuvo en lo que me senté. No muy buena idea, pensé. Me va a amarrar a ella. "Perdóname un momento," dijo Jurgen y fue hasta el cuarto. Me quedé sentada en la orilla de la silla, a menos de diez pies de la puerta entreabierta, calculando el mejor momento para escapar. Salía y entraba de mi campo de visión según se iba cambiando la camisa, poniéndose la corbata, la chaqueta. Cada vez que yo estaba a punto de brincar y salir corriendo, él se viraba y me sonreía. Se peinó y se paró en la puerta del pasillo. "Vamos a decírselo a Donny," me dijo.
Brinqué de la silla al pasillo, confundida, pero contenta de que pronto estaríamos afuera y yo podría salir corriendo. Cuando estábamos bajando, Jurgen habló del tiempo que llevaba esperando a la chica ideal y lo afortunado que era de haberme encontrado. Declaró haberse enamorado de mí mientras yo estaba parada en la esquina de la Quinta, cerca de la 48. "Yo no soy hombre impulsivo," insistía, "pero yo sigo, ¿cómo dice? instinto."
Nos casaríamos en los Estados Unidos, sugirió Jurgen, viajaríamos a Alemania a conocer a su familia y después nos iríamos a vivir a Egipto. Era la conversación más surrealista que había tenido con alguien que no viviera dentro de mi cabeza. Cada fantasía del príncipe azul que había imaginado se estaba haciendo realidad. Como si de verdad existiera el amor a primera vista, el romance, hombres inteligentes y encantadores con dinero, dispuestos a gastarlo en mí, hasta a casarse conmigo. "¿Tú sabes bailar?" pregunté, segura de que había alguna falla en esta trama demasiado perfecta. Para probar que sabía, Jurgen me "tangueó" por la puerta de entrada al restaurante donde estaba todavía Donny detrás del bar sirviéndole a unos cansados hombres de negocio.
"¡Casarse!" Donny alzó las cejas tan alto que se le desaparecieron dentro del pelo negro. Cuando se recuperó, felicitó a Jurgen. "Te dije que era un hombre decente," guiñó un ojo. "Ahora tengo que proponerle matrimonio a Laryssa," dijo haciendo una mueca, y nos reímos.
Jurgen tenía que irse a su reunión y me pidió que lo esperara allí con Donny hasta que regresara y entonces celebraríamos nuestro compromiso con una cena y champán.
"Jurgen," empecé, a punto de decirle que el juego ya había ido demasiado lejos, que yo no quería casarme con él —ni con nadie—a quien sólo conocía hacía, déjame ver, cuatro horas. Lo que me salió fue, "Tengo que llamar a mi mamá."
"¿Hablo con ella?" me ofreció Jurgen con ojos serios. Fue entonces que supe que su petición no era juego.
Jurgen se quedó delante de mí esperando una respuesta. Tenía los ojos verdes de Naftalí y su voz callada, la altura, el colorido y el acento de Otto, el físico perfecto de Sr. Grunwald. Hasta de Avery Lee tenía algo —su misma sonrisa pícara y aire resuelto. Me tomó sólo un segundo transformar a Jurgen en la personificación de todos los hombres que había amado. Me rendí.
Nos paramos junto al teléfono en la parte de atrás del restaurante, componiendo el cuento que le íbamos a hacer a Mami. Nos conocíamos hacía un año, nos presentó la mismísima Sandy Dennis en el set de _Up the Down Staircase_ , nos habíamos vuelto a encontrar y decidimos que no podíamos vivir el uno sin el otro. "Te va a querer conocer," le advertí y se ofreció a recogerme en casa el domingo cuando fuéramos a la playa con Donny y Laryssa.
Cuando llamé a Mami para decirle que me había comprometido tuvo sus dudas, hizo las preguntas esperadas y escuchó cuidadosamente mis respuestas. Jurgen se puso al teléfono y le dijo, " _I_ mucho _love your daughter. Very_ mucho." Cuando me devolvió el teléfono, ella aceptó que sonaba simpático.
"¿Lo vas a traer a casa ahora?"
"No, Mami, mañana. Vamos a ir a la playa. Él me va a ir a recoger, y entonces lo conoces."
Tan pronto colgué, me arrepentí de haber llamado. Antes de llamar todavía estaba a tiempo de haber cambiar de opinión, de decirle adiós a Jurgen, de haberle dado un número de teléfono falso, de haberme mantenido lejos del centro un par de días en lo que Jurgen volaba y se iba. Jurgen se dio cuenta de mi estado de ánimo.
"Ven conmigo," me dijo. Me pareció raro que me llevara a una cita de negocios, pero en este punto todo era tan irreal que nada me sorprendía. Cogimos un taxi hasta un negocio de carros de lujo que había en la Décima Avenida. Carros deportivos y sedánes brillaban detrás de una enorme plancha de cristal, algunos de ellos con las puertas abiertas para mostrar los interiores. Cuando entramos, un hombre altísimo y de lo más emperifollado se nos acercó y a mí se me hacía difícil imaginarlo sentado al volante del Porsche que nos mostró. Por su conversación estaba claro que a Jurgen le interesaba un Porsche como el azul que estaba en la vitrina. Quería probarlo y después de presentar a "su prometida," Jurgen sacó el carro para darle una vuelta por las congestionadas calles de Manhattan, por donde no podía correr más rápido que un Ford. El hombre emperifollado estaba esperándonos en la puerta del concesionario cuando regresamos, y se pasó dándole coba a Jurgen que le habló de caballos de fuerza y de torsión mientras yo me preguntaba cómo era que nueve horas antes yo había salido de Brooklyn sufriendo todavía el rechazo de Avery Lee, y ahora estaba sentada en las oficinas de un concesionario de Porsche con mi futuro esposo.
Cenamos temprano y después Jurgen me llevó a ver la producción negra de _Hello Dolly_ con Pearl Bailey. Después del teatro quería acompañarme a casa, pero yo le convencí de que no era necesario. Caminó conmigo hasta la estación del tren, me dio el teléfono de Donny e insistió en que lo llamara tan pronto llegara a casa para estar seguro de que había llegado bien.
En el tren, camino a casa, me maravillé de lo extraño que había sido el día. De acuerdo a los periódicos, la mitad de mi generación supuestamente estaba con una _nota_ por LSD o alguna otra droga alucinógena. Yo no había tomado nada más fuerte que café y un par de copas de vino en la cena, pero sentía que estaba "tripeando." En cualquier momento, despertaría en mi cama, en el cuarto de atrás de nuestra casa de Glenmore Street, en el East New York de Brooklyn y el día entero sería sólo un sueño. O quizás había muerto y este era el paraíso. O quizás era el infierno y mi castigo, por no ser religiosa, era pasarme la eternidad divertiéndome por la tarde como la prometida de un hombre guapo y rico que podía comprar Porsches y teniendo que regresar por las noches a casa en Brooklyn. Tan pronto entré por la puerta llamé a Jurgen para asegurarme de que existía y que la tarde entera no había sido una fantasía extendida. Se oía aliviado de que lo hubiera llamado, me dijo que me quería, y me pidió que le dijera cómo llegar a casa al otro día. Fabriqué una historia sobre Jurgen y yo para Mami, Tata, Don Carlos, Don Julio y la hermanas y los hermanos que se habían quedado despiertos esperándome. Cuando me acosté finalmente, me creía cada palabra de la mentira que les había inventado. Jurgen y yo estábamos enamorados, nos casaríamos, viajaríamos a Alemania y después a Egipto, donde viviríamos felices para siempre a la sombra de las pirámides.
A la hora en punto en que había quedado en recogerme, el acelerador de un carro deportivo atrajo a mis hermanos al patio de cemento que dividía nuestra casa de la acera. La familia llevaba horas levantada, limpiando y recogiendo para la inminente llegada de mi prometido. Llegó en un Porsche negro, no el mismo que habíamos visto en el _showroom_ ni el que había probado. Tenía miedo que no lo fuera a reconocer cuando lo volviera a ver pero no había cómo confundir su tez clara y su sonrisa traviesa. Donny estaba en el asiento del pasajero. Los dos me besaron los dos cachetes y yo se los presenté a mi familia. No había vuelto a traer a ningún hombre a casa desde Otto, así es que velé a Mami para ver su reacción. Se la ganó la galantería de Jurgen, su suave encanto, el ramo de flores que me dio a mí, las cerezas cubiertas de chocolate que le trajo a ella, y que lo congraciaron enseguida con mis hermanas y hermanos. La única con la cara fruncida era Tata que se había visto obligada a cambiarse su cómoda bata de algodón de estar en la casa por el vestido de encaje negro que se había puesto para recibirnos en el aeropuerto el día que llegamos hacía siete años.
Al entrar a la casa, cada superficie relucía y olía a Pine Sol, a Pledge, o a Windex. Jurgen y Donny se sentaron en la orilla del sofá cubierto de plástico que se había comprado recientemente, frente al nuevo televisor de consola. Con ayuda de los que teníamos edad para trabajar, Mami había logrado decorar su casa a su gusto con muebles nuevos, cortinas bonitas, un mantel de encaje para la mesa de comedor más grande que encontró, con sillas de espaldar alto sellados en plástico.
Jurgen y Mami hablaron a través de mí o de mis hermanas y hermanos.Ella le hizo las mismas preguntas que me había hecho a mí el día antes cuando hablamos por teléfono y un par más, basada en la información que yo me había inventado la noche antes. Jurgen estaba tranquilo y relajado y se hacía el que su inglés era peor de lo que realmente era cuando no sabía qué le había dicho yo a Mami. Ella estaba a la vez confundida y encantada con él, pero cuando Jurgen formalmente le pidió mi mano, se la concedió con una sonrisa.
Jurgen nos informó que debido a su itinerario de viaje, la boda tendría que ser en menos de un mes. Yo casi me caigo de la butaca, pero Mami se quedó como si nada ante el reto. "Tenemos que ordenar tu vestido mañana," me dijo.
A pesar de que Mami y Tata habían preparado comida, yo quería salir de allí antes de que Mami descubriera la verdad. Les recordé a Jurgen y a Donny que teníamos que recoger a Laryssa. El Porsche había atraído a los vecinos curiosos a la acera y a las ventanas. Salimos de la casa seguidos por mi mamá y mis hermanas y hermanos, y yo no podía ocultar el orgullo que sentía. Si hubiera sido uno de los vecinos hubiera estado celosa de mí cuando me monté en el asiento, al lado de mi novio tan guapo. Donny apretujó su cuerpo rechoncho en el asientito de atrás y en un par de segundos Jurgen había acelerado los muchos caballos de su carroza y alzamos vuelo desparramando polvo y basura en las aceras de "East New York."
Era imposible mantener una conversación en el Porsche. Era un carro ruidoso, especialmente con la capota baja. Jurgen subió el volumen del radio. Diana Ross gemía que no nacería nunca un niño de su amor mientras Jurgen volaba de Brooklyn a Long Island. El pelo me golpeaba los cachetes, los ojos; me hundí más en mi asiento pero no me ayudó. Cada vez que me movía, se me metía el pelo en la boca.
La casa de Laryssa estaba en el medio de un bosque de pinos en una comunidad de casas que eran copias unas de otras, excepto por el arreglo paisajista y el color. Cuando entramos a la casa, un gato se escurrió debajo de una butaca de posiciones LA-Z-BOY que estaba frente a un televisor de consola mucho más grande que el nuestro. Laryssa nos recibió vestida con un _top_ amarillo y unos _shorts_ turquesa y el pelo rubio recogido en un rabo de caballo largo. Nos ofreció té frío y sándwiches de atún en una cocina soleada que tenía una puerta de cristal que daba a un patio donde se veía una piscina. Había dos personas reposando al lado de la piscina que no vinieron a saludar ni Laryssa nos llevó afuera para conocerlos. Nos dejó en la cocina y se fue a cambiar. Una muchacha salió de uno de los cuartos, en rolos, su cuerpo esbelto y bronceando, vestida con una _babydoll_ transparente y unos pantaloncitos de volantes, Donny y Jurgen se miraron.
"Hola," ronroneó. "Soy Jen, la hermana de Laryssa." Los hombres se pusieron de pie para saludarla y después se acordaron y me presentaron a mí. Jen se sirvió un vaso de té, se excusó y se fue por el pasillo por el que había desaparecido Laryssa. Unos segundos después oíamos gritar que Laryssa le debió haber dicho a Jen que tenía visita y a Laryssa contestando que Jen no debía andar por ahí medio desnuda. "Y encima de eso, en pleno día," chillaba Laryssa. Los hombres y yo masticábamos nuestros sándwiches y sorbíamos nuestro té helado en silencio, con la mirada fija en la puerta cerrada. Las dos hermanas siguieron la pelea que terminó cuando Laryssa salió del cuarto con su bolsa de playa colgada del hombro. "Vámonos," dijo. Antes de salir gritó a través de la puerta. "¡Nos vemos más tarde, Mom, Dad!" Las dos figuras al lado de la piscina saludaron con la mano sin virarse.
Laryssa y Donny se fueron en su VW Beetle, Jurgen y yo los seguimos y sobre el motor del Porsche y del viento, Jurgen comentó, "Eso es lo que a mí no me gusta de las muchachas americanas." No elaboró, así es que me dejó pensando si se refería a la semi-desnudez de Jen, a la discusión entre las dos hermanas o a los padres que no les importaba con quiénes salían las hijas mientras no los molestaran a ellos. En la playa dimos vueltas por el estacionamiento un par de veces, le pasamos por el lado a unos cuantos sitios buenos hasta que Jurgen se alineó al lado de un Jaguar donde había un muchacho profundamente dormido en el asiento del conductor. Jurgen se bajó del Porsche y golpeó con el puño la naríz del Jaguar. El muchacho se despertó asustado, sus ojos negros redondos de pánico. Cuando vio a Jurgen, salió volando del carro y lo abrazó con cariño y los dos conversaron en alemán hasta que Jurgen se acordó de que yo estaba allí.
El amigo se llamaba Felipe, pero todo el mundo le decía Flip. "¿Eres español?" le pregunté.
"Mexicano," me dijo. Tenía el pelo negro, lacio, y los ojos achinados, la piel más marrón que la mía, un cuerpo musculoso con las piernas ligeramente arqueadas. Tenía puestas chancletas de goma y caminaba moviéndose de lado a lado afirmando con el lado de afuera del pie. Por lo que yo podía ver Jurgen, Donny y Flip habían quedado en encontrarse en Jones Beach. Flip había guiado el Jaguar desde California, lo que explicaba por qué había queda'o como muerto en una toalla de playa debajo de la sombrilla de Laryssa y durmió el resto del día. Jurgen inspeccionó el carro con el mismo cuidado con que lo hizo con el Porsche el día anterior. Levantó el bonete, miró el motor, abrió y cerró las puertas, examinó la carrocería, abrió el baúl y lo inspeccionó.
"Es un buen carro," dijo finalmente y le dio la mano a Flip.
Yo tenía puesto el bikini que me había comprado Avery Lee, amarillo con cuadritos blancos, ni remotamente tan diminuto como el de Laryssa. Los hombres tenían también unos bikinis pequeñitos, lo que me hacía sentir a mí sobrevestida, a pesar de que era la primera vez que mostraba mi abdomen en público. Como no sabía nadar, me quedé sentada en la orilla mientras Donny, Laryssa y Jurgen nadaban elegantemente entre las olas y regresaban flotando. Dentro y fuera del agua, Laryssa y Donny se pasaron todo el tiempo sobeteándose. Jurgen me besó un par de veces, pero a mí me daba pachó hacerlo en público y medio desnuda, así es que él desistió.
Al final del día, Laryssa regresó a su casa en su carro. Nosotros llegamos a mi casa, Jurgen y yo en el Porsche y Donny y Flip en el Jaguar. Mami salió a las escaleras del frente a recibirnos y Flip le gritó a Jurgen. "Prepárate Jurgen, dicen que las hijas terminan pareciéndose a las madres." Los tres hombres se rieron y Mami, que entendió, enfurruñó la cara en su mueca más fea. Yo fulminé a Flip con la mirada y él encogió los hombros avergonzado.
"Siempre hace chistes," dijo Jurgen. Caminó conmigo hasta la entrada. Habíamos quedado en vernos en la ciudad al día siguiente y lo confirmó delante de Mami. Regresó al automóvil, se despidió con la mano desde el asiento del conductor y se fue seguido de Flip y Donny que se reían a carcajadas de algo que alguno de ellos dijo en alemán y que yo no entendí.
"¿Por qué no lo invitaste a entrar?" me preguntó Mami cuando lo vio irse.
"Estoy llena de arena, tengo picor y estoy cansada," me quejé. "Necesito un baño."
Me molestaba que Flip hubiera dicho algo tan hiriente frente a Jurgen y que él lo hubiera tolerado. ¿Y quién era ese tal Flip? Decía ser mexicano, hablaba el alemán de corrido, pero, más allá de su nombre, no me dijo una palabra en español. Cruzó el país guiando un Jaguar, llegó esa misma mañana a Jones Beach —de todos los sitios— y cuando Jurgen abrió el baúl no había nada. ¿Dónde estaba su equipaje? No tenía ni una muda de ropa.
¿Y de dónde sacó Jurgen el Porsche? No lo tenía el día anterior. ¿Por qué quería ir a comprar otro si ya tenía uno? El Porsche podría ser de Donny. ¿Los _bartenders_ hacían tantos chavos como para tener carros deportivos último modelo? Sentía confusión y desconfianza, estaba segura de que algo raro estaba pasando pero no tenía idea de qué.
Al día siguiente, Jurgen y yo paseamos por Central Park camino al Lago. Él tenía en mente alquilar un bote y llevarme a dar un paseo. Le recordé que no sabía nadar, le tenía miedo al agua y me aterraba la idea de estar alejada más de dos o tres pulgadas de tierra firme. Pero él fue inflexible. Teníamos que hacerlo, me dijo, porque remar era uno de sus deportes favoritos. Estaba en un equipo, dijo y cómo íbamos a ser marido y mujer yo tenía que aprender a disfrutar de las cosas que él disfrutaba.
"¿Puedo usar un salvavidas?"
"No necesitas ninguno," rió. "Soy un excelente nadador."
"¡Pero yo no!"
Me dijo que le preocupaba que yo no le confiara mi vida. "Yo te salvaré, lo prometo."
Una vez nos metimos, el bote se vio más grande. Jurgen me puso su chaqueta encima de la falda, se subió las mangas de la camisa y dejó el muelle.
"Relájate," rió. "Suelta los bordes."
Remó hasta el centro del lago, aseguró los remos y se echó para atrás con un suspiro de satisfacción. Se veía muy cómodo rodeado de agua; su pelo relucía como oro bajo los rayos del sol; sus ojos se volvieron un profundo azul gris, como un océano sin fondo. Me agarré de los lados del bote. "¿Nos podemos ir ya?"
"Todavía no," me dijo. La gravedad de su voz, el modo en que su cuerpo se tensó, me dio escalofríos. "Tú has sido clara conmigo," murmuró. "Me has presentado a tu familia. Son personas honestas. Tu mamá es una buena mujer. Pero no me has preguntado sobre mí."
"No digas nada más." Me cubrí la cara con las manos. El cuento de hadas estaba a punto de terminar. Ahora, pensé, me va a confesar que él en realidad es un mozo en el restaurante donde trabaja Donny y que tiene una esposa y cinco hijos en Hamburgo. "No necesito saber." Flotamos en silencio un rato, yo con las manos en la cara todavía. Sentí sus ojos sobre mí y cuando los miré había una expresión de preocupación en ellos. Tantas preguntas se agolpaban en mi mente que no sabía por dónde empezar. "¿Estás casado?" pregunté finalmente. Se rió con tal fuerza que se meció el bote. Entonces notó que yo estaba seria.
"No, _liebchen_ , no estoy casado," dijo suavemente.
Hice la pantomima de tomar una libreta en la mano, una pluma; me ajusté unos espejuelos invisibles. Apreté los labios y fingí una voz chillona. "Muy bien, señor. ¿Tiene usted hijos?"
Él me siguió la corriente. "No, señora."
"¿Es usted realmente de Hamburgo?"
"Sí señora, lo soy."
"¿Cuál es su fecha de nacimiento, señor?"
Estuvimos bromeando un rato y entonces Jurgen cogió los remos y me regresó a tierra firme. Caminando por un sendero sombreado le hice otra pregunta. "¿Qué es lo que haces? De trabajo, quiero decir."
Él se detuvo, se volvió hacia mi, buscó mis ojos. "¿De verdad quieres saber?"
El modo en que me lo preguntó me hizo desear no saberlo. Asentí con la cabeza.
"Vuelo aviones," dijo, y empezó a caminar de nuevo.
Supuse que quería decir que era piloto, pero lo negó con la cabeza. ¿Piloteaba para una compañia de carga? No. ¿Para la Fuerza Aérea? Tampoco. Me di por vencida.
"Robo aviones," declaró.
Solté una carcajada. Él sonrió vagamente.
"¿Qué haces con los aviones robados?" seguí riéndome sin poderme contener. "¿Los escondes en tu garaje?"
"Los vendo."
Según Jurgen era fácil robarse un avión. "Los pequeños, no los Jumbo Jet," aclaró. Se ponía un uniforme de piloto, entraba al hangar, escogía un avión, lo volaba hasta México y lo vendía.
"Entonces, tú robas aviones y los vendes en México," dije con una risita tonta.
"O en otros lugares, depende de quien me lo encargue."
"Tú has visto demasiadas películas del Agente 007."
Jurgen sonrió. Estaba segura de que me estaba tomando el pelo. "¿Y Flip y Donny están en eso también?" le seguí el juego.
"No. Ellos prefieren los carros." Jurgen pasó a decirme que desde niño había soñado con volar y que había aprendido a hacerlo siendo un joven. Cuando se llevaba los aviones, volaba bajito para evitar los radares, me dijo. A veces volando sobre el mar, veía enormes escuelas de peces, ballenas, delfines. Había volado por todo el mundo y me describió corrientes de aire peligrosas alrededor de las montañas, bolsillos de aire repentinos, tormentas eléctricas dentro de las que había caído, y que le habían hecho pensar que lo harían caer para siempre. Sonaba como si hubiera estado narrando un sueño, pero según hablaba, mi escepticismo fue cediendo hasta que caí en cuenta que estaba a punto de casarme con un hombre que robaba aviones para vivir.
"No lo puedo creer," gemí. Caminé hasta un banco que estaba cerca y me senté porque las rodillas ya no me sostenían. Jurgen me pasó el brazo por los hombros.
"No te preocupes," me murmuró en el pelo.
"¿Qué no me preocupe? Jurgen, ahora que yo se esto, soy una criminal también. Se supone que yo vaya a la policía o algo..."
Me acarició el cachete, me juró que tenía que ser honesto conmigo, que no sería justo que no lo fuera. De todos modos, una vez nos casáramos no me podían obligar a testificar en contra de él si la cosa llegaba a ese punto. Quería preguntarle por qué no había esperado para decírmelo hasta después de la boda, pero me di cuenta que ese no era el problema.
"Voy para Los Ángeles," me dijo. "Cuando regrese me caso contigo si todavía me quieres." Era sincero. Lo oía en su voz. De alguna manera torcida su confesión era un acto responsable. Sin embargo, una parte de mí todavía se preguntaba si habría fabricado esta historia para espantarme porque el asunto del matrimonio había ido demasiado lejos y no quería lastimar mis sentimientos.
Me escoltó hasta la estación de tren. Me había estado enseñando unas palabras en alemán y me las fue practicando según caminábamos. Fingimos que nuestros planes de casarnos no habían cambiado después de su revelación. En el tren de regreso a Brooklyn, decidí que Jurgen quería probarme. No tenía mucho sentido que si de verdad robaba aviones, se lo dijera a alguien que apenas conocía. Pero bueno, éste era el mismo hombre que me había propuesto matrimonio tres horas después de conocerme. "No soy hombre impulsivo," me había dicho un par de días antes. Hice una nota mental de buscar la palabra en el diccionario para ver si había algún significado que se me hubiera escapado la primera vez.
¿Debía llamar a la policía? La historia era increíble. No tenía pruebas. Imaginaba la escena: una muchacha puertorriqueña entra al cuartel de la policía, alega que un alemán que conoció en la calle, y con quien aceptó casarse tres horas después, le confesó que roba aviones y sus amigos roban carros de lujo. Podía oír las carcajadas.
Jurgen partió para Los Ángeles. No me dejó un teléfono donde conseguirlo, pero me prometió llamarme todos los días. No le creí, así es que fue una sorpresa cuando llegó un ramo de rosas y esa noche, cuando el teléfono sonó era Jurgen. "Hola, _liebchen_ ," susurró. "¿Todavía quieres casarte conmigo?"
# "Tu cara ya no es inocente."
#
Me pasé la semana siguiente buscando mi vestido de novia con Mami. Las llamadas diarias de Jurgen fueron agotando mi resistencia y convenciéndome de que estábamos hechos la una para el otro. Me juró que no robaría más aviones. Había estado considerando una oferta de trabajo que tenía en Egipto para pilotear el avión privado de un príncipe árabe y había decidido aceptarla. Según Jurgen, su vida se había transformado gracias a mí. La mía estaba a punto de ser tranformada por él. Era un intercambio justo. Yo lo salvaría de la vida en la cárcel. Él me salvaría de la vida en Brooklyn.
Mami y yo escogimos un conjunto de vestido y chaqueta en seda de moiré color champán para el día en que fuera a conocer al papá y a la mamá de Jurgen. Tal y como Mami lo había planificado desde hacía tanto tiempo, mis hermanas serían las damas y mis hermanos, los ujieres. Franky, que tenía cinco años, portaría los anillos; Donny, el amigo de Jurgen, sería el padrino. Escogí a La Muda de madrina. Papi vendría desde Puerto Rico a entregarme. Mami encontró a un cura que nos casaría, aunque nunca había estado en su iglesia. Jurgen se mantenía al tanto de todo a través de sus llamadas diarias. Me preocupaba el costo de la boda, sobre todo, porque habíamos dado tantos depósitos que no eran reembolsables. Pero de acuerdo a los libros de etiqueta que había consultado, la familia de la novia corría con los gastos.
Cuando no andaba comprando mi ajuar, andaba repitiendo frases de los discos del curso de Berlitz "Enséñese Alemán" que había encontrado en la biblioteca. En la pared tenía un mapa de Egipto con un círculo rojo bien grande alrededor de Alejandría, donde me había dicho Jurgen que íbamos a vivir, y no en el Cairo, como yo había pensado. Yo no creía en el Karma, la astrología, la lectura de mano, el análisis de la escritura, la reencarnación, la percepción extrasensorial, la proyección astral, la meditación trascendental, en Nostradamus, en los Chariots of the Gods, ni en ningún otro embeleco con el que toda persona joven en los Estados Unidos estaba obsesionada en 1968. Pero, ¿cómo explicar el hecho de que yo, que me había pasado tres años perfeccionando el papel de Cleopatra, estaba a punto de mudarme al lugar de su nacimiento y de su temprana muerte?
A pesar de que mi matrimonio con Jurgen parecía predestinado, la duda me atormentaba. No parecía ser un hombre violento, pero él mismo había admitido que era un criminal. ¿Y si había hecho cosas peores y no me las había dicho porque yo no le había hecho las preguntas correctas? Me aterraba pensar que me llevara para Egipto y después, yo tuviera que quedarme allí sembrá', sin nadie que me ayudara si resultaba ser un abusador o un borrachón.
Había otra cosa que me molestaba. No lograba convencerme de que amaba a Jurgen. ¿No era una locura pensar que podía estar enamorada de un hombre que había visto sólo unas cuantas veces? Me molestaba el que a pesar de que esperaba con agrado sus llamadas, había olvidado cómo era. ¿Cómo era la forma de sus ojos? ¿Cuál era el color de su pelo? Si de verdad lo quería, sus ragos deberían haber estado grabados en mi memoria. ¿Cuánto medía? ¿Escribía con la mano derecha o con la izquierda? No sabía si tenía algún lunar de nacimiento o dónde se hacía la partidura. Según se fue acercando el día del regreso de Jurgen, me fui poniendo nerviosa y deseaba estirar el tiempo para que no estuviera a punto de llegar en dos semanas, diez días, cinco días, tres.
"No puedo hacerlo," lloraba por teléfono dos días antes de su regreso.
"¿Qué quieres decir?" Sabía exactamente lo que quería decir.
"Esto va demasiado rápido. No estoy lista..."
"¿Es que tú no me quieres?"
Me aterraba esa pregunta. En las dos semanas que llevábamos comprometidos, nadie me la había hecho, ni siquiera Jurgen. Fue mi silencio lo que se lo confirmó, el hecho de que no lo interrumpí para decirle, "No, no es eso, no tiene nada que ver con eso."
"Ya veo," dijo después de un rato.
"Quizás si tuviéramos más tiempo para conocernos," dije, no muy convencida.
Jurgen oyó la incertidumbre en mi voz y no trató de hacerme cambiar de opinión. Si hubiera tratado, a lo mejor hubiera flaqueado, aunque fuera por algún tiempo. "Tantos planes que teníamos," me dijo con tristeza; las mismas palabras de Mami cuando le dije que había suspendido la boda, aunque ella estaba más enojada que triste.
Perdimos varios cientos de dólares en depósitos para el traje de novia, el salón de la recepción, los trajes de las damas. Le avisé a la Sra. Davis en el Advertising Checking Bureau que no le diera mi puesto a nadie porque no me mudaba para Egipto. Al principio me daba vergüenza tener que explicarle a la gente que me había arrepentido, pero después de un tiempo, me sentí orgullosa de mi misma. Me había salvado, pensaba. Había hecho algo que la mayoría de las mujeres no hacen hasta que es demasiado tarde.
Shoshana se había pasado todo el verano en Israel. "¿Todavía eres virgen?" me preguntó tan pronto nos volvimos a ver, y tuve que admitir que lo era, y ella también. "No que no haya tenido un montón de oportunidades," me aclaró, lo que me llevó a contarle de mis aventuras con Avery Lee y Jurgen.
"Chica, pero, ¿qué es lo tuyo con los alemanes?" quiso saber.
"No soy yo la que los escojo," me defendí, "ellos me escogen a mí."
Shoshana se matriculó en unos cursos en Manhattan Community College, pero yo no porque quería dejar los días libres para el Children's Theater International. Para complementar mi salario de tiempo parcial en el Advertising Checking Bureau, conseguí un trabajo repartiendo volantes frente a un banco en Park Avenue. Un día, una mujer con un afro bien acicalado y un vestido africano estampado, se detuvo a hablarme. Tenía una agencia de modelos "exóticas" y quería saber si me interesaba. Hicimos una cita para el día siguiente y yo me aparecí en su oficina de la Sexta Avenida con la 40, con mi portafolio de fotografías hechas por Shanti. La puerta estaba cerrada con llave. De vez en cuando sonaba un teléfono adentro, pero nadie lo contestaba. Esperé en el pasillo durante media hora, hasta que me di por vencida, molesta por haber perdido inútilmente una tarde de trabajo.
Caminé hasta el Woolworth's de la Quinta Avenida, donde los teléfonos estaban en cabinas hechas de caoba con puertas de cristal que cerraban bien para asegurar la privacidad. Cuando me estaba acomodando en la primera cabina, un hombre se asomó, pero cuando levanté la vista, siguió caminando. Que espere, pensé. Llamé a la agencia de trabajos temporeros para decirles que estaría disponible durante los próximos días. Luego, llamé a Mami para decirle que iba a llegar temprano para empacar mis cosas porque estábamos de mudanza de nuevo, esta vez a la Calle Fulton en la sección de Brownsville en Brooklyn. Mami estaba entusiasmada porque Titi Ana había aceptado alquilar un apartamento de la casa, lo que quería decir que Mami podría comprar el edificio. Nuestras primas, Alma y Corazón, vivirían con nosotros. Salí de la cabina con mejor humor del que había entrado.
"Disculpe," una voz me sobresaltó y cuando me viré, allí estaba el hombre que se había asomado en la cabina telefónica. Estaba segura de que se iba a quejar porque me había tardado mucho hablando, pero se sonrió y señaló mi portafolio. "¿Es modelo?"
"Tratando," sonreí.
"Yo director de cine," me dijo. "Estoy buscando primera actriz para mi película." Tenía un acento pesado, y vacilaba entre palabras como para estar seguro de la pronunciación correcta.
"¿Dónde son las audiciones?" le pregunté entusiasmada, pero tratando de parecer profesional.
"Yo escribo para usted." Arrancó un pedacito de papel de una nota que tenía en el bolsillo, apuntó un nombre y un teléfono y me lo dio.
"Ulvi Dogan," leí.
_"Dawn,"_ me corrigió. "Como la mañana."
"¿De dónde es usted?" le pregunté.
"Turquía. ¿Y usted?"
"De Puerto Rico." Me presenté y prometí llamarlo al día siguiente.
"Muy bien," afirmó con la cabeza. "En la tarde. Estaré allí."
Esa noche llamé a Shoshana para decirle que un director de cine turco quería audicionarme. "Ven conmigo," le pedí.
"¿Y si no hay ninguna parte para mí en la película?"
"Ven y me acompañas."
La cita era en la East 58 Street, lejísimo del distrito de los teatros y de los estudios que se usaban para audiciones. "Esto está medio raro," le dije a Shoshana, frente al edificio residencial de ladrillos blancos. A ella le pareció que a lo mejor la compañía de cine había alquilado un apartamento para hacer allí las entrevistas.
El portero llamó, dio mi nombre y nos dirigió hacia arriba. Tocamos el timbre de la puerta que daba al pasillo, directamente frente al ascensor como nos había indicado. La abrió el hombre de Woolworth's, cuya amplia sonrisa se apagó tan pronto vio a Shoshana. Nos invitó a pasar.
"Espero que no le moleste," me disculpé, "pero mi amiga es actriz también. Por si acaso necesita _extras_..." Asintió con un gesto.
La sala donde entramos estaba decorada en negro y blanco. Dos butacas en piel negra, un sofá que hacía juego y lo que parecía ser una mesa con un tope en piel negra estaban organizados alrededor de una alfombra peluda con un diseño en cuadros negros y blancos. En las paredes blancas y desnudas había cuatro enormes carteles, unos _close-ups_ en blanco y negro de una misma mujer en le deleite del éxtasis sexual. Shoshana y yo nos miramos.
"Sr. Dogan," empecé a excusarme para poder salir de allí volando.
"Llámeme Ulvi, por favor. Siéntense, por favor." Nos acompañó hasta las butacas. "¿Les puedo ofrecer una Coca-Cola?"
Shoshana aceptó y yo la fulminé con la mirada. Mientras Ulvi abría la nevera, sacaba hielo de las cubetas, abría y servía las _Coca-Colas_ en la cocina, Shoshana y yo nos secreteamos. A ella le pareció que el apartamento estaba decorado con buen gusto y que las fotos eran tan artísticas como las de nuestros portafolios. "Se está tocando," protesté.
Ulvi regresó con nuestros refrescos. Se sentó en el sofá, se echó para atrás y cruzó las piernas. Tenía puestos unos mocasines de piel marrón, sin medias. Shoshana también se fijó. Mientras tomábamos nuestra sodas, nos dijo que iba a filmar su película en Long Island. Le pedí ver el guión y puso la mano sobre unos papeles bien organizados que estaban encima de la mesita de centro. "No está listo todavía," nos dijo. Nos preguntó acerca de nuestra experiencia en la actuación. Shoshana había participado en un par de obras en la escuela superior. Yo le enumeré mis credenciales y quedó impresionado. Shoshana tenía curiosidad por saber qué había dirigido él y Ulvi le contó que su película había ganado el primer premio en el Festival de Cine de Berlín. Los carteles de la pared empezaron a parecerme más artísticos.
Ulvi se inclinó hacia mí, me tocó la mano. "Estoy seguro de que puedo usarte en mi película," dijo. "Pero hay que hacer una prueba. ¿Sí?"
"Sí, claro," respondí.
Se echó para atrás, juntó las manos por las puntas de los dedos y dijo que también tenía un papel para Shoshana, pero menor. Shoshana irradió gratitud. Le pregunté cuándo sería la prueba y me dijo que todavía tenía que hacer los arreglos, pero que necesitaba mi teléfono para poder avisarme. No pidió el de Shoshana. Nos acompañó a la puerta y se quedó en el pasillo hasta que llegó el ascensor.
"¡Vas a ser una estrella!" gritaba Shoshana cuando íbamos por la calle.
"La prueba puede salir malísimamente mal..."
"¿Viste cómo te miraba? Cada movimiento tuyo... ¡te miraba con tanta atención!"
"No me di cuenta."
"Voy a poder decir que te conocía cuando...," reía entusiamada.
No me atreví a tener esperanzas. Ulvi hablaba como un director y Shoshana me señaló que sería fácil averiguar si realmente había ganado en Berlín. Caminamos hasta la biblioteca y, efectivamente, allí estaba, en la página 42 del _New York Times_ , del 8 de julio de 1964. El público se había sorprendido de que se le otorgara el premio a _Dry Summer_ , una película turca. Se describía a Ulvi como su "productor juvenil" y Shoshana y yo estuvimos de acuerdo en que si no joven, por lo menos lucía juvenil.
Nos separamos en la estación del _subway_ , con Shoshana totalmente convencida de que Ulvi sería mi gran oportunidad.
Llamó para decirme que la prueba sería el domingo, así es que me vestí con mi mejor ajuar y me presenté en su casa. No había cámaras en el apartamento, ni luces, ni personal o equipo de filmación. Me pregunté si habría llegado muy temprano, pero Ulvi me aseguró que no, que el camarógrafo se había retrasado. Pregunté, "¿Vuelvo más tarde?", pero él sugirió que nos quedáramos hablando hasta que llegara el _crew_.
Quise saber si el guión estaba listo. "Mi guionista es muy lento," me dijo, con una sonrisa indulgente y un movimiento de hombros.
Después de cinco minutos de plática sobre quién yo era, qué hacía y dónde vivía, sonó el teléfono. Habló en turco, una lengua que nunca había oído antes. El sonido era tranquilizante, por lo menos como él lo hablaba, en una voz sosegada, íntima, un susurro áspero. De cuando en cuando, mientras escuchaba al que lo había llamado, levantaba la mano hacia mí en un gesto de "espera un momentito."
Tenía unas enormes pupilas marrón oscuro; pelo negro, finito; una frente ancha. Unas líneas profundas iban de la nariz a los labios que parecían dibujados en su rostro, su forma precisa, aplastada. Su nariz parecía una línea recta que salía de su frente y se abría para terminar en una base ancha. De perfil, recordaba los frescos de los jinetes etruscos que había en los museos o a un rey de Mesopotámia. Su aire majestuoso se intensificaba con sus movimientos lentos, estudiados, como si tuviera que tener cuidado de no tumbar nada.
Cuando colgó me preguntó sobre Puerto Rico. No había estado nunca allí, pero había asistido a festivales de cine en Cartagena, Colombia y en Venezuela, Costa Rica y México. En el trayecto, había recogido dos o tres palabritas en español. "Señorita," me dijo. "¿Cómo está?" Lo felicité por su acento excelente. "Es con inglés que tengo problemas," sonrió.
Le aseguré que hablaba bien y que yo entendía todo lo que decía. Con otro gesto sobrio, me dio las gracias. El teléfono volvió a sonar, pero esta vez habló en alemán. A pesar de que yo no entendía las palabras, no sentía titubeo en su voz como cuando hablaba inglés.
Después que terminó de hablar, discutimos la película que había ganado el _Golden Bear_ en Berlín. Me contó que era una historia de amor y que él la había producido y dirigido. También había hecho el papel del galán romántico. La primera actriz era ahora toda una estrella en Turquía. "Pero yo la descubrí," subrayó.
Me dijo que la había visto sentada en la escalera frente al edificio donde él tenía la oficina. Estaba esperando a su mamá, la señora de la limpieza. Nunca había trabajado en una película, pero tan pronto él la vio, se dio cuenta que tenía potencial de estrella. Cuando me vio en Woolworth's reconoció en mí las mismas cualidades que había visto en ella.
Me sentía halagada, pero también estaba consciente de que estábamos solos en su apartamento decorado con mujeres que estaban masturbándose. Artístico o no, era imposible mirar para ningún sitio en el apartamento y no ver un pezón, un ombligo brotado o vello púbico. Después de media hora, cuando no apareció el _crew_ por ninguna parte, me levanté. "Debo irme."
Ulvi sugirió que diéramos un paseo. "Estarán aquí cuando regresemos," me prometió.
Paseando por la 58 hasta la Quinta Avenida le pregunté dónde había aprendido alemán.
Abrió los ojos. "¿Tú también hablas?"
"No," me reí. "Algunos de mis amigos..." y agité la mano para restarle importancia.
Asintió con la cabeza. Había vivido en Alemania muchos años y hablaba la lengua con fluidez. "Mejor que el turco, a veces," se rió. Nos lamentamos de lo difícil que era conservar la lengua vernácula cuando había tan pocas oportunidades de practicarla. Coincidimos en que el anhelo de regresar a la patria —aún después de años de estar tan lejos — no desaparecía nunca.
"Pero cuando uno regresa," dijo, "no aprecian." Se había convertido en una celebridad en Turquía después que su película ganó los premios. Pero la prensa lo atacó. Levantó las manos con las palmas al aire e hizo unos movimientos de sube y baja. "Me criticaban por esto, por aquello, por nada." Agradecí sus respuestas bien pensadas, disfruté nuestra conversación que pasaba de un tema a otro. Su voz suave y su manera tan calmada eran reconfortantes. Como bien había notado Shoshana, me miraba con interés. Al principio me sentía incómoda con la intensidad de su atención. Pero entonces, me di cuenta de que tenía que hacerlo porque leía los labios. No porque tuviera problemas auditivos, sino porque así encontraba otras pistas para interpretar lo que yo decía. Yo hacía lo mismo. Aún después de siete años de inglés intensivo, me enfocaba en la persona que me estuviera hablando para encontrar otras claves, más allá del lenguaje, que me ayudaran a comprender. Cuando hablaba, todavía traducía simultáneamente del español y estaba segura de que Ulvi hacía lo mismo, del turco vía alemán. No en balde hablaba tan despacio.
En la Quinta Avenida nos topamos con una parada. Las bandas marchaban seguidas de unas _cheerleaders_ acróbatas que agitaban pompones de colores. Unas carrozas adornadas de banderines de papel crepé se movían lentamente llevando unas mujeres jóvenes vestidas en trajes de noche. Sus guantes blancos parecían limpiar el aire en arcos perfectos según saludaban, primero a la derecha, después a la izquierda y a la derecha otra vez.
Más tarde me preguntaba en qué momento durante la parada me había cogido la mano. Y por qué cuando pasó la carroza con los bailarines de polka me rodeó con el brazo. Y cómo fue que cuando pasó la Banda del Middletown Police Athletic League tocando su versión de Winchester Cathedral, mi cara estaba contra su pecho y sentía su olor, limpio, en realidad un "sin-olor" cautivante.
Regresamos a su apartamento. Aunque me habían besado y tocado antes, y conocía el contorno del cuerpo masculino a través de la ropa, fue diferente cuando estuvimos desnudos. Su piel era del color de las nueces tostadas. Ni una sola línea de bronceado dañaba el tono uniforme de su piel, desde la punta del pelo hasta la planta del pie. Su pecho estaba forrado con un corazón de vello negro y lacio que terminaba en punta, debajo de su costillas. Su abdomen era plano, pero suave, sin músculos definidos, una lisa mesa de carne donde yo recostaba mi cabeza para escuchar su vida. Cuando me le acomodaba más arriba, su corazón palpitaba contra mi oído y me arrullaba hasta que me adormecía en su pecho. O si me acurrucaba cerca de su ombligo, el ruido de un arroyo alborotoso gorgojeaba intermitentemente. Corrí mis dedos desde la punta de la cabeza, cruzando por su frente ancha atravesé la arruga grabada de sien a sien, hasta su nariz, una pirámide de base ancha sobre unos labios suaves y frescos. En reposo, lucía triste y solemne, pero yo lo hacía sonreír. Tenía un hoyuelo en la barbilla, una hendidura llana, donde yo metía la lengua para sentir los pinchazos de unos toconcitos finos. Su cuello era largo, con dos surcos profundos entretejidos de oreja a oreja, como cicatrices. Acaricié su pecho, el corazón de pelo negro y lacio, un corazón sobre su corazón. Y la expansión plana de su barriga inmaculada.
Después de hacer el amor, hizo unas llamadas telefónicas. Desnuda, me enrosqué en su cuerpo, su brazo izquierdo debajo de mi cabeza, el mío sobre su pecho. Me le pegué bien cerquita hasta que nuestras pieles morenas fueron una. No podía pensar en lo que acababa de hacer, me negaba a contestarle a la voz que me preguntaba, ¿Por qué el? ¿Por qué no Otto o Avery Lee o Jurgen? ¿Por qué no me había resistido, por qué, de hecho, había tirado alegremente mi ropa en el respaldar de la butaca de piel negra?
Hablaba en su lengua extranjera y yo escuchaba las palabras, su risa ahogada, sus susurros. En una, me acercó el teléfono a los labios y me dijo "saluda" y yo dije "hola," sin saber a quien estaba saludando. Después de un rato, me puse celosa. Me retorcí de un lado a otro, me calenté, me le trepé encima, rodé de nuevo y me acomodé de lado hasta que él me aplastó, hasta que sentí su peso, hasta que me hundí bajo su cuerpo largo y oscuro, hasta que no podía respirar. Fue cuando lo empujé suavemente, con un quejido, que él se movió, me acarició la cara y me llamó "Chiquita."
"¿Quién?" me incorporé apoyada en el codo y busqué su rostro.
"Tú eres chiquita," sonrió, "mi nena pequeña. ¿Eso es lo que quiere decir la palabra 'chiquita' en español?"
"Sí," le dije más tranquila. "Pequeña. Nena chiquita." Y hechas las paces, me recosté otra vez.
Saciada, regresé a Brooklyn, con el cuerpo hormigueando de secretos. A esa hora, el tren A venía lleno de trabajadores que regresaban después de terminar el turno de la noche. La mayoría dormitaba o leía el periódico con mucha cautela como temiendo confirmar sus peores temores. Un hombre en un mono de mahón dormía en el asiento que estaba pegado a la cabina del conductor. Una mujer vestida de enfermera agarró bien la cartera cuando me le senté al frente. Otra, delgada y nerviosa, se halaba los rizos cansados alrededor de sus hombros; la ventana detrás de mí era un espejo mugriento en el que se miraba con desesperación. De vez en cuando, las dos mujeres me miraban y enseguida cambiaban la vista. ¿Se darían cuenta ellas de lo que había estado haciendo? me preguntaba. ¿Tendría señas que me delataran?
No, él había tenido cuidado de no marcarme. Mi piel se sentía más caliente ahora que cuando llegué, pero no había marcas ni señas de que habíamos estado desnudos por horas. Busqué su olor en mi piel, pero también se había ido. Ulvi había insistido en que nos ducháramos después de tener sexo. Nos bañamos juntos, mi pelo envuelto en una toalla sedienta y él de rodillas enjabonando y enjuagando. Sus manos, a veces una caricia, a veces un sondeo, borraron todo rastro de que habíamos hecho el amor, toda evidencia de que había estado conmigo, dentro de mí. Con el agua caliente golpeándome la espalda, cerré los ojos y dejé que sus dedos resbalosos de jabón me exploraran, entre los dedos del pie, detrás de las rodillas, por las nalgas. Chorreaba de deseo, pero él murmuró, "Ahora no, no más. Basta por hoy." Me envolvió en sus enormes toallas negras y frotó mi piel con las puntas hasta que secó cada gotita. Regresé a casa de Mami hambrienta, sedienta, impaciente porque llegara el próximo día, cuando regresaría al apartamento negro y blanco y me desnudaría y lo dejaría tocarme otra vez donde nadie lo había hecho jamás.
"¿Que hiciste qué?" Shoshana batió sus larga pestañas. "¿Cuándo? ¿Cómo?"
Era difícil explicar _cómo_ había pasado. No la mecánica del sexo, sino cómo pasé de ser actriz de cine novata a la... ¿qué del director? No podía ni nombrar en qué me había convertido.
"¿Por qué él?" quería saber Shoshana.
No había manera de contestar esa pregunta tampoco. No, no era tan guapo como Neftalí, Otto, Avery Lee o Jurgen. En la semana que habíamos estado juntos no me había llevado a ningún restaurante, ni al teatro, ni siquiera al cine. No gastaba dinero en mí. No me había pedido que fuera su novia, su amante o su esposa. No hizo promesas de ningún tipo. No parecía tener ninguna expectativa, excepto que yo me presentara en su apartamento todos los días a la hora acordada. Cuando le sugerí que yo tenía que conseguir algún método anticonceptivo me dijo que no me preocupara. "Yo me encargo," dijo, y lo hizo.
"Y," sonrió Shoshana con malicia, "¿conseguiste el papel?"
Ulvi admitió que no había tenido la intención de usarme en ninguna película. "Te quiero para mí," me dijo, "nadie más."
_"¡Wow!"_ Shoshana estaba impresionada.
Para tener más tiempo para estar con Ulvi, cambié mi horario de trabajo. Cotejaba los anuncios por la mañana y me pasaba la tarde con Ulvi. Generalmente, llegaba a casa a la hora de comer y entonces, me metía en el cuarto que compartía con Delsa en nuestra nueva casa de la Fulton. No quería darle la oportunidad a Mami de que me mirara demasiado, por el miedo que me daba de que fuera a sospechar, de que mi vida con Ulvi se notara en la forma en que me movía o me comportaba.
Un día, me encontré con Shanti en la Quinta Avenida. Había estado por llamarme, me dijo, porque quería tomarme unas fotos a color. Tomó mi barbilla y movió mi cara de lado a lado para capturar la luz. "Tu cara ya no es inocente," concluyó.
"Ni yo tampoco," le solté. Se estremeció. "Tengo que irme." Con la garganta trinca, lo dejé parado en la esquina de la Quinta con 48. Corrí hasta el Hotel Algonquin, atravesé por la barra, bajé corriendo hasta el baño de damas atestado. Me miré en el espejo mucho rato, pero no podía ver lo que vio él. ¿Es que era visible sólo para los demás?
Vera llamó para que discutiéramos la próxima temporada del Children's Theater International. Me querían otra vez de Soni en _Babu_ y como princesa japonesa en una obra inspirada en el teatro _kabuki_. Después de hacer la lectura de la obra en el estudio de ensayos en Christopher Street, Bill nos dio pon a unos cuantos hasta el área residencial de la ciudad. Iba camino a Ulvi.
"¿Cómo estuvo el verano?" preguntó Bill y los demás intercambiaron historias de repertorios de verano y salas de teatro con restaurantes. Fui la última en bajarme de la guagua VW. "Has estado callada," comentó Bill, al detenerse en la esquina de la 58 con Tercera Avenida. Estuve a punto de contarle a dónde iba, pero no pude pronunciar su nombre.
"No sé cómo empezar," tartamudeé.
Me apretó la mano. "Es sorprendente cómo un verano puede cambiarle a uno la vida." Me acerqué y lo besé en la mejilla. Lo quería tanto; quería a Allan, quería a tanta gente. ¿Quería a Ulvi? Tenía que quererlo para haberme entregado a él con tanto gusto. Sin embargo, lo que sentía por él no era nada parecido a lo que sentía por Bill y Allan, por mi familia, por Shoshana. De ellos, podía decir fácilmente que los quería. De Ulvi, lo más que podía decir era: "Hice el amor con él."
¿En qué me convertía eso? Después de años de fantasear con el amor romántico, había venido a caer en las sábanas blancas y negras de un hombre que no era romántico en el sentido tradicional de la palabra. Cero flores, cero cenas a la luz de la vela, cero hablar del futuro más allá del día siguiente cuando nos volveríamos a ver. Estar con Ulvi era como estar suspendida en el tiempo. Después de aquella primera conversación larga, no habíamos vuelto a hablar sobre nada de nuestras vidas.
"No me interesan ni tu familia, ni tus amigos," me dijo cuando traté de contarle. "Te quiero sólo a ti." Era liberador no tener un pasado con él. Pero me molestaba que si a él no le interesaba mi vida lejos de él, yo no podría justificar preguntarle de su vida lejos de mí.
Un lunes por la tarde me encontré con Shoshana en el Automat. Metimos las monedas en la ranura y las puertas se abrieron para poder sacar nuestros platos de macarrones con queso.
"Te tengo que contar," me dijo Shoshana excitada. Tan pronto me senté frente a ella, me espepitó la noticia. "Lo hice."
No fue muy difícil entender a qué se refería el "lo." "¿Cuándo? ¿Quién?"
Había ido a una fiesta ese fin de semana, conoció a un turco. "Más joven que el tuyo," añadió. "Ahora las dos perdimos la virginidad con turcos," rió.
Cuando se lo conté a Ulvi, pensando que le daría gracia la coincidencia, las cejas se le enfurruñaron y apretó lo labios. "Es una chica muy tonta," dijo.
"¿Qué quieres decir? ¿Debe tener cuidado con Ali? ¿Lo conoces?"
"Hay miles de Alis," gritó Ulvi. Nunca lo había oído levantar la voz más alto que un murmullo y ese cambio me asustó. "¿Cómo puedes tener una amiga así?" continuó. No entendí lo que quería decir. Cuando le señalé que no había hecho nada peor de lo que hacíamos nosotros, me miró con severidad. "No es lo mismo. Ella es chica barata."
Quedé pasmada. Su opinión de Shoshana era tan injusta, le argumenté. Sólo la había visto una vez. Era una persona maravillosa, cariñosa, graciosa, inteligente. ¿Cómo podía decir algo así?
"Conozco millones de chicas así," y el desprecio en su voz me dio escalofríos. Me tomó entre sus brazos, me acarició el pelo, me besó. "Eres niña tan ingenua," me dijo. "Hay tantas cosas que tengo que enseñar, Chiquita." Fue tierno, delicado. El círculo de sus brazos, un mundo en el que me sentía protegida, un lugar donde podía admitir mi ignorancia. Sí; era ingenua, pero en sus brazos mi inocencia era atesorada. En sus brazos no tenía que pensar, no tenía que hacer planes, no tenía que hacer otra cosa que no fuera responder a sus caricias. Cuando me abrazaba, no lo cuestionaba ni lo retaba, porque yo no sabía nada. Ni siquiera conocía la verdadera naturaleza de mi mejor amiga.
Tarde en la noche, la Calle Fulton estaba tranquila, las sombras sólidas como muros. Caminaba por el lado de la acera más cerca del parque, donde la verja se extendía alta e imponente entre los columpios, las chorreras, las barras y yo. A mi derecha, los carros estaban cerrados con llave y acomodados para la noche, pero en algunos, unas sombras humanas se movían en cuidadosa anticipación. Cuando les pasé por el lado, el corazón me latió con más rapidez de lo que podían andar mis pies. Me mantuve pegada de la verja, mirando hacia el frente, pero alerta a cualquier movimiento inesperado. Una voz grave murmuró "Hola, mi vida," otra "Mira, mamita" y el peligro me impulsó, casi me levantó de la acera, pero no correría, no a menos que me persiguieran. Si corría sin una clara provocación, se darían cuenta de lo asustada que estaba, así es que caminé —ligero, pero confiada de que llegaría a la puerta de casa, que me daría tiempo a meter la llave en la cerradura, quitar el seguro, empujar la puerta pesada y entrar, antes de que alguien me alcanzara.
Una vez a salvo, me recosté contra la puerta y respiré hasta que dejó de hormiguearme la espina dorsal y el corazón me volvió a su ritmo normal, hasta que me sentí tan compuesta por dentro, como se veía mi cara —libre ya del miedo, hasta que las manos me dejaron de temblar y las rodillas se me estabilizaron. Quité el seguro y abrí la puerta interior que daba al pasillo oscuro donde esperaba escuchar los pasos de Mami y encontrarme, al final del pasillo, con su desaprobación; sus ojos oscuros, tristes, cargados de desilusión y reproches. Pero, era demasiado tarde. Estaba dormida en la orilla de la cama, con las chinelas puestas todavía, con las rodillas dobladas y la bata de nilón encaramada en las caderas. Tenía el brazo derecho doblado sobre sus ojos, como para espantar las pesadillas, y el izquierdo agarraba la baranda de la cuna de Ciro, que dormía profundamente hecho una bolita.
Pasé en puntillas por su puerta hasta el cuarto que tenía las literas contra una pared y la cama que Delsa y yo compartíamos contra la otra. En la oscuridad asfixiante sentí otra vez la emoción del peligro, sólo que esta vez no era el miedo a un asaltante desconocido. El recuerdo de las manos de Ulvi dejaba rastros en mi cuerpo, cargaba mi piel con una energía que estaba segura que cualquiera podía ver, podía sentir. Cuando me metí en la cama al lado de mi hermana, Delsa se movió, y me pareció natural acurrucarme contra su cuerpo como lo hacía con él. Pero ella era mi hermana; si la hubiera despertado para abrazarla, hubiera pateado y maldecido y me hubiera empujado de la cama. Me quedé bocarriba, con los brazos tendidos a lo largo del cuerpo, tomando el menor espacio posible en la cama estrecha. La respiración profunda y uniforme de mis hermanas y hermanos dormidos era una de los sonidos más tranquilizantes que había oído, pero no me indujo al sueño, como había hecho antes. Estaba demasiado consciente de esa otra respiración, a millas de distancia, en el apartamento escasamente amueblado de mi amante. ¡Mi amante! Otra vez sentí deseos de virarme y abrazar el cuerpo junto al mío. Pero, era Delsa. En vez de hacerlo, me abracé a mí misma, cerré los ojos, imaginé que mis brazos eran los suyos, que estaba en la enorme cama de sábanas negras, donde el acercarse buscando calor era recibido con un gemido de placer, no con molestia.
Por la mañana, Mami deliberadamente no me quitó los ojos de encima mientras yo salía y entraba al baño, pero no me preguntó dónde había estado. Cuando abrí la tabla de planchar, se salió del medio sin decir palabra. Su serenidad era desconcertante en medio del caos de prepararles el desayuno a mis hermanas y hermanos y ayudarles a organizarse para irse a la escuela. "Vélame los nenes," dijo y señaló a Charlie y a Cibi que estaban atrapados en el corral. Acompañó a Raymond y a Franky hasta la puerta, esperó allí hasta que doblaron la esquina y regresó a rescatar a Ciro de su cuna, donde llevaba un rato lloriqueando. Planché mi ropa velándola con el rabo del ojo, consciente de que su silencio podía explotar en una discusión de grandes proporciones a la menor provocación. Caminaba con pesadez, arrastrando los pies por el linóleo. Tres bebés en dos años la habían dejado fofa y rellena, con los ojos permanentemente hinchados por la falta de sueño, las facciones laxas, como si sus músculos no tuvieran la energía para animarle el rostro. Quité la vista de su figura exhausta, abochornada de estar añadiendo peso a su carga.
Jugué con los bebés, manteniéndome lo más lejos de Mami que me permitía la cocina congestionada y entonces me metí en el cuarto a cambiarme. Cuando salí, Mami estaba sentada en la mesa con su taza de café negro matutino frente a ella y Ciro en la falda.
"Me voy a trabajar y después tengo ensayo," le dije.
Me miró, frunció los labios, asintió. Agradecía su censura silenciosa, un día más en que no me confrontaba con sus sospechas, y salí del apartamento con un sentido de triunfo hueco, porque ella se negaba a pelear.
# "¿Dónde tú estabas anoche?"
#
La opinión de Ulvi sobre Shoshana no me impidió seguirla viendo cada vez que podía. No había nadie, incluyéndolo a él, con quien me sintiera más cómoda o con quien me divirtiera tanto. Nos reuníamos para almorzar, visitábamos museos, dábamos largos paseos por la Quinta Avenida charlando sobre nuestras vidas amorosas. La relación de Shoshana con Ali no duró mucho, pero tampoco fue muy lamentada. Al dejar de ser su estudiante, Shoshana comenzó una relación con el Sr. Arthur Delmar, el Profesor de Fundamentos de la Publicidad.
"¿Por qué no me dijiste," me cuestionó Shoshana, "que el sexo con un hombre mayor es mucho mejor?"
"No tengo base para comparar," le recordé.
"¿Cuántos años tendrá?" se preguntó en tono reflexivo. "Espero que no tantos como mi papá."
Papi era mayor que Mami, que tenía treinta y siete. Hacía siete años que no lo veía y se me hacía difícil hacerme una imagen de él. ¿Estaría viejo y arrugado? ¿Estaría barrigón? ¿Usaría espejuelos? Ulvi se veía más joven que Mami pero no quería decirme su edad. Arthur tenía el pelo gris, así es que suponíamos que era mayor que Ulvi, pero Shoshana no estaba en las de preguntarle directamente. "No quiero saber," dijo agitando la mano.
No nos hacíamos ilusiones de que haríamos nuestras vidas con Ulvi o con Arthur. Aunque Arthur le propusiera matrimonio, Shoshana nunca se casaría con él porque no era judío. "Tengo que pensar en el futuro de Israel," decía con seriedad.
Yo no tenía una nación entera que dependiera de mi selección de esposo, pero tampoco esperaba que Ulvi se casara conmigo. Era cortante cuando me decía que no quería involucrarme en su vida. "¿Por qué no?"
"Es complicado," me respondía y entonces, besaba mi frente ansiosa. "No te preocupes," me decía, "no es nada para tu preocupación." Si le hacía más preguntas, me acallaba con caricias. "Nunca te haré daño," me aseguraba, y hasta ahora, no lo había hecho.
Ulvi insistía en que nuestras vidas fuera de su cama fueran privadas, lo que me hacía sospechar que tenía secretos peores que los de Jurgen. Una tarde en que tuvo que asistir a una reunión me pidió que lo esperara en el apartamento. Era la primera vez que estaba sola en su apartamento y decidí aprovechar la oportunidad. Si encontraba algo ilegal o que lo incriminara, juré que me iría y no volvería nunca. Era tan meticuloso que me tomó muchísimo rato inspeccionar el apartamento porque tenía que dejarlo todo exactamente igual a como lo había encontrado. Sus pertenencias estaban acomodadas en un orden preciso impuesto a cada tablilla, gaveta o gabinete. Las toallas negras estaban dobladas a lo largo en tres partes y en tres partes de nuevo, y acomodadas de modo que las orillas no sobresalieran. No había nada entre las toallas, ni detrás, ni debajo. No usaba calzoncillos y contrario a mi primera impresión, sí usaba medias que estaban emparejadas y dobladas en fila en el fondo de la gaveta del tocador. No había un revólver, ni un paquete de marijuana, ni cartas de amor. Sus camisas, pantalones y chaquetas estaban colgadas por color, cada una en su sección del _closet_. No había una pared falsa ni una caja de seguridad detrás. Los zapatos estaban alineados en el piso, cada uno con un pie de cedro adentro. Nada se les cayó de adentro cuando los viré uno por uno. No usaba prendas, usaba una rasuradora eléctrica, no se untaba loción para después de afeitarse. No había drogas en el botiquín, ni siquiera una aspirina. En los gabinetes de cocina había una vajilla para cuatro, platos blancos con borde negro. Había dieciséis vasos, cuatro de cada tamaño en orden descendente, adornados con motivos de barajas —sota, reina, rey, y el as de copa. En la nevera había unos cuantos vegetales, un envase de jugo de china, mantequilla, dos o tres huevos. Aparte de las mujeres masturbándose en las paredes, no había retratos en ningún otro sitio, ni premios por sus películas, ni recortes de periódicos, ni comunicados de prensa. No encontré recibos de tarjetas de crédito, ni libretas de cuentas de ahorro, ni chequera.
Todo en el apartamento era nuevo, cuidadosamente escogido para que combinara. Caro, también. Las toallas eran gruesas y mullidas. Su ropa era de marca de diseñador y estaba hecha de telas finas; lana, seda, casimir. Sus zapatos eran de suela fina, forrados en piel, suaves. En la tablilla de arriba de su _closet_ había un juego de maletas hechas de cuero negro con guarniciones de bronce y un candado de combinación incrustrados en el medio. Había algo en la maleta más grande, pero no la abrí porque pesaba demasiado para yo bajarla.
Acostumbrada al caos de mi casa de Brooklyn, el apartamento de Ulvi me pareció estéril, su orden, siniestro. Estaba tan limpio, tan ordenado. Ni en las esquinas quedaba algún rastro de polvo o migajas o algún hilo suelto. Después de examinarlo todo y de no encontrar nada que me resultara sospechoso, abrí el sofá-cama de piel y me acosté pensando en lo que eso significaba. Él era un hombre sin historia, sin edad, lo suficientemente rico como para vivir a dos bloques de Bloomingdale's, pero no tanto para derrochar. Había tantas preguntas que quería hacerle, pero cada vez que trataba, él desviaba mis dudas con besos. Decidí que el único modo de hacerlo hablar sería en público, donde no podría distraerme con sus caricias.
Cuando regresó de su reunión, lo primero que hizo fue buscar en el _closet_ donde guardaba sus toallas. Me pregunté si habría pasado algo por alto, pero ya pa' qué, ya era muy tarde. No me preguntó si había rebuscado sus cosas, pero tuve la impresión de que él sabía aunque no dijo una palabra. Nunca volvió a dejarme sola en el apartamento.
Mi prima Alma vivía con su mamá y su hermana en un apartamento en el segundo piso de la casa de Mami en la Fulton. Cuando vivían más lejos, pasaba más tiempo con Alma porque hacíamos el esfuerzo de reunirnos para comer. Sin embargo, desde que había conocido a Ulvi, mi amistad con Shoshana se había fortalecido y mi relación con Alma se había enfriado. Tenía más de qué hablar con Shoshana, sin el peligro de que mi vida secreta llegara a oídos de Mami.
Alma y yo habíamos pasado horas hablando de conseguir un apartamento para mudarnos juntas. Nuestras madres se aseguraron de que eso no pasara. Mi meta era ahora conseguirme un sitio donde pudiera ir y venir a mi antojo. A los veinte años, le discutía, tenía edad suficiente para cuidarme. Mami insistía que la única manera en que me dejaría ir sería del brazo de un hombre, un esposo legal, preferiblemente. Mami me hacía ver que Alma, que era un año mayor que yo, todavía vivía en su casa. Con Titi Ana y Mami apoyándose una a la otra, no había manera de que Alma y yo consiguiéramos lo que queríamos.
Shoshana me decía que yo tomaba demasiado en cuenta los deseos de Mami. Para convertirme en mujer, afirmaba, tenía que rebelarme en contra de Mami. Lo que me decía tenía sentido y yo, hasta llegué a discutirlo con Alma, que estuvo de acuerdo con Shoshana. Aun así, no veía como desafiar a Mami, como tampoco Alma se oponía a Titi Ana, ni Shoshana confrontaba a sus papás. Alma estaba dedicada a su trabajo; a su hermana, Corazón; a sus libros. Shoshana y yo maquinábamos, planificábamos, soñábamos, alimentábamos secretos. Pero, ninguna de nosotras se enfrentó a su mamá y le dijo: "Te dejo. Puedo sostenerme sola. Es tiempo de vivir mi vida."
Pasábamos el menor tiempo posible en la casa. Shoshana tenía el Manhattan Community College, un trabajo en una tienda de zapatos en la calle 34 y a Arthur, que la mantenían ocupada. Yo tenía el Children's Theater International, el Advertising Checking Bureau y a Ulvi. Juntas, Shoshana y yo teníamos también nuestras citas.
Lo mismo en la calle que en un restaurante, con frecuencia, a Shoshana y a mí se nos acercaban hombres deseosos de invitarnos a salir. La mayor parte de las veces aceptábamos, pero seguíamos unas reglas estrictas para estas citas inesperadas. Solamente íbamos a cenar a restaurantes finos, nunca a bares. Rechazábamos las bebidas alcohólicas. Acordábamos una hora y aunque los hombres fueran fascinantes, nos íbamos a la hora en punto. Llegábamos juntas y nos íbamos juntas. Una nunca dejaba a la otra sola con un hombre. La mayoría de los hombres se contentaban con hablar, pero algunos nos ofrecían dinero a cambio de sexo. Cuando eso ocurría, Shoshana y yo ejecutábamos una salida dramática. Después de una señal preacordada, nos levantábamos de la mesa como si fuéramos una y salíamos furiosas. El noventa por ciento de las veces, los hombres quedaban tan sorprendidos que se quedaban sentados con la boca abierta, mientras el resto de los comensales se nos quedaban mirando. Una vez un hombre nos gritó un chorro de obscenidades, lo que nos confirmó que habíamos hecho bien en salir de allí lo antes posible.
No pensábamos que lo que estábamos haciendo fuera malo o que estuviéramos engañando a Ulvi o a Arthur. Ellos nunca nos sacaban a ningún sitio, como si les diera miedo que nos vieran con ellos. Los extraños que nos invitaban a cenar estaban encantados de tenernos de acompañantes, nos llevaban a restaurantes elegantes e insistían en que pidiéramos los platos más caros. Eramos —lo sabíamos— un adorno, una línea en sus informes de gastos. Pero, no nos importaba. Sus conversaciones, que a sus esposas o novias podían parecerles estupefacientes, eran fascinantes para nosotras que nunca habíamos conocido a un contable de Peoria o a un jefe de personal de Alburquerque.
Lo primero que averiguábamos del individuo era donde vivía. Preferíamos que no fuera de Nueva York para no corrernos el riesgo de volvernos a tropezar con él. Entonces, le preguntábamos si era casado, si tenía hijos. Si mentía, se engañaba a sí mismo. Si compartía con nosotras fotos de su esposa e hijos y cuentos de los juegos de pequeñas ligas y de funciones de teatro escolar, le rendíamos un servicio a su familia al mitigar su soledad y evitar que fuera a hacer algo de lo que tuviera que arrepentirse. No teníamos nada que perder, disfrutábamos de una cena agradable con conversación interesante y nos sentíamos virtuosas porque estábamos salvando una familia mientras nos manteníamos leales a nuestros novios.
Aprendimos a identificar a los echones y a los presuntuosos, cuyas mentiras y exageraciones contribuían a nuestra risería cuando al día siguiente intercambiábamos impresiones de la noche anterior. Así como mis hermanas y yo les teníamos nombres a los hombres con quienes bailábamos en los clubes —"los rompemedias" y "los pulpos"— Shoshana y yo también les teníamos nombres en clave a nuestros acompañantes.
Primero estaban los _"Groovies,"_ quienes trataban de impresionarnos con su labia, utilizando expresiones juveniles de moda a cada momento. Como Shoshana y yo teníamos otra lengua vernácula, no manejábamos la jerga americana con la facilidad de los angloparlantes. La mayor parte del tiempo, el uso de la jerga de los _Groovies_ era como otro idioma para nosotras y escuchábamos en rapto, cómo las frases idiomáticas que surgían de nuestra propia generación nos distanciaban de ellos. Como habíamos aprendido el inglés como segundo idioma, Shoshana y yo estábamos obsesionadas con la corrección del lenguaje. Hablábamos inglés de libro de gramática y mirábamos con desprecio al segundo tipo, los "TS's", aquellos que no podían formar una oración sin meterle el "tú sabes" entre frase y frase. El _Groovy_ nos trataba con condescendencia, un TS no era específico nunca, dejaba que las oraciones se fueran desintegrando en generalidades. Si nos sentíamos especialmente malvadas ese día, empujábamos a los TS's a que nos dijeran más, hasta que les quedaba claro que no, que no sabíamos. Shoshana sostenía que los TS's se sentían amenazados por la verdadera ignorancia, porque al decir "tú sabes," evitaban mostrar la suya.
El tercer grupo, los "Papitos," eran hombres mayores que, a veces, durante la cena nos comparaban con algunas de sus hijas. "Tú me recuerdas tanto a Lindy," le dijo uno a Shoshana y ella lo animó a que le describiera a Lindy. En un dos por tres, nos habíamos enterado de la historia de su vida con detalles sobre sus parientes políticos, sus mejores amigos, los pagos de pensiones alimentarias, los derechos de visita. Los Papitos eran los más dados a querernos ver de nuevo, pero otra de nuestras reglas era no salir dos veces con la misma persona. Los _Groovies_ asumían que les podíamos conseguir alguna conexión de drogas. Los TS's eran los más inclinados a ofrecernos dinero a cambio de sexo.
A veces, Shoshana le contaba a Arthur de nuestras citas. A Ulvi, que no tenía ningún interés en mi vida, no le contaba nada. Nuestra relación era una burbuja aislada del resto de nuestra existencia, confinada entre las paredes blancas de su pulcro apartamento de una habitación.
Los ensayos para la nueva producción de Children's Theater International, inspirada en un motivo japonés, eran por las noches y los fines de semana, mientras que las representaciones de _Babu_ nos cogían dos o tres mañanas a la semana. Con excepción de Tom y yo, muchos de los actores de la temporada anterior de _Babu_ tuvieron que ser reemplazados, porque tenían otros compromisos. Allan se había incorporado al elenco de Broadway de _Fiddler on the Roof_ , así es que un nuevo actor, Jaime, tomó su lugar en el repertorio. Jaime era puertorriqueño, como yo, pero nacido en los Estados Unidos. Nos dábamos cuenta de la ironía envuelta en tener a dos puertorriqueños haciendo el papel de realeza india.
"Esto no está bien," se quejaba Jaime. "Deberíamos estar luchando por los derechos de nuestra gente."
Jaime estaba orgulloso de su herencia y decidido a hacer lo que pudiera para preservar la cultura puertorriqueña en Nueva York. En el Barrio, en el Bronx, en partes de Brooklyn, otros puertorriqueños, algunos miembros de los Young Lords, montaban campañas para mejorar la vida de sus compatriotas. Mi prima Corazón trabajaba con un grupo en el Lower East Side que ofrecía clases de arte y fotografía a estudiantes puertorriqueños de escuela superior. Mi hermano Héctor y mi hermana Delsa estaban involucrados en organizaciones de jóvenes de nuestra comunidad.
Mi propia conciencia social estaba patéticamente subdesarrollada. No sentía ninguna obligación con "nuestra gente" en abstracto. Más bien, me sentía aplastada por mi deber con mi gente en concreto: Mami, Tata, mis diez hermanas y hermanos.
"Eso es una excusa para no comprometerte." Jaime alegaba que yo usaba a mi familia como una excusa para evadir la lucha puertorriqueña. "¿Y qué es eso de tanto baile indio?" me regañó. "Tenemos que promover nuestro arte y teatro. Deja que los hindúes se preocupen por el suyo."
Mi devoción por la danza india, argumentaba, no era parte de ninguna conspiración para promover su cultura sobre la puertorriqueña. Mi amor por la danza clásica india no se extendía a ningún otro aspecto de la cultura del subcontinente. No me gustaba el _curry_ , ni la comida con pique, no usaba saris, no le rezaba a Krishna, a Shiva o a Ganesh, estornudaba cuando me prendían incienso cerca.
"Tú no entiendes," seguía diciendo Jaime, "si los puertorriqueños se pasan a otras culturas, perdemos la cultura puertorriqueña."
"¿Qué tú crees que nos pasa aquí?" le replicaba. "¿Tú crees que somos tan puertorriqueños en los Estados Unidos como somos en la isla?"
"Más," insistía. "Aquí cuesta más trabajo serlo."
Veía su punto, pero no me incitaba a correr al centro comunitario más cercano a bailar plena. ¿Por qué iba a ser yo menos puertorriqueña porque bailara Bharata Natyam? Los bailarines de ballet de la isla, ¿eran menos puertorriqueños porque su arte se originó en Francia? ¿Y los pianistas que tocabar a Beethoven? ¿O la gente que leía a Nietzche? Era inútil discutir con él. Aunque le ganara, el juicio que hacía Jaime sobre mí, despiadado y consistente, me hacía cuestionarme mi lealtad hacia mi gente.
A pesar de las acusaciones de Jaime, de que las usaba de excusa, todavía definía "mi" gente como: Tata, Mami, Delsa, Norma, Héctor, Alicia, Edna, Raymond, Franky, Charlie, Cibi, Ciro. En la periferia estaban también: Papi, Don Carlos, Don Julio, La Muda, Tía Ana, Alma, Corazón y las muchas tías, tíos y primos en Nueva York y en la isla.
Desde que tenía memoria, se me había dicho que yo tenía que ser ejemplo para mis hermanas y hermanos. Era un peso tremendo, especialmente, según seguía creciendo la familia, pero asumía la tarea con seriedad, determinada a enseñarles a mis hermanas y hermanos que no teníamos que rendirnos jamás a las bajas expectativas. Para evitar el estigma de chica fácil, me vestía mona, pero conservadora. No fumaba ni bebía. Si me encontraba en una situación donde se estuviera usando drogas, yo buscaba la manera de alejarme para que no se confirmara el estereotipo de puertorriqueño tecato. Había suficientes alcohólicos en mi familia para saber que beber no era ni lindo ni divertido y que lo que los borrachos querían eliminar con el licor, no se iba nunca.
El primer puertorriqueño adicto a drogas que conocí fue Neftalí, que pagó el serlo con su vida. Quizás se sentía bien después de inyectarse ese veneno por las venas, como se sentía bien Tata cuando bebía cerveza. Desde mi perspectiva, sobria y estricta, la "nota" que cogía no valía el "bajón" que venía después y que para Tata llegaba todos los días cada vez más temprano.
Los únicos que yo conocía que usaban drogas eran los estudiantes universitarios americanos. Tenían sus sesiones de fumadera en una esquina del Manhattan Community College o se pasaban dando vueltas en grupos, todos desaliñados, por las calles del Village cerca de NYU y de los estudios de ensayo. Ofrecían "compartirla" conmigo, pero yo la rechazaba. No tenía ningún deseo de alterar mi consciencia, ni de escapar de la realidad. Si me iba en un solo "viaje," no regresaría jamás. Con obstinación, observaba cada segundo de mi vida "square," sentía cada punzada de dolor, aguantaba humillaciones, sucumbía a la alegría, me lanzaba a la pasión.
Mami me inculcó que yo tenía una sola virtud. No era la más linda de sus hijas, ni la más fuerte de su prole; pero era, decía ella con frecuencia, inteligente. Era en el poder de esa inteligencia que yo confiaba. Si esa única virtud me iba a servir de algo, mi cerebro tenía que mantenerse claro y enfocado. Mi ensimismamiento y claridad mental me mantuvieron sobria. También, me convencieron de que a pesar de la censura de Jaime, no podía ser de ninguna utilidad para "mi gente" hasta que me ayudara a mí misma.
Jaime y yo éramos demasiado profesionales para permitir que la relación tirante que teníamos fuera de escena afectara nuestro trabajo en el mundo del teatro infantil, donde se vive feliz para siempre. Pero, nunca fuimos tan apegados como había sido con Allan, que me había exigido menos y me había aceptado como era. Sin Allan en el elenco y para evitar los frecuentes reproches de Jaime, me acerqué más a Tom, el otro actor que quedaba de la producción de Broadway. Era fácil estar con él, era gracioso, un buen actor, bailarín ágil. Me aclaró desde el principio que su amistad conmigo no era tan desinteresada como la de Allan y Bill. Sin embargo, cuando le dije que tenía una relación con otra persona, me confesó que estaba enamorado de una bailarina. "Pero, tenía que intentarlo," me dijo con una sonrisa traviesa.
Para la obra nueva, _A Box of Tears_ , Robert De Mora, quien también había trabajado en _Babu_ , diseñó una escenografía espectacular y un vestuario ingenioso. Yo hacía de una sirena que se disfrazó de tortuga y quedó atrapada en las redes de un pescador. Después de una serie de aventuras, el pescador se convirtió en príncipe y yo en princesa y vivimos felices para siempre. Mi traje de sirena era todo bordado en lentejuelas verde esmeralda y provocaba aplausos del público cuando hacía mi entrada. Usaba una peluca de pelo largo, verde, hecho de un material tan finito que flotaba a mi alrededor cuando me movía, produciendo la impresión de que estábamos debajo del agua. Al convertirme en princesa, mi kimono era tradicional en su diseño y mi peluca, elaboradamente peinada y decorada.
Trajeron una consultora para que nos enseñara a movernos como japoneses, incluyendo cómo hacer correctamente la reverencia del saludo tradicional. También me hizo una demostración de cómo ponerme las tres capas de kimonos y me dio unas sugerencias de cómo caminar en zapatos japoneses sin caerme (dando pasitos bien cortos). Durante la función usábamos el maquillaje al estilo Kabuki. Llegaba al teatro dos horas antes de empezar la obra para transformarme de bailarina clásica india puertorriqueña a sirena japonesa. Primero me aplicaba en la cara una pasta blanca que me borraba las facciones. Entonces me pintaba los ojos sesgados, las cejas rectas y la boca de piquito, como la de la foto de guía que me había dado Kyoko.
En una de las escenas, tenía que actuar para el Rey del Océano en mi traje de sirena. Kyoko me enseñó cómo cantar _"Sakura"_ en japonés y coreografió un baile utilizando abanicos para contar la historia de cómo el pescador atrapó la tortuga/sirena. Después que me oyeron cantar, Bill y Vera decidieron que lo mejor era que yo moviera los labios mientras Kyoko cantaba y tocaba la _koto_.
Me encantaba la obra, las libertades tan extravagantes que De Mora se tomaba con el vestuario y la escenografía, las bromas que nos hacíamos en escena unos a otros, para tratar de desconcentrarnos. Decía mis primeras palabras fuera de escena, a través de un micrófono, mientras era todavía una tortuga en las manos de Tom, el pescador. En vez de decir mis líneas, como esperaba Tom, hice unos gorgoritos aguachosos que resonaron por todo el teatro. La primera vez que lo hice, Tom se desorientó y miró para todos lados como si la voz hubiera venido del cielo. En una escena posterior, mientras yo estaba en pleno baile de los abanicos, Tom permanecía de espaldas al público. De vez en cuando, me hacía muecas, mientras yo trataba de mantener la dignidad de un Buda.
Mis hermanas y hermanos no podían venir a las funciones porque eran durante las horas de clase, pero yo estaba disfrutándolas tanto que quería compartir mi alegría con alguien. Muy consciente de que Ulvi prefería mantener nuestras vidas privadas separadas, insistí sin embargo en que Ulvi viniera a verme actuar en la Y de la Calle 92. El enorme auditorio resonaba con el bullicio de niñas y niños que eran traídos en guaguas desde las diferentes escuelas de la ciudad. No fue hasta que me estaba maquillando que imaginé a Ulvi en el público, rodeado de inquietos, habladores y precoces escolares de la ciudad de Nueva York. Quisquilloso por naturaleza, posiblemente notaría el agrio olor que despide un salón lleno de niños. Posiblemente, montaría cara por sus voces chillonas, por su modo de corretear por los pasillos para lograr un asiento junto al mejor amigo. Con Ulvi entre el público se me hizo difícil concentrarme porque me preocupaba que el contexto en que se hacía la presentación le impidiera disfrutarla.
Quedamos en encontrarnos en su apartamento después de la función y cuando entré al apartamento ni me saludó. Me tomó en sus brazos y me amó y supe que él supo que mi actuación había sido para él. Que cada poro estaba enfocado, no en los niños y las niñas que eran el público primario, sino en el oscuro rostro malhumorado que ahora cubría mis pechos de besos.
En el Advertising Checking Bureau, la Sra. Davis me llamó a la oficina de uno de los gerentes porque tenía que hablarme en privado. Le habían informado, me dijo en su mejor voz de supervisora, que ya no era estudiante del Manhattan Community College. Mi trabajo caía bajo el programa de "Educación Cooperativa," que quería decir que recibía crédito por trabajar, pero sólo si estaba matriculada en alguna institución académica. Como no lo estaba, la Sra. Davis me sugirió que solicitara un trabajo a tiempo completo en otra división de la Compañía para ella poder contratar a otro estudiante. Por mi horario en el teatro no podía trabajar cuarenta horas a la semana, así es que renuncié al Advertising Checking Bureau. Sin embargo, según se fue acercando la Navidad, se hizo evidente que con lo que ganaba en el teatro infantil no me daba para cubrir mis gastos, aún después de haber dejado las clases y los talleres de baile. Bill y Vera me prometieron que habría más funciones en la primavera y también una gira, pero no podían asegurarme cuántas serían, en qué fechas y cuándo empezaría la gira. Justo al empezar el año, me vi forzada a dejar Children's Theater International y a buscar un trabajo de verdad. Les sollocé mi adiós a Bill, que se iba también a San Francisco, a Vera, a Tom y a Jaime. Era difícil pensar que ya no usaría más el extravagante vestido de sirena, que ya no habría más cadenas arrastrándome fuera del escenario. En el año y medio que estuve trabajando para el teatro infantil, llegué a encariñarme con las respuestas entusiastas de nuestro público: la tensión cuando el héroe o la heroína estaban en peligro, las carcajadas con que recibían las changuerías del mono-dios y el regocijo al final cuando el príncipe y la princesa aparecían en todo su esplendor, con la certeza de un futuro feliz.
Mi propio futuro no parecía muy alentador que digamos. Enero no era el mejor mes para buscar trabajo. Dondequiera que iba me decían que el negocio estaba en receso luego del ajetreo de la Navidad. Lo más que podían hacer las agencias de empleo era mandarme a hacer inventarios en las tiendas por departamentos que tenían que inventariar su mercancía antes de empezar la temporada de especiales. Era un trabajo tedioso y me fastidiaba tener que contar miles de zapatos, trajes, abrigos, que no me podía comprar ni siquiera a precio de descuento.
Shoshana llegó en mi auxilio. Había dejado el _college_ y estaba a punto de empezar un trabajo como modelo para un fabricante de vestidos y faldas talla _junior_. Habló con el dueño de la tienda de zapatos donde trabajaba cerca del Empire State Building y lo convenció de que debía reemplazarla conmigo. Pero yo no era muy buena vendiendo zapatos. Cuando una señora me preguntó cómo se le veían unas botas blancas a gogo, me incliné hacia la verdad —una virtud en la vida pero no en las ventas. El Sr. Zuckerman me sugirió que me buscara otro tipo de trabajo. Después de un montón de entrevistas en oficinas que requerirían más destrezas de las que yo podía ofrecerles, me cogieron en _Lady Manhattan_. Cuando se lo dije a Mami, se disgustó. "¿Tanta educación pa'acabar trabajando en una fábrica?" protestaba y yo le aseguraba que iba a estar en una oficina, no en un taller.
Ahora que estaba empleada de nueve a cinco, Ulvi y yo cambiamos nuestros encuentros amorosos de por la tarde a las noches y los fines de semana. Cuando no estaba con él, me encontraba con Shoshana o iba al cine o me reunía con Alma para comer. Alma acababa de empezar un trabajo como secretaria en NBC donde la mayoría de los ujieres que conocía habían progresado a posiciones mejor pagadas como asistentes de producción y escritores. Alma admiraba a su jefe de quien se decía que estaba destinado a alcanzar grandes logros en la compañía. Lo vi una vez en la oficina de ella. Era como una versión joven del señor Rosenberg, el productor del teatro yídish donde había sido ujier, solo que más nervioso.
Mi jefa en _Lady Manhattan_ era Iris, una mujer treintona de unos amables ojos verdegris, pelo corto de un castaño rojizo y un cuerpo que Shoshana describía como _zeftig_ — no gordo, pero tampoco flaco. Como asistente suya tenía derecho a mi propia oficina aparte de la de ella donde mi trabajo consistía en mantener sus archivos en orden, contestar el teléfono, estar pendiente de su calendario de citas, pedirle el almuerzo, buscarle café, atender su correspondencia. Cuando me entrevistó, Iris no me hizo una prueba de mecanografía y no fue hasta después de una semana de estar trabajando que se dio cuenta de que lo debió haber hecho. Me tomó una mañana entera escribir una simple carta de un párrafo con su duplicado. Cada vez que me equivocaba, sacaba el original y el papel carbón que tenía detrás y volvía a empezar para que los dos me quedaran perfectos. Iris le echó una mirada a la pila de papel timbrado de _Lady Manhattan_ y de papel carbón arrugado en el zafacón y sacudió los hombros.
"No te apures, déjalo," me dijo. "Yo misma lo paso."
En su oficina, Iris tenía un tablón de edictos tamaño pared en el que pegaba retazos de color para las diferentes estaciones de la moda: la anterior, la actual y las próximas dos. Su trabajo consistía en comprar las telas que los diseñadores usaban para las blusas que _Lady Manhattan_ fabricaba. Era buena en su trabajo, me dijo Iris sin que se lo perguntara, y si yo era lista y prestaba atención, podría aprender mucho de ella. Me hacía estar presente en las reuniones, supuestamente para tomar notas, pero ella me admitió que era para que fuera aprendiendo los trucos del oficio. Juntas fuimos decidiendo los nombres de los colores para la próxima estación. Yo le sugerí azul-gris, ella le llamó azul Mediterráneo. Como nunca había visto ese mar, no podía discutir con ella. Cuando yo le ofrecí "naranja oscuro", ella me respondió con "pastel de calabaza." Si yo veía azul marino, ella imaginaba "medianoche." Era obvio para todo el mundo en _Lady Manhattan_ , aunque no para la generosa Iris, que yo no gozaba del instinto poético o hiperbólico necesario para tener éxito en la industria de la ropa.
El edificio donde trabajaba quedaba a siete bloques del teatro de Broadway donde Allan trabajaba en _Fiddler on the Roof_ , en la que Henry Goz hacía el papel estelar. Allan estaba en el coro y hacía de suplente en el papel del estudiante idealista que se casaba con una de las hijas, que en ese momento lo hacía Adrienne Barbeau. Un día Allan me llamó a la oficina para decirme que esa noche iba a estar a cargo del papel y que podía conseguirme un boleto para que fuera a verlo. Llamé a Ulvi para cancelar nuestra cita. No me preguntó por qué y yo no entré en detalles.
Yo había visto _Fiddler on the Roof_ cuando Allan fue seleccionado para trabajar en la obra. Era maravilloso ver cuánto había madurado su papel. Le había añadido un aire juvenil, amuchachado, al rol romántico, una inocencia que hacía que el público se encariñara con él. Después fui a verlo al camerino y me presentó a Adrienne y a Harry Goz; a Florence Stanley, a quien había conocido en _Up the Down Staircase_ y que hacía de Yenta; a Bette Midler, que hacía de hija mayor. Después que todos firmaron autógrafos en la puerta de los camerinos, cruzamos la calle y fuimos a comer a un restaurante largo como un túnel con las paredes decoradas con fotos firmadas por los actores de Broadway. El bar estaba lleno de humo y de actores habladores, todavía excitados después de la función. En la vellonera, Diana Ross proclamaba que algún día estaríamos juntos, y el coro era repetido una y otra vez por un grupo lloroso y afligido que, en una esquina, estaba despidiendo a uno de sus miembros.
Era tardísimo cuando llegué a casa. Mami alzó la cabeza de la almohada, me saludó con la mano y se volvió a dormir. Al día siguiente en el trabajo, me sentía exhausta y cuando Iris se fue a una reunión en New Jersey, le pedí a la operadora del cuadro que atendiera las llamadas, me encerré en la oficina de Iris y dormí en el piso dos horas. Cuando cotejé, encontré varios mensajes de Ulvi. Estaba en su casa y aunque no era una de nuestras noches, insistió que fuera hasta su apartamento porque tenía que hablar conmigo. Estaba furioso, lo sentía en su voz.
"¿Qué es lo que pasa?" le pregunté, pero se negó a discutirlo por teléfono. "Ven después del trabajo," me dijo.
Me preparó comida, como hacía con frecuencia. Sus inventos eran simples —vegetales salteados con un poco de queso feta, ensalada, espinaca al vapor con un huevo pasado por agua en el medio, berenjena asada. Llegué a apreciar los sutiles y delicados sabores de los vegetales frescos que casi nunca se servían en casa y que eran básicos en la dieta de Ulvi. Esta vez preparó coliflor al vapor aderezada con generosas porciones de salsa holandesa de pote, mi contribución a su dieta. A él no le gustaba ese plato tanto como a mí, así es que me impresionó que se tomara la molestia de preparármelo. Comimos en silencio, en la mesa de centro con tope en piel. Me daba cuenta de que algo pasaba; sentía en el aire una tensión tan sólida como las cuatro paredes. Tan pronto recogimos los platos me llevó hasta el sofá-cama, se sentó en una punta y me señaló la otra, donde me senté encima de mi pierna doblada.
"Chiquita," empezó, "¿Dónde tú estabas anoche?"
Le conté de la obra, de Allan, del resto del elenco, del restaurante ruidoso y lleno de humo. Me escuchó con atención, me preguntó sobre Allan. ¿Cuándo lo conocí? ¿Dónde? ¿Era Allan mi novio antes de él conocerme?
"Ay, no," me reí, "no es así con Allan. Lo quiero muchísimo pero no de esa manera. Somos amigos."
Ulvi asintió con la cabeza, un dedo doblado cerca de los labios. "Dime, Chiquita, ¿tú tienes muchos amigos?"
"Sí," le contesté con honestidad.
Se levantó y de tres zancadas llegó a la puerta. "¡Lárgate!" Tenía tanto coraje que estaba rojo encendido.
Quedé atónita, incapaz de moverme, muda. Abrió la puerta y repitió la palabra con tal veneno que no tuve más alternativa que arrastrarme del sofá, recoger mi cartera e irme del apartamento. Tiró la puerta detrás de mí. En el ascensor, en el _lobby_ , fuera del edificio, por la Tercera Avenida, iba sostenida por la fuerza de su rabia. ¿Qué había hecho? Caminé hasta la estación del tren, esperé detrás de una columna con mis pensamientos enfocados en cada palabra que había pronunciado en su presencia. Al tramitar mi inglés a través del español, ¿se había perdido algo? ¿Malinterpretó lo que yo dije según fue traducido del inglés, al alemán, al turco? ¿O había roto yo algún tabú turco al salir una noche con mis amigos? Turquía era un país musulmán. Los seguidores del Islam no bebían. ¿Se habría ofendido porque fui a un bar? No. Tenía coraje porque yo tenía amigos varones. ¿Estaba eso prohibido para las mujeres en Turquía? ¿Fue por eso que reaccionó tan violentamente? Aguanté el llanto hasta que llegué a casa —más temprano de lo usual como bien puntualizó Mami al alzar las cejas. Me encerré en el baño, llené la bañera con agua hirviendo y estuve en remojo y sollozando durante una hora mientras afuera, periódicamente un hermano o una hermana tocaba a la puerta porque tenía que orinar.
Se acabó, sin más ni más. La mayor cantidad de palabras que Ulvi y yo habíamos intercambiado desde el día de la parada nos había llevado a nuestra primera y única pelea. ¿En realidad fue una pelea? ¿No se suponía que para pelear se necesitaban por lo menos dos? Había sido todo tan unilateral. No tuve oportunidad de defenderme. ¿De qué? ¿Había hecho algo para merecerme este trato? Su furia fue tan inesperada, tan rápida como la picada de un escorpión e igual de dolorosa. "Lárgate," me dijo. Con una simple palabra me pateó de su vida. Mientras me retorcía en la cama al lado de Delsa, gemía y me quejaba tan fuerte que Mami vino a ver qué me pasaba.
"Algo que comí," le dije, "me cayó mal." Cinco minutos después me trajo un té de manzanilla con miel. Me lo tomé a sorbitos frente a ella y de vez en cuando, contraída por los sollozos, empujaba los brazos contra el estómago para aquietar el dolor que latía, no allí sino un poquito más alto, hacia la izquierda.
# "Para ese aire de niña-esclava."
#
Iris se dio cuenta de que me veía pálida y exhausta al día siguiente. Me llamó a su oficina y me preguntó qué me pasaba. Fue imposible contener las lágrimas que acechaban tan cerca de la superficie y que brotaron en contra de mi voluntad.
"Ay, pobrecita," me dijo Iris. "Un hombre te hizo esto." Asentí con la cara escondida entre las manos. Se levantó de su escritorio y vino hasta donde estaba yo sentada y me abrazó. Me frotó la espalda, me ofreció pañuelitos desechables, me quitó el pelo húmedo de los cachetes. Pero ninguna de las palabras nacidas de su sabiduría de mujer podían quitarme la pena. "Es un perro," dijo finalmente, aunque nunca lo había visto. Asentí otra vez. Me dio libre el resto del día.
Llamé a Shoshana a su trabajo y quedamos en encontrarnos para comer. Salí de la oficina y caminé rápidito por la Séptima Avenida hasta Central Park. Era un día frío y lluvioso y mis ojos rojos, mi cara hinchada y los sollozos ocasionales pasaron inadvertidos para la gente en la calle. Al llegar al sitio donde Jurgen me había confesado a qué se dedicaba, mi dolor se volvió rabia. ¿Cómo se atrevía Ulvi a botarme de su apartamento sin ninguna explicación? ¿Qué clase de estúpida era yo que le había hecho caso? ¿Qué habría pasado si hubiera discutido con él? Un par de veces viré para ir hasta su apartamento. Pero según fui representando en mi mente lo que podía pasar, me pareció melodramático, demasiado parecido a una telenovela, demasiado cerca de lo que se esperaba de una apasionada puertorriqueña agraviada. Reprimí el deseo de matarlo y seguí caminando hasta que llegué al Metropolitan Museum of Art dejando que las lloviznas me empaparan. Me detuve frente al cuadro de Seurat donde había conocido a Avery Lee y me arrepentí de haberle dicho que no. Si lo hubiera aceptado, ahora estaría viviendo rodeada de lujo en El Paso, donde nunca llovía. Avery Lee no esperaba de mi más de lo que yo le daba a Ulvi. Y sabía inglés. Observé la pintura un largo rato pero todavía no logré encontrarle más sentido que cuando la vi por primera vez.
En la cena, Shoshana insistió que me comiera otro plato de sopa de pollo para espantar los mocos y el lloriqueo que ya no eran causados por las lágrimas de Ulvi sino por el catarro que había cogido caminando en la lluvia. Cuando finalmente le conté la historia, ya no me dolía tanto hablar de él. "Esto se te va a pasar pronto," predijo. "Tú no eres de las que sufren por mucho tiempo." Tenía la cabeza tan pesada que no podía pensar con la rapidez necesaria para estar o no de acuerdo con ella. En el _subway_ , camino a Brooklyn, anoté sus palabras en un pedazo de papel y lo guardé doblado dentro del monedero. No, yo no era de las que me aferraba al dolor mucho tiempo. ¿Para qué? Ya vendría por ahí un nuevo revés.
Falté tres días al trabajo. Tata y Mami me cuidaron dándome caldos y el temido _tutumá_ que no sabía mejor ahora, que tenía veinte años, que cuando tenía trece y Mami lo inventó. Dormía arrullada por el rrorró de las máquinas de coser. Mami tenía ahora un negocio en casa. De la fábrica, traía ropa en piezas y entonces ella, Titi Ana y otras mujeres terminaban de montarla en las máquinas de coser que estaban en la sala. Tata atendía a Charlie, Cibi y Ciro que a veces venían a la fábrica improvisada buscando la admiración, y los arrullos y arrumacos de las mujeres.
El sábado por la mañana me sentía mejor pero me quedé leyendo en la cama. Shoshana me llamó para saber cómo me sentía. Cuando se fue a despedir me dijo la verdadera razón de su llamada.
"No te quise decir nada el otro día, que te sentías tan mal..." Regresaba a Israel para cumplir con el Servicio Militar que había estado posponiendo. Estaría fuera varios meses. "Si los árabes no hacen alguna locura, a lo mejor regreso," bromeó, pero yo no me reí. No tenía mucho tiempo, pero quedamos en encontrarnos para una última cena juntas. Perder a mi mejor amiga a la vez que perdía a mi amante me devolvió a la cama un día más, pero el lunes me arrastré a trabajar. Había un montón de mensajes encima del escritorio, la mayoría para Iris pero algunos para mí. "Ulvi llamó," había escrito la recepcionista que atendía las llamadas en por lo menos cinco papelitos rosa. En el último había escrito "¡Urgente!" y al lado en paréntesis, "Me pidió que escribiera esto," con una flecha señalando la palabra. No lo llamé. Cada vez que miraba los papelitos me acordaba de su furia, del gesto cruel en sus labios cuando me pidió que me largara de su apartamento.
Me reuní con Shoshana para cenar y después caminamos por la Quinta Avenida como habíamos hecho tantas veces hasta el Plaza, intentando volver sobre nuestros pasos hasta Grand Central y los _subways_. Pero frente a la fuente se nos acercó un hombre. Era un productor, nos dijo. Estaban proyectando su película en el Paris Cinema al cruzar la calle. Nos preguntó si nos gustaría verla.
Era una impenetrable película en blanco y negro en una lengua eslava que ninguna de las dos podíamos identificar. Los subtítulos tampoco nos ayudaron. Shoshana y yo nos reímos durante toda la película mientras los amantes del cine serios nos mandaban a callar y el pobre productor subía y bajaba nerviosamente por los pasillos, escudriñando las caras del público, tratando de identificar a las atrevidas que osaban reírse de su obra magna. Sin poder aguantar más la risa, salimos corriendo de allí tan pronto empezaron a pasar los créditos. Mientras caminamos tomadas del brazo hasta Grand Central, supimos que nuestros días de libertad y aventuras terminaban esa noche. Pasarían meses antes de que Shoshana regresara a los Estados Unidos y para entonces quién sabe dónde estaría yo. Nos abrazamos y Shoshana prometió escribirme tan pronto llegara a Haifa. No le recordé que siempre que se iba a Israel prometía escribirme pero nunca lo hacía. Minutos después de despedirnos, sentí la pérdida de la mejor amiga que había tenido, la única persona, creía yo, que realmente me conocía.
No había olvidado a Ulvi; pero en los cinco días que habían pasado después que me tiró la puerta, se había convertido en un dolor parecido al que deja una cortadura. La mayor parte del tiempo no lo sentía hasta que me tropezaba con algo. El día después de mi cena con Shoshana, contesté el teléfono de Iris.
"¿Chiquita?" Su voz suave, vacilante me mareó un poco y mi instinto inicial fue colgar el teléfono. Primero se disculpó por su comportamiento de la semana anterior. Después me dijo que quería verme porque tenía algo que explicarme. "Quizás tú no entiendas," conjeturó "por qué estoy molesto."
El uso del presente no me desconcertó porque Ulvi con frecuencia confundía el presente con el pasado. Insistí en que nos reuniéramos en un restaurante y no en su apartamento. "Si quieres hablar," le dije "es mejor así." Cuando lo vi frente al _Magic Pan_ en East 57 Street por poco me tiro en sus brazos pero me contuve, dejé que me besara el cachete y me separé antes de que fuera a flaquear en mi determinación.
Pensó sobre lo que había pasado la semana pasada. "Tú eres una niña," dijo. "Lo olvido a veces."
Le recordé que tenía veinte años y medio, no diez. Sonrió indulgente. "Para mí, tú eres mi Chiquita," dijo. "Siempre."
Tenía coraje esa noche, me dijo, porque teníamos una cita y yo la había cancelado a la última hora para estar con otro hombre. Le expliqué que Allan no era "otro hombre" en el sentido que yo creía que lo entendía Ulvi; sino un amigo muy querido. La razón para haber cancelado a última hora fue que Allan, como actor suplente, no siempre sabía con anticipación cuándo tendría que actuar.
"¿Por qué no me dijiste?" preguntó Ulvi.
"Porque no me lo preguntaste. Nunca lo haces. Tú no quieres saber nada de mi vida privada, ¿recuerdas?" Fue imposible disimular el resentimiento que había en mi voz, el sarcasmo que se filtró en las últimas tres sílabas. Ulvi se encogió. Permanecimos sentados en silencio unos minutos. Lo sentí luchar con su respuesta. No había posibilidad de que borrara con besos lo que sentía, allí, en un restaurante repleto, frente a comida francesa de embuste. Ni podía tranquilizarme con promesas de que nunca me lastimaría. Ya lo había hecho. Le tomó mucho rato formular lo que quería decir. Él pensaba que era el único hombre en mi vida. Le inquietaba que yo me sintiera libre para ver otros hombres.
Le contesté que nunca habíamos hablado de nuestra "relación" en términos que me llevaran a pensar que no podía salir con otros hombres. Si eso lo hacía sentir mejor, le aseguré, nunca había tenido relaciones sexuales con otro hombre que no fuera él. El alivio que cubrió su rostro me sorprendió. ¿Qué esperaba?
Salimos del restaurante, le dimos la vuelta a la manzana y vinimos a parar a su calle. Se sonrió, me atrajo hacia él, me besó y yo me disolví dentro de mi pesado abrigo de invierno. Sus brazos se sentían familiares, sus labios como los míos. Tenía la estatura perfecta, yo no tenía que estirarme, ni agacharme para descansar la cabeza en su hombro, o para que él me pasara el brazo por la cintura. Caminamos con el mismo paso decidido, nuestros pies tocaban la acera a la vez, sincronizados por un mecanismo interno que ninguno de los dos controlaba. Hacer el amor fue danzar; cada parte de nuestro cuerpo en armonía con su complemento en el otro, como si no fuéramos dos, sino uno. Después, mientras reposaba satisfecha en sus brazos, me dijo que quería tenerme siempre a su lado. No fue una promesa, ni una proposición de matrimonio ni una declaración de amor pero yo entendí todas esas cosas.
Durante los días y las noches que siguieron, nos acercamos más. A regañadientes Ulvi se fue abriendo y me habló más de sí mismo y de su obsesión con la película _Dry Summer_. El éxito obtenido lo había sorprendido porque en Berlín, la favorita para primer premio había sido _The Pawnbroker_ con Rod Steiger. "Nadie espera que yo gane," rió, "ni yo." Se hizo famoso de un día para otro, viajó alrededor del mundo para asistir a festivales y competencias, hizo dinero. Compró un Rolls Royce blanco que guió de Nueva York a Hollywood. Allí conoció a Kim Novak y a Angie Dickinson. El productor de Hollywood, Sid Solow le prestó su casa de huéspedes por un par de semanas en lo que lograba cerrar un trato para exhibir la película. Pero a pesar de sus esfuerzos, _Dry Summer_ no encontró un distribuidor en América del Norte. Donde quiera que iba, le decían que la película era hermosa pero que si quería presentarla en los Estados Unidos necesitaba añadirle más sexo y mejor música. Vendió el Rolls Royce e invirtió el dinero en la película. Contrató al compositor griego Manos Hadjidakis que había escrito la música de _Never on Sunday_ , para que le compusiera una música nueva para _Dry Summer_. Ulvi encontró una muchacha que se parecía a Hulya Kocigit, su interés romántico en la película. Viajó con un cinematógrafo hasta Long Island para filmar escenas nuevas. Ahora estaba reeditando la película para incorporarle las escenas de sexo.
"¿Hulya accedió a eso?" pregunté.
"Es una estrella famosa en Turquía ahora. No tiene tiempo."
La _suite_ de edición quedaba en el segundo piso de un edificio desvencijado a medio bloque del Woolworth's donde Ulvi me había visto por primera vez. Su editor era un hombre mayor, de piernas largas, de pelo blanco, ojos tristes y un rostro de arrugas profundas que apenas sonreía. Hans me recordaba a Bela Lugosi, tanto en su físico como en su modo de hablar con su acento pesado. Trabajaba con una Movieola vertical, sus dedos volando de la máquina de edición al cenicero que tenía al lado. En el cuarto de atrás había otra Movieola que se le había alquilado a otro cineasta.
Todos los días después del trabajo, me reunía con Ulvi en el cuarto de edición a un par de bloques de mi oficina. Cuando terminaba, nos íbamos hasta el apartamento, comíamos, dábamos largos, y helados paseos por la Quinta Avenida a través de Central Park y hablábamos —o mejor dicho, hablaba él. Yo atesoraba cada palabra suya, su entonación, las pausas y vacilaciones de su hablar. Empezaba cada una de sus muchas confidencias con "Esto yo no lo digo a nadie, Chiquita," lo que me hacía sentir incluida en su vida, conocedora de sus secretos.
Ulvi contrató un escritor para que le hiciera los subtítulos nuevos a la película. Se preocupaba de que el dinero que le quedaba se le estuviera yendo como agua entre los dedos en su afán de que la película le resultara atractiva a los norteamericanos. Por las noches, después que me acompañaba a la estación del tren, se iba hasta el apartamento de Manos, en los altos del restaurante Acropolis en West 57 Street, para trabajar en la partitura. Según Ulvi, a Manos no le gustaba componer de día, así es que sus sesiones de trabajo empezaban después de las once de la noche y terminaban en la madrugada. Era un horario extenuante pero Manos sostenía que su creatividad llegaba a su mayor intensidad tarde en la noche. Como Manos se había ganado un premio de la Academia con _Never on Sunday_ , Ulvi sentía que tenía que complacerlo. Tenía la esperanza de que la partitura de Manos aumentará el interés en su película.
Cuando le expresaba preocupación por su salud debido a lo mucho que trabajaba, Ulvi me lo agradecía pero me decía que no tenía otro remedio. "Esta es mi única oportunidad, Chiquita," me decía en tono de confidencia.
Durante las semanas siguientes, mis días, noches y fines de semana fueron consumidos por Ulvi. No pasaba tiempo con nadie más, ni con mi familia, ni con mis amistades, ni con mi prima Alma. Tenía dinero de nuevo para pagar mis clases de baile pero las dejé después de que Ulvi y yo fuimos a ver una película de Satyajit Ray.
"Esa es la clase de baile que hago," le dije refiriéndome a la secuencia de Bharata Natyam al empezar la película.
"Es baile ridículo" fue la opinión de Ulvi. "No para ti."
La próxima vez que asistí a un taller de baile, me miré en el espejo, consciente de los movimientos estilizados, las expresiones faciales afectadas, la música atonal. Me veía ridícula en mi sari, con campanas en los tobillos, el punto rojo en el medio de la frente.
De ahí en adelante, le dediqué cada minuto libre a él. Mientras Ulvi trabajaba en su película yo leía en una esquina del cuarto de ediciones. A veces me mandaban a buscar café o almuerzo o a recoger o a entregar algún paquete. Johan, que rentaba la otra Movieola que había en la _suite_ , me pidió que le tradujera su película. Él y su hermano Fritz habían documentado una expedición arqueológica en la jungla Colombiana. Muchas de las escenas eran en español, un idioma que ni John ni Fritz entendían. Traduje al inglés las escenas que eran en español, y entonces traduje toda la película al español para que tuvieran una versión en cada idioma.
Mientras él y Hans editaban, vi escenas de un joven y ardiente Ulvi pero nunca vi la película entera de principio a fin. Las escenas de los desnudos fueron hábilmente tomadas, una en un maizal y la otra frente a un telar que tenía colocada una alfombra tensada a medio terminar. La actriz se parecía lo suficiente a Hulya para que a través de un manejo hábil de la luz y de las posiciones de su cara, las transiciones fluyeran con suavidad aunque no perfectamente.
En la oficina, hubo muchos días en que no logré inventar evocadores nombres para los colores primarios. No podía pasar una carta a maquinilla sin desperdiciar diez hojas de papel. El itinerario tan ocupado de Iris, hacía que con frecuencia me quedara sola en la oficina, agobiada por el aburrimiento, rogando que el teléfono sonara para poder tomar algún mensaje. Me sentía mal de estar cobrando un salario cuando no tenía nada que hacer así es que decidí mejorar mis destrezas secretariales. Compré el libro _Teach Yourself the Gregg Shorthand Method_ pero, entre el español, el inglés, el espanglés, el francés de escuela superior, el turco que iba y venía en el Movieola, y el alemán que Ulvi, Hans y Johan hablaban entre ellos, no me quedaba espacio en la cabeza para otro idioma.
Mami ya no me preguntaba dónde había estado, con quién, que había estado haciendo. Era como si, con los otros diez hijos que tenía que cuidar, mis actividades fueran para ella de poca importancia mientras yo volviera a casa todas las noches. Apenas veía a mis hermanas y hermanos porque procuraba irme de la casa lo más temprano posible y regresar mucho después de que todo el mundo se hubiera acostado. Un domingo, Ulvi no podía verme por que tenía que visitar unos amigos en Long Island. Había planificado quedarme en casa ese día pero dos horas después de levantarme, salí de la casa sofocada por el revolú, el desorden, la confusión, el entra y sale de gente corriendo de un cuarto a otro, subiendo y bajando la escalera. Anhelaba el silencioso y austero apartamento de Ulvi, su orden sistemático que ya no me parecía siniestro, sino reconfortante. Sin embargo sin él allí, no podía ir al apartamento. Me pasé el día en el cine, vi la doble tanda dos veces y regresé a casa a la misma hora de siempre.
Ulvi me sorprendió un día. Me invitó a cenar en un restaurante caro porque quería celebrar que había encontrado una gente que se había interesado en su película. Habían filmado un par de películas de poco presupuesto en Italia y estaban interesados en Ulvi como director para otra película que querían producir. Ulvi me dijo que estaban impresionados con su reputación como director de películas de arte, una imagen que ellos también estaban tratando de crearse.
"Es una buena posibilidad," dijo Ulvi como restándole importancia, pero yo sabía que se sentía aliviado. Le había quitado el peso de tener que financiar él solo cada aspecto de _Dry Summer_ , justo cuando se estaba quedando sin dinero. "Me queda suficiente para un mes" me dijo y yo me pregunté que habría pasado si no hubieran aparecido los socios.
Cuando regresamos al apartamento, Ulvi me tomó la mano y me la besó. Me pidió que cerrará los ojos y me enganchó una pulsera en la muñeca. Era una malla de oro pesado, como de una pulgada y media de ancha con un trenzado de oro en las orillas. Quedé sin habla, cortada por la extravagancia del regalo.
"¿Por qué es esto?" gagueé.
"Has sido una niña buena," murmuró.
"Pero es demasiado caro para ti."
"No te preocupes," me dijo.
Más tarde, desnuda excepto por la pulsera, moví la muñeca de aquí, para allá, para ver el reflejo del oro contra mi piel. Me pregunté cuánto costaría la pulsera, si cientos o miles de dólares.
"Me siento rara aceptándola cuando sé que necesitas el dinero," le dije apenada.
"Dije no te preocupes," respondió molesto.
De regreso a casa esa noche me halé la manga del abrigo sobre la pulsera para que nadie en el tren la viera y me la fuera a robar. Como sabía que Mami se fijaría, le inventé una historia de una amiga que quería venderla. ¿Valdría los veinticinco dólares que me pidió? A Mami le pareció que el precio era más que razonable.
Me ponía la pulsera para ir a todas partes, con ropa casual o de vestir, porque a Ulvi le gustaba vérmela puesta. Cuando se la enseñé a Iris me dijo que le recordaba los grilletes. "Para ese aire de niña-esclava" añadió.
En las semanas siguientes Ulvi me hizo otros regalos caros, una agenda de piel de Hermés, un llavero de plata en forma de corazón de Tiffany's. También empezó a mostrar interés en la forma en que me vestía. Insistía en que no me maquillara cuando estábamos juntos. "No me gustan las mujeres pintadas," decía. Me acompañaba a comprar ropa. Antes de hacer películas, había trabajado de ingeniero textil en Alemania y era muy exigente con lo que le tocaba la piel. Cuando yo escogía alguna pieza, la frotaba entre los dedos, se la pasaba por la palma de la mano, viraba la pieza al revés para saber si el diseño estaba estampado o tejido en la tela y para examinar las terminaciones de las costuras. Descartaba la mayor parte de mis selecciones. "Esto es para chica barata," decía despectivamente y escogía otra cosa. "Este mejor para ti."
"Chica barata" era su mayor insulto, exactamente lo opuesto a la "chica elegante," que vestía bien y se comportaba apropiadamente de acuerdo a un complicado sistema de reglas de etiqueta y modales que Ulvi juraba que yo tenía que dominar. "Si vas a estar conmigo, tienes que aprender."
Quería estar con él, así es que presté atención a sus lecciones. Cuando salíamos, yo tenía que imitar cada uno de sus movimientos para no pasar una vergüenza. Comería si él comía, con los cubiertos que él usara, hablaría menos, escucharía más, me reservaría mis opiniones. Me hizo estar consciente de mis limitaciones y me prometió ayudarme a superarlas. "Tú eras pobre niña con mente estrecha," me dijo una vez y lo repetía con frecuencia. Cuando se daba cuenta de que me había ofendido, me explicaba que lo que quería decir no era que yo fuera estúpida sino poco sofisticada porque me habían sobreprotegido.
"Es lo que me encanta de ti, Chiquita," me dijo. "Te puedo enseñar todo." Quería ser Pigmalión y yo me convertí en la piedra en que esculpió a Galatea. Cuando sentía que me estaba controlando demasiado la vida, me quejaba pero él me acallaba con una caricia y una promesa. "Estarás junto a mí pero tienes que hacer lo que yo te diga." Para estar con él tuve que desechar lo que yo era para convertirme en la mujer con quien él quería estar. "Tengo miles de novias," se jactaba, "pero tú eres la única que me importa." Fue lo más cerca que llegó de decirme que me quería, pero para mí fue suficiente.
Poco a poco me presentó alguna gente de su vida. Cada encuentro era una prueba que tenía que pasar para moverme al próximo nivel. Primero conocí a Hans, Johan y Fritz y me comporté bien en el sofocante cuarto de edición. Entonces me presentó a Bruce, el escritor que lo ayudó con los subtítulos y a su delicada esposa, Diana. Peter, el camarógrafo iraní que filmó las escenas de sexo, y su esposa Bárbara fueron los próximos. Cuando me presentó a Tarik, el hombre a quien llamaba su mejor amigo, supe que confiaba en mí. Cada pedacito de su vida que me permitía compartir me sabía a victoria porque me había ganado el derecho a estar con él, a su lado.
Manos terminó la mayor parte de la partitura y pensaba grabarla con un grupo de músicos adiestrados en Julliard que actuaban bajo el nombre de The New York Rock and Roll Ensemble. Llegué al estudio de grabación y conocí a Manos por primera vez. Era enorme, tenía una sonrisa cautivante; manos pequeñas de dedos cortos y gordos; risueños ojos ébanos.
Después de las dos primeras sesiones en el estudio de grabaciones, Ulvi me dijo que no volviera. Los músicos y sus novias fumaban pasto continuamente y a Ulvi le preocupaba que estuvieran usando otras drogas. "No te quiero cerca de eso," dijo y yo le agradecí su preocupación.
El día que me dijo su edad entendí por que la había mantenido en secreto tanto tiempo. Tenía treinta y siete años, la misma edad de Mami, diecisiete años más que yo. Había estudiado psicología en Manhattan Community College y estaba consciente de que Ulvi era el clásico sustituto del padre, pero no me importaba. Me cuidaba como nadie me había cuidado. En sus brazos me sentía segura y protegida. Arropada en su abrazo no tenía mayor responsabilidad que hacer lo que él dijera. "No te preocupes", me tranquilizaba, "yo me hago cargo de todo." Él tenía claro lo que esperaba de mí. A diferencia de los demás adultos en mi vida, no decía una cosa y hacía otra. Si no quería que yo bebiera era porque él no bebía. Si se oponía al cigarrillo, él no fumaba. No quería niños, él se hacía cargo de que yo no saliera encinta.
Él necesitaba una discípula; yo necesitaba que me guiarán. Sentía que me sumergía como una piedrita en un lago. Sin resistencia, sin rastro alguno de que hubiera estado en otro sitio, de que hubiera sido una persona, sin él. Con Ulvi ya no era la hija de un padre ausente, la mayor de once hijos, la modelo para diez hermanas y hermanos, la traductora de mi mamá. No era Esmeralda fracasada actriz/bailarina/secretaria. Con mi cabeza recostada en el pecho de Ulvi, mis brazos alrededor de su cuello, yo era lo que había dejado de ser el día que me monté en el avión de hélices en Isla Verde para salir a la lluviosa noche de Brooklyn. Después de siete años en los Estados Unidos, me había convertido en lo que había dejado de ser cuando dejé Puerto Rico. Me había convertido en Chiquita —pequeña, niña, niñita.
# "Así tiene que ser."
#
En abril, Ulvi y yo dimos largos paseos en Central Park por caminos que el parecía conocer íntimamente. "Yo corro por aquí," me dijo, lo que me sorprendió porque nunca me había dicho que corría. De vez en cuando le gustaba dejar los caminos y andar por la hierba, sus ojos escudriñando el verdor en busca de un trébol de cuatro hojas. Me impresionaba que siempre encontraba alguno, lo arrancaba y lo aplastaba entre los pliegues de un dólar. Más tarde en el apartamento, lo pegaba con cinta adhesiva y luego lo recortaba cuidadosamente. Se los mandaba a los amigos, me dijo, y una vez me regaló uno a mí. "Te traerá buena suerte," me aseguró. Lo guardé en mi monedero, como me había indicado, pero no noté ninguna diferencia en mi fortuna.
"Quizás soy inmune a la buena suerte," bromeé, pero no lo cogió.
Nuestros paseos por el parque acababan siempre en el zoológico, donde visitábamos las tristes criaturas enjauladas que caminaban de un lado a otro con la misma tenacidad que hubieran mostrado si realmente hubieran ido para algún sitio. Nos deteníamos para mirar a las focas deslizarse en las aguas turbias, su piel brillante. Muchísima gente se arremolinaba por el estanque cuando las alimentaban, pero a mí, sus monerías a cambio de un pescado muerto, me parecían lastimosas.
Shanti me había tomado numerosas fotografías en el zoológico, cerca de la jaula de los monos o en un banco rodeada de palomas que una viejita alimentaba de un cubito plástico que tenía al lado. Cada vez que pasaba por esos lugares recordaba los ojos marrón-Crayola de Shanti, su modo de inclinar la cabeza para mostrarme cómo debía colocar la mía.
Cuando Ulvi y yo caminábamos alrededor del lago, recordaba la mirada de satisfacción de Jurgen cuando levantaba los remos, los metía en el agua, los traía hasta su pecho para impulsarnos hasta cruzar el lago. A Ulvi le gustaba sentarse en los recibidores de los hoteles elegantes como el Plaza, el St. Regis, el Waldorf. No le conté de las cenas de Shoshana y mías en los restaurantes de esos hoteles, ni de los hombres solitarios dispuestos a gastar su dinero y su tiempo en un par de muchachas ávidas de compañia masculina.
El restaurante de Rockefeller Center donde Jurgen y yo hablamos por primera vez, todavía servía comidas caras bajo sombrillas brillantes, pero no le conté a Ulvi que una vez me senté bajo su sombra a llorar en el hombro de un desconocido porque Avery Lee me había pedido que fuera su amante. Ulvi, que hablaba como si yo nunca fuera a recordar una palabra de lo que me decía, no me preguntaba nunca sobre mí. Mientras menos interés mostraba en mi vida, más avergonzada de ella me sentía, avergonzada de tener una vida antes de él, una vida sin él.
Un par de semanas después que cumplí los veintiún años, estábamos caminando por el parque cuando Ulvi se encogió de dolor. Lo ayudé a llegar a un banco cerca y estuvo sentado un rato, pero rechazó mi ofrecimiento de acompañarlo a una sala de emergencias. "No es nada Chiquita. Yo he tenido esto antes," me dijo y yo no lo presioné más, pero insistí en que tomáramos un taxi hasta su apartamento. Se acostó un ratito y me dijo que se sentía mejor. Antes de irme, me prometió que vería a un médico.
"Necesito operación," me dijo unos días más tarde.
Las únicas personas cerca de mí que habían necesitado una operación, eran Raymond, cuyo pie herido y las subsiguientes operaciones habían sido la causa de nuestro viaje de ida a Brooklyn, y Francisco, el amor de Mami, el papá de mi hermano Franky. Las operaciones de Raymond le habían salvado el pie. Francisco había sufrido innumerables procedimientos que no pudieron salvarlo. Instintivamente, hice caso omiso de los éxitos y me enfoqué en los fracasos de la medicina. Al igual que Francisco, Ulvi entraría a la sala de operaciones, y ya no lo vería más. Yo no era fuerte como Mami, no podría sobrevivir a los meses de negra desesperación.
"No esté tan asustado, Chiquita," Ulvi me tomaba en sus brazos y me abrazaba fuerte mientras yo sollozaba en su pecho. "Es sólo una operación de hernia. Nada de preocupar."
Pero, no pudo convencerme. Mi cabeza se llenaba de imágenes de Ulvi muerto. Ulvi, un fantasma que me rondaría para siempre como Francisco había rondado a Mami. Me tomó mucho tiempo calmarme, y entonces, Ulvi me dijo el resto. Era cirugía menor, no estaría en el hospital más de un par de días, pero no tenía seguro médico y el dinero se le había acabado. Los distribuidores que querían lanzar _Dry Summer_ hicieron los arreglos de hospital y con un médico que le haría la operación gratuitamente, en Fort Lauderdale.
"¿Por qué tan lejos?" me quejé.
"Así tiene que ser." Esperó a que yo empezara a discutirle y cuando vio que no lo hice, continuó. A pesar de que estaba sufriendo, quería terminar la película antes de irse de Nueva York. Había que añadirle los subtítulos, y hacer el _remix_ para incorporarle la música de Manos y otros efectos de sonido. Necesitaba un par de semanas y entonces volaría hasta Fort Lauderdale para operarse. No sabía cuando o si regresaría a Nueva York. Habló en tono práctico, en su inglés simple y declarativo; cada palabra cuidadosamente escogida. Estaba sentada con las piernas dobladas debajo de mí, con las manos apretadas entre mis muslos. Hacía tiempo que temía una conversación como ésta, que sabía que un día Ulvi saldría de mi vida con la misma rapidez con que entró. Me alegraba que no fuera a morir, que sólo se iba a la Florida. Según hablaba, yo me forcé a retraerme hasta que estuvimos, no en los dos extremos de su sofá de cuero negro, sino en dos continentes diferentes.
Cuando terminó de exponerme sus planes, Ulvi se hundió aún más en su esquina del sofá, juntó las manos, tocó sus labios con las puntas de los dedos y dijo bajito, tan bajito que tuve que esforzarme para oírlo. "Puedes venir conmigo."
Había esperado, deseado esas palabras, segura de que nunca las oiría, aliviada cuando las dijo. Me sorprendí entonces, cuando mi respuesta fue que mi mamá nunca me dejaría ir.
"Entonces, debes dejarla," sentenció.
No había modo de explicarle a Ulvi, que no conocía a Mami, por qué la idea de dejar a mi mamá para irme a Fort Lauderdale con mi amante me aterraba. Él no había estado cuando ella se me apareció en Long Island, en medio de una tormenta de nieve, para rescatarme de que tuviera sexo con Otto. No había escuchado el dolor en su voz cuando lamentaba haber dejado incompleta su educación, haber sido madre joven y soltera, los hombres que la traicionaron. No había estado con ella en la oficina del _welfare_ , no había estado allí, serio y asustado, mientras ella se humillaba ante una gente que conquistaría su orgullo porque nunca podría derrotar su espíritu. Nunca había recostado su cabeza en su falda, nunca la había escuchado cuando revelaba sus sueños para sus hijos, que ella esperaba fueran más listos en la vida de lo que había sido ella. No le había visto el rostro iluminado cuando pensaba en mí, su hija mayor, vestida con un traje de novia blanco, camino a la Catedral.
"Quizás, si nos casamos," le sugerí, patética hasta para mí misma.
Ulvi, movió la cabeza, "No, no nos podemos casar."
No dio explicaciones para su negativa y yo no las pedí. Estaba sombrío, paciente. Sus ojos me miraron con la misma intensidad con que me habían mirado hacía unos meses, cuando Shoshana y yo estuvimos en esa misma sala, tratando de impresionarlo para que nos diera un papel en su película. Ésta era —lo sabía— una prueba de lealtad. Si me negaba a seguirle a Florida, fracasaría.
Durante los siete meses que nos habíamos conocido, le había cedido mi voluntad a él. Había dejado de ver a mis amistades, había dejado de ir a bailar, salía corriendo del trabajo a sus brazos. Pero todavía, me iba todas las noches a dormir bajo el techo de Mami. Sin decir las palabras, Ulvi me estaba pidiendo que la dejara también, que escogiera entre los dos.
En todo el tiempo que habíamos sido amantes, no se me ocurrió nunca que alguna vez tuviera que escoger. Algún día, Ulvi regresaría a Turquía, o a Alemania, o quién sabe y a quién le importa dónde. Sería Ulvi el que saldría de mi vida, no Mami. Después de años de ver a Mami, a La Muda, a mis tías y primas —cómo amaban, fracasaban, volvían a amar—, había aprendido que del amor una se recuperaba. Si Ulvi se iba, vendría otro hombre, pero nunca habría otra Mami.
"Piénsalo," me recomendó Ulvi, cuando no le contesté enseguida. Por primera vez desde la primera vez, me fui de su apartamento sin quitarme la ropa ni una vez. Cogí el tren a Brooklyn. El aire pesado y polvoriento de los _subways_ era sofocante, me hacía imposible respirar, nublaba mis pensamientos hasta que no supe dónde estaba, a dónde iba ni por qué. El tren expreso volaba por Nostrand Avenue, Utica Avenue, Broadway-East New York. Me bajé del tren y cambié al tren local para hacer una parada a Liberty Avenue, a un bloque de la casa de Mami. Era tarde, pero más temprano de lo que usualmente regresaba. Mami y Tata habían cocinado un ollón de arroz con pollo y habichuelas guisadas.
"Ay, Negi, llegaste temprano. Qué bueno," dijo Mami. "Déjame que te sirva."
Estaba contenta porque era sábado y ella cobraba los viernes, lo que quería decir que había una compra generosa en la alacena. Las máquinas de coser de la sala estaban calladas, cubiertas con sábanas para que los nenes más chiquitos no las tocaran. Héctor, Raymond y Franky estaban en el patio de cemento jugando con una bola. Un perrito me mordió los tobillos. ¿De dónde había salido? ¿Era nuestro? ¿Cuánto tiempo lo habíamos tenido? Me puse ropa cómoda de estar en la casa y me senté con Mami y con Tata en la cocina y las tres comimos su comida sabrosa, mientras en el otro cuarto la televisión vociferaba un espectáculo de variedades.
Mis hermanas y hermanos se tiraban en el piso o en los muebles cubiertos de plástico y se reían y se burlaban de la ropa y de las artistas estrambóticas. Uno de los bebés lloró, otro chilló, el perro ladró, Tata prendió un cigarrillo y abrió una cerveza. Mami le gritó a Edna que cogiera a Ciro para que dejara de llorar. Me levanté, puse los platos en el fregadero y me refugié en el cuarto que compartía con Delsa, en la cama que compartía con Delsa. Arropada hasta la cabeza para aislar el ruido, la confusión, el drama de la vida de mi familia, supe, como lo sabía Ulvi cuando preguntó, que mi decisión ya estaba tomada.
# Reconocimientos
#
Esto es lo que recuerdo, como lo recuerdo. Frases memorables, confesiones apremiantes y preguntas fascinantes han contribuido a las conversaciones recreadas en algunas escenas.
Los nombres de mi familia inmediata son verdaderos pero las circunstancias me han llevado a cambiar otros. Por ejemplo, hasta ahora, ha habido doce Franks y cinco Normas en mi vida. Si bien es cierto que yo puedo distinguirlos, es más difícil lograrlo en la página. Para evitar confusiones les he puesto un apodo o les he cambiado el nombre.
También está la gente cuyos nombres no recuerdo. Algunos podrían ser personajes secundarios en una novela, pero en la vida real si se recuerdan no son secundarios en absoluto. Les pido perdón a los que se reconozcan pero tengan un nombre diferente en estas páginas. Por favor, comprendan que aunque haya olvidado sus nombres todavía los recuerdo a ustedes.
Varias personas me han ayudado a darle forma a este libro. Estoy particularmente en deuda con mi editora Merloyd Lawrence, cuya confianza y estímulo son los mayores motivadores que cualquier escritor o escritora pueda desear. Las veces que me sentí abrumada por la emociones que esta memoria me hizo evocar, llamaba a mi amiga y agente Molly Friedrich, cuya seguridad y confianza me ayudaban a mantenerme en ruta. Mis compañeros y compañeras en la sanadora tarea de escribir, Terry Bazes, Ben Cheever, Joie Davidow, Audrey Glassman, Marilyn Johnson y Mary Breasted, generosamente dejaron a un lado su propio trabajo para leer este manuscrito en sus diferentes etapas. Mil gracias, queridos amigos.
El clan Santiago/Cortez/Martínez me ha sostenido con su benevolencia y cariño, aunque a veces esté en desacuerdo con mi versión de los hechos. Individual y colectivamente, mi mamá, mi papá, mis hermanas y hermanos, son una manifestación de lo que respeto y dignidad significan para una familia puertorriqueña.
Y finalmente mi esposo Frank Cantor, mi hijo Lucas y mi hija Ila, han logrado entender cuándo necesito estar sola y cuándo necesito un abrazo. Ustedes me hacen cantar. (Pero no se preocupen, no lo voy a hacer en público.)
# TAMBIÉN DE ESMERALDA SANTIAGO
CUANDO ERA PUERTORRIQUEÑA
La historia de Esmeralda Santiago comienza en la parte rural de Puerto Rico, donde sus padres y siete hermanos, en continuas luchas los unos con los otros, vivían una vida alborotada pero llena de amor y ternura. De niña, Esmeralda aprendió a apreciar cómo se come una guayaba, a distinguir la canción del coquí, a identificar los ingredientes en las morcillas y a ayudar a que el alma de un bebé muerto subiera al Cielo. Pero precisamente cuando Esmeralda parecía haberlo aprendido todo sobre su cultura, la llevaron a Nueva York, donde las reglas —y el idioma— eran no sólo diferentes, sino también desconcertantes. Cómo Esmeralda superó la adversidad, se ganó entrada a la Performing Arts High School y después continuó a Harvard, de donde se graduó con altos honores, es el relato de la tremenda trayectoria de una mujer verdaderamente extraordinaria.
Autobiografía/Estudios Latinos
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Tobias Erler (17 de maig de 1979) va ser un ciclista alemany, professional del 2006 al 2013. Els seus principals èxits els aconseguí en proves del calendari de l'UCI Àsia Tour.
Palmarès
2003
Vencedor de 2 etapes al Tour de Taiwan
2005
Vencedor de 4 etapes al Tour de Taiwan
2006
1r al Tour de Corea i vencedor d'una etapa
2007
1r al Rund um den Sachsenring
2010
Vencedor de 2 etapes a l'International Presidency Tour
2011
1r al Tour de Tailàndia i vencedor d'una etapa
Vencedor de 2 etapes al Tour de Corea
Enllaços externs
Fitxa a sitiodeciclismo.net
Fitxa a cyclebase.nl
Fitxa a museociclismo.it
Fitxa a procyclingstats.com
Ciclistes alemanys | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,482 |
\section{Introduction}
In recent years holographic models in which translations are broken either spontaneously or pseudo-spontaneously have been intensively studied. These models include Q-lattices \cite{Amoretti:2016bxs,Amoretti:2017frz,Amoretti:2017axe,Amoretti:2018tzw,Donos:2018kkm,Donos:2019hpp,Amoretti_2019,Amoretti:2019kuf,Amoretti:2020ica,Donos:2021ueh}, massive gravity \cite{Amoretti:2014zha,Baggioli:2014roa,Amoretti:2014mma,Amoretti:2015gna,Amoretti:2017xto,Alberte:2017oqx,Alberte:2017cch,Ammon:2019wci,Baggioli:2020edn,Baggioli:2021xuv} and phases with spatially modulated charge density \cite{Donos:2011qt,Andrade:2017cnc,Andrade:2020hpu}.
On a parallel side, hydrodynamic models have been developed which describe the physical behavior, at late times and large distances, of charged fluids in the presence of (pseudo)-Goldstone modes related to the (pseudo)-spontaneous breaking of translations \cite{Delacretaz:2016ivq,Delacretaz:2017zxd,Delacretaz:2019wzh,amoretti:hydrodynamicmagnetotransport,Armas:2019sbe,Armas:2020bmo}. The dynamics predicted by these effective theories has been tested, in the appropriate regime, against holographic models and been found to concur with a great accuracy.
Of particular relevance for experiments, is the study of the thermo-electric transport properties of these systems in the presence of an external magnetic field, as this is a very common feature in many experimental setups testing the transport behavior of strongly coupled condensed matter materials. This includes for example high-temperature superconductors \cite{Delacretaz:2016ivq,Amoretti:2019buu}. Along this line, hydrodynamic models describing the magneto-transport properties of these systems in vacua that minimize the free energy have been developed in \cite{Delacretaz:2019wzh,amoretti:hydrodynamicmagnetotransport}. On the holographic side, some of the consequences of the presence of an external magnetic field on the (pseudo)-spontaneous breaking of translations have been analyzed in the massive gravity context in \cite{Baggioli:2020edn} and in Q-lattices in \cite{Donos:2021ueh}. Still a complete study of the transport properties of these holographic models in the hydrodynamic regime with an external magnetic field is missing.
In this paper we generalize the holographic Q-lattice model with a (pseudo)-spontaneous symmetry breaking of translation, initially described in \cite{Amoretti:2017frz,Amoretti:2017axe}, in order to include an external magnetic field. We will take this magnetic field to be non-zero in the thermodynamics and thus at the same order as the chemical potential in the derivative counting of hydrodynamics. We analyze the model's thermo-electric transport properties. As already observed in \cite{Amoretti:2017frz,Amoretti:2017axe,Amoretti:2018tzw,Amoretti:2019kuf}, these kind of models correctly describe the physical behavior of systems exhibiting a spontaneous or pseudo-spontaneous symmetry breaking of translations even though they are metastable; namely, the background vacuum does not minimize the free energy. Due to this fact, it was pointed out in \cite{Armas:2019sbe,Ammon:2019apj,Armas:2020bmo,Ammon:2020xyv} that existing hydrodynamic approaches (see e.g. \cite{Delacretaz:2016ivq,Delacretaz:2017zxd} and references therein) do not apply straightforwardly to these models, and a more general formulation must be followed \cite{Armas:2019sbe}. This is due to the appearance of an additional term - the lattice pressure - in the thermodynamics. In stable systems which minimize the free energy this lattice pressure, $P_{l}$, vanishes \cite{Donos:2018kkm,Armas:2020bmo} and its thermodynamic derivatives ($\partial_{\mu} P_{l}$, $\partial_{T} P_{l}$) can be absorbed into re-definitions of the transport coefficients. In such situations we can then employ the expressions of \cite{amoretti:magnetothermaltransporta}. This will not be the case here and thus one of the main results of the paper is the generalization of the formalism of \cite{Armas:2019sbe} to include the effects of pseudo-spontaneous symmetry breaking and an external magnetic field. Moreover, by combining the hydrodynamic correlators obtained in this way with the method outlined in \cite{amoretti:magnetothermaltransporta,amoretti:hydrodynamicmagnetotransport}, we will provide an expression for the hydrodynamic AC correlators in terms of the DC conductivities of the model. The latter quantities can be computed analytically for the model at hand and have been known for a long time \cite{Amoretti:2015gna,Blake:2015ina}. Thus, we can eventually provide an analytic expression for the holographic AC thermo-electric correlators in terms of the horizon data up to one coefficient, the pinning frequency, which has to be determined numerically. Finally, we show that our analytic result is in excellent agreement with the numerically computed holographic correlators.
The paper is organized as follows. In Section \ref{sec:2} we generalize the hydrodynamic method of \cite{Armas:2019sbe} to take into account pseudo-spontaneous symmetry breaking of translations and an external magnetic field. Combining the results with the method outlined in \cite{amoretti:magnetothermaltransporta,amoretti:hydrodynamicmagnetotransport} we provide a closed form for the AC hydrodynamic correlators which depends solely on their DC values and the pinning frequency. In Section \ref{sec:holomodel} we compute the same correlators in the holographic Q-lattice model with an external magnetic field. Using the known analytic results for the DC conductivities we provide a closed form for the thermo-electric correlators in terms of horizon data and one undetermined parameter, the pinning frequency, which we determine numerically finding excellent agreement with the expected result. Finally we comment upon some features of the model and we conclude the paper in Section \ref{sec:conclusions}.
\section{Broken translation invariance with non-zero lattice pressure}
\label{sec:2}
{\ The equations of motion for the (almost-)conserved hydrodynamic charges are the conservation equations for the stress tensor and the charge current. For our system, which includes the presence of translation breaking scalar operators $O_{I}$, these take the form
\begin{eqnarray}
\label{Eq:ConservationEquations2}
\partial_{\mu} \langle T^{\mu \nu}\rangle = F^{\nu \mu}\langle J_{\mu}\rangle
- \left(\partial^{\nu} \Phi^{I}(x_i)\right) \langle O_{I}\rangle \; , \qquad \partial_{\mu} \langle J^{\mu}\rangle
= 0 \, ,
\end{eqnarray}
where $T^{\mu \nu} = \langle T^{\mu \nu}\rangle$ is the stress tensor, $F^{\mu \nu}$ an external electromagnetic field strength, $J^{\mu} = \langle J^{\mu}\rangle$ a $U(1)$ charge current and $\Phi^{I}(x)$ are spatially modulated sources for the scalars $O_{I}$. In addition we will need the ``Josephson relation'' which can be thought of as generating the evolution of the translation breaking scalars. This latter relation must be derived order by order in derivatives and doing so in the presence of an external magnetic field, and in the case of explicit breaking an additional non-zero phase relaxation, is one of the main thrusts of this section.}
\subsection{Homogeneity and the Ward identities}
{\ The conservation equations \eqref{Eq:ConservationEquations2} allow one to consider a broad range of translation breaking scenarios. Here we will restrict ourselves to systems where the vev of the operators $O_{I}$ are proportional to the spatial coordinates, $\langle O_{I} \rangle \propto x^{i} \delta_{iI}$ (in the spontaneous case). On the other hand, for explicit breaking, we take the source to be proportional to the spatial coordinates $\Phi_{I}(x) = \varphi x^{i} \delta_{iI}$ with $\varphi$ a constant. Consequently the space-time derivative of the sources in the ground state of our system are constants. This symmetry breaking pattern can be realised in models with spatial translation invariance and where the scalar operators $O_{I}$ have a constant shift symmetry such that the diagonal subgroup of this pair of symmetries remains unbroken. This ensures homogeneity of our equations of motion. Indeed, this kind of breaking has been extensively considered in the literature as a description of various systems including lattice phonons \cite{Leutwyler:1993gf,Nicolis:2017eqo}, classifications of solid state phases \cite{Nicolis:2015sra} and hydrodynamic \cite{Delacretaz:2017zxd,Delacretaz:2016ivq} and holographic \cite{Andrade:2017cnc,Alberte:2017oqx,Amoretti:2017frz,Amoretti:2017axe,Alberte:2017cch,Amoretti:2018tzw,Andrade:2018gqk,Amoretti:2019kuf,Amoretti_2019,Ammon:2020xyv,Amoretti:2020ica,Andrade:2020hpu,Baggioli:2020nay,Baggioli:2020edn,Baggioli:2021xuv} constructions of charge density wave state effective field theories.}
{\ Regarding the constant $\varphi$ setting the value of the
scalar operator source,
we can use it to qualitatively classify the explicitly broken regime into two cases: pseudo-spontaneous and truly explicit. The first case consists of situations where $\varphi \ll |\partial_i \langle O_i \rangle|$ so that the Goldstone bosons of spontaneously broken translation invariance have acquired a small mass and can be thought of as pseudo-Goldstone bosons. This small mass is called the ``pinning frequency'' which we shall denote by $\omega_{0}^2$. On the other hand, a truly explicit case occurs when $\varphi \gtrsim |\partial_i \langle O_i \rangle|$. This happens for example in the models of \cite{Andrade:2013gsa}, where only the source is non-zero and the vev is vanishing.}
{\ With these restrictions and definitions in place, we note that from the one-point function Ward identities \eqref{Eq:ConservationEquations2} we can also derive Ward identities for the two-point functions. In particular, imposing homogeneity of the source term for the scalar operators $\partial_i \Phi^I= \varphi \delta_i^I$, and employing the convention
\begin{eqnarray}
f(t,\vec{x}) = \int \frac{d^{2}k d\omega}{(2 \pi)^3} f(\omega,\vec{k}) e^{- i ( \omega t - i \vec{k} \cdot \vec{x} )} \; ,
\end{eqnarray}
one finds at zero-wavevector ($\vec{k}=0$) the following relations
\begin{subequations}
\label{Eq:2ptWard}
\begin{eqnarray}
\label{Eq:ExplicitWardIdentity1}
i \omega \langle Q^{i} Q^{j} \rangle &=& - \left( i \omega \mu \delta\indices{^{i}_{k}} - F\indices{^{i}_{k}} \right) \langle Q^{k} J^{j} \rangle + \varphi \langle Q^{i} O^{J} \rangle \delta^{jJ}
- i \omega \left( \chi_{\pi \pi} - \mu n \right) \delta^{ij} \; , \qquad \\
\label{Eq:ExplicitWardIdentity2}
i \omega \langle Q^{i} J^{j} \rangle &=& - \left( i \omega \mu \delta\indices{^{i}_{k}} - F\indices{^{i}_{k}} \right) \langle J^{k} J^{j} \rangle + \varphi\langle J^{i} O^{J} \rangle \delta^{jJ} - i \omega n \delta^{ij} \; , \\
\label{Eq:ExplicitWardIdentity3}
i \omega \langle Q^{i} O^{J} \rangle &=& - \left( i \omega \mu \delta\indices{^{i}_{k}} - F\indices{^{i}_{k}} \right) \langle J^{k} O^{J} \rangle - \varphi\langle O^{I} O^{J} \rangle \delta^{iI} + \delta^{iJ} \; ,
\end{eqnarray}
\end{subequations}
where $Q^{i} = T^{it} - \mu J^{i}$ is the canonical heat current. These will be crucial in our second aim, deriving analytic expressions for the hydrodynamic transport coefficients, following the approach of \cite{amoretti:magnetothermaltransporta} and \cite{amoretti:hydrodynamicmagnetotransport}.}
{\ In brief, the method for generating these analytic expressions for the transport coefficients relies on the existence of a ladder structure in \eqref{Eq:2ptWard} which reduces the number of independent correlators from six to three: $\langle O^{I} O^{J} \rangle$, $\langle J^{i} O^{J} \rangle$ and $\langle J^{i} J^{j} \rangle$. As a consequence one finds that the leading terms in the $\omega\to0$ limit
of the original six correlators are all contained in the low frequency expansion of the independent correlators. Hence, knowing the DC values of all the correlators is as good as knowing the low frequency expansion of the independent correlators. Comparing the hydrodynamic expressions at low frequencies with what is imposed by the Ward identities \eqref{Eq:2ptWard} allows us to fix the hydrodynamic transport coefficients analytically when we know the DC terms analytically (such as in our holographic model).}
\subsection{Spontaneous case}
{\ Given that the formalism is slightly simpler we shall first consider the case of spontaneous breaking ($\varphi=0$). In the spontaneous case fluctuations of the scalar operators $O^I$ correspond to the Goldstone modes of spontaneously broken translation symmetry.}
\subsubsection{Constitutive relation}
{\ We employ the formalism of \cite{Armas:2020bmo}, making minor appropriate changes to account for the presence of a magnetic field. The indices $I$ represent coordinates on the unbroken $ISO(2)$ manifold (the ``crystal'' in the terminology of \cite{Armas:2019sbe,Armas:2020bmo}) and we define $e^I_{\mu}=\partial_{\mu}O^I$ to be the pullback map from the $(2+1)$-dimensional spacetime onto this $2$-dimensional ``crystal manifold''. Subsequently we can define an inverse metric on the crystal $h^{IJ}=g^{\mu\nu}e^I_{\mu}e^J_{\nu}$ using the inverse spacetime metric $g^{\mu \nu}$. The tensor $h^{IJ}$ can be used to raise crystal indices. We further adopt the convention of \cite{Armas:2020bmo} and define the lower index tensor $h_{IJ}$ by $h_{IJ}=(h^{-1})_{IJ}$. Crystal indices will be lowered with respect to $h_{IJ}$.}
{\ The non-linear strain tensor $u_{IJ}$ measures the distortions of the crystal from a reference ``rest'' configuration denoted $\mathds{h}_{IJ}$. This non-linear strain tensor is defined as the difference between $h_{IJ}$ and this reference configuration i.e.~$u_{IJ}=(h_{IJ}-\mathds{h}_{IJ})/2$. For our purposes we choose the reference configuration $\mathds{h}_{IJ}=\delta_{IJ}/\alpha^2$ because it respects homogeneity and spatial isotropy. We interpret the constant parameter $\alpha$ to be the inverse size of the crystal. We make a choice to set it to one in the following since situations where $\alpha\neq1$ can be obtained by a trivial rescaling of the fields $O^I\rightarrow \alpha O^I$ \cite{Armas:2019sbe}.}
{\ Assuming small strain we can construct the free energy $F$ of the crystal plus fluid order by order in the amplitude of $u_{IJ}$. This free energy is the integral over the total pressure $F=\int\dif^2x\sqrt{-g}P$ with the total pressure up to and including quadratic terms in the strain given by
\begin{equation}\label{eqn:pressure}
P=P_f - m B +P_l\left(u^I_I+u^{IJ}u_{IJ}\right)-\frac{1}{2}K\left(u^I_I\right)^2-G\left(u^{IJ}u_{IJ}-\frac{1}{2}\left(u^I_I\right)^2\right)+\mathcal{O}(u^3) \; .
\end{equation}
In the above expression, $P_f$ is the thermodynamic fluid pressure, $m$ the magnetisation density, $P_l$ the lattice pressure and $K$ and $G$ are respectively the bulk and shear modulus. This should be compared to the total pressure $P$ as reported in \cite{amoretti:hydrodynamicmagnetotransport} with the major difference between our current system and those considered in \cite{amoretti:hydrodynamicmagnetotransport} being the non-zero lattice pressure ($P_{l}$) term.}
{\ With the free energy to hand \eqref{eqn:pressure} we can now order by order in derivatives construct the constitutive relations. To keep our notation compact we will use the projectors $P^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$ and $P^{I\mu}=P^{\mu\nu}e^I_{\nu}$ and define the electric field by $E_{\mu}=F_{\mu\nu}u^{\nu}$. From here, the constitutive relations for an isotropic fluid in the Landau frame are
\begin{subequations}
\label{eqn:constitutive_relations}
\begin{eqnarray}
\label{eqn:constitutive_relations1}
J^{\mu}&=&n u^{\mu}-P^{I\mu}\sigma_{IJ}P^{J\nu}\left(T\partial_{\nu}\frac{\mu}{T}-E_{\nu}\right)-P^{I\mu}\gamma_{IJ}u^{\nu}e^J_{\nu} \; , \\
\label{eqn:constitutive_relations2}
T^{\mu\nu}&=&\left(\epsilon+P\right)u^{\mu}u^{\nu}+Pg^{\mu\nu}-r_{IJ}e^{I\mu}e^{J\nu}-P^{I(\mu}P^{J\nu)}\eta_{IJKL}P^{K(\rho}P^{L\sigma)}\nabla_{\rho}u_{\sigma} \; , \qquad
\end{eqnarray}
\end{subequations}
where $P$ is the total pressure of \eqref{eqn:pressure}, $\epsilon$, $n$ and $s$ are the total energy, charge and entropy densities and $r_{IJ}$ is a thus-far undetermined quantity - the elastic stress tensor. These quantities are related by the thermodynamic relations
\begin{eqnarray}
\dif P=s\dif T+n\dif\mu + m \dif B +\frac{1}{2}r_{IJ}h^{IJ} \; , \qquad \epsilon+P=sT+n\mu \; ,
\end{eqnarray}
which defines $r_{IJ}$ in terms of the derivative of $P$. The total charge density and entropy can further be decomposed into free quantities, given by variation of $P_{f}$ with respect to the thermodynamic parameters $\dif P_f = s_f\dif T+n_f\dif\mu + m \dif B$, and the lattice contributions $\dif P_l=s_l\dif T+n_l\dif\mu$. We will assume that both the free and lattice thermodynamic quantities have formally similar integrated first laws, up to contribution of the magnetisation, i.e.~$\epsilon_f+P_f=s_fT+n_f\mu + mB$ and $\epsilon_l+P_l=s_lT+n_l\mu$.}
{\ In addition to the constitutive relations for the (almost-)conserved currents $J^{\mu}$ and $T^{\mu \nu}$, as discussed above, we must supply an evolution equation for the crystal (or Goldstone) fields. At zeroth order in hydrodynamics such equations correspond to constancy of the scalars along a fluid worldline. At first order in derivatives one finds
\begin{equation}
\label{eqn:configuration_equation}
\sigma^{\phi}_{IJ}u^{\mu}e^I_{\mu}+\gamma'_{JK}P^{K\mu}\left(T\partial_{\mu}\frac{\mu}{T}-E_{\mu}\right)+\nabla_{\mu}\left(r_{JK}e^{K\mu}\right)=K^{\text{ext}}_J\;,
\end{equation}
where $\sigma_{IJ}$, $\gamma_{IJ}$, $\gamma'_{IJ}$, $\sigma^{\phi}_{IJ}$ and $\eta_{IJKL}$ are all dissipative transport matrices and $K^\text{ext}_J$ is an external background source coupled to $O^I$ that will be zero in global thermodynamic equilibrium. We will restrict ourselves to models which have time-reversal symmetry, such that the Onsager relation $\langle O^I J^j\rangle=-\langle J^i O^J\rangle$ requires that $\gamma'_{IJ}=-\gamma_{IJ}$ \cite{Armas:2019sbe}.}
{\ The form of the constitutive relations \eqref{eqn:constitutive_relations} is the same for all values of $u_{IJ}$ - in particular one can in principle use the complete pressure and not its small amplitude expansion \eqref{eqn:pressure}. However, in practice, it is sufficient to consider fluctuations about a state of global thermodynamic equilibrium and linearise in small amplitude $u_{IJ}$. In this small strain regime the transport coefficient matrices can be taken to be strain independent at first order in derivatives. Additionally, in the presence of a constant, background, external magnetic field, we must allow for the possibility of Hall transport coefficients \cite{amoretti:magnetothermaltransporta} and subsequently decompose the transport matrices as
\begin{equation}\label{eqn:transport_coef_decomposition}
\left(\gamma,\sigma,\sigma^{\phi}\right)_{IJ}=\left(\gamma,\sigma,\sigma^{\phi}\right)_{(\mathrm{L})}\delta_{IJ}+\left(\gamma,\sigma,\sigma^{\phi}\right)_{(\mathrm{H})}F_{IJ}\;,
\end{equation}
where we have defined $F_{IJ}=F_{\mu\nu}e^{\mu}_Ie^{\nu}_J$. As we decompose with respect to $F_{IJ}$ rather than $\epsilon_{IJ}$ the Hall coefficients are spatial parity invariant. We identify $\sigma_{(\mathrm{L},\mathrm{H})}$ to be the longitudinal and Hall components of the charge conductivity, $\sigma^\phi_{(\mathrm{L},\mathrm{H})}$ to be the crystal diffusivity components and $\gamma_{(\mathrm{L},\mathrm{H})}$ the mixed scalar-charge conductivities. In principle we could also decompose $\eta_{IJKL}$ in terms of longitudinal and Hall bulk and shear viscosities, but because we are only interested in the diffusive sectors at zero wave-vector the viscous terms will not be relevant.}
\subsubsection{AC conductivities}
{\ We will fluctuate about a flat background $g_{\mu\nu}=\eta_{\mu\nu}$, with a constant magnetic field $F^{12}=B$ and vanishing external source for the Goldstone field $K^{\text{ext}}_I=0$. The corresponding equilibrium configuration has a fixed temperature $T=T_0$ and chemical potential $\mu=\mu_0$, no spatial velocity $u^{\mu}=(1,\vect{0})$ and a uniform value for the scalars $O^I=x^I$. We linearise the constitutive relations around this equilibrium configuration\footnote{Note that the sign of the $\delta O^{I}$ is the opposite to that used in \cite{amoretti:hydrodynamicmagnetotransport} to match the conventions of \cite{Armas:2020bmo}.},
\begin{align}\label{eqn:fluctuations}
T&\rightarrow T_0+\delta T\;,
&\mu&\rightarrow\mu_0+\delta\mu\;,\nonumber\\
u^{\mu}&\rightarrow (1,v^i)\;,
& O^I&\rightarrow x^I-\delta O^I\;,
\end{align}
and find the two-point functions by solving the non-conservation equations for fluctuations of the hydrodynamic variables in the presence of plain wave sources for the $U(1)$ field strength $\delta F^{0i}\sim\exp(-i\omega t+ik_jx^j)$ and the source terms $\delta K^{\text{ext}}_I$ and $\delta g_{\mu\nu}$. Following this procedure allows us to find the independent correlators: $\langle J^iJ^j\rangle$, $\langle J^iO^J\rangle$ and $\langle O^IO^J\rangle$. Subsequently, by applying the Ward identities \eqref{Eq:2ptWard} we can derive expressions for correlators involving the heat current\footnote{The interested reader could also readily obtain some correlators involving the thermodynamic heat current by computing the variation of the $U(1)$ charge current and scalar with respect to metric.}. This approach should be contrasted to the more standard Martin-Kadanoff method which is made difficult in the presence of lattice pressure due to the appearance of double time and mixed space/time derivatives in the constitutive relations (this is hidden in the $r_{IJ}$ containing terms of \eqref{eqn:constitutive_relations1} and \eqref{eqn:configuration_equation}).}
{\ We define the following AC conductivities in the spontaneous case\footnote{We explicitly include an argument in the AC conductivities of \eqref{Eq:DefinitionsofDC1} and \eqref{Eq:DefinitionsofDC2} to differentiate them from hydrodynamic transport coefficients as there is some notational overlap between such quantities in the literature e.g.~the hydrodynamic electric conductivity $\sigma^{ij}$ and the AC electric conductivity $\sigma^{ij}(\omega)$.}
\begin{eqnarray}
\label{Eq:DefinitionsofDC1}
\left(\sigma^{ij}, \alpha^{ij}, \gamma^{iJ}\right)(\omega) &=& \left( \frac{1}{i\omega} \langle J^{i} J^{j} \rangle , \frac{1}{i\omega} \langle Q^{i} J^{j} \rangle, \langle J^{i} O^{J} \rangle \right) \; , \\
\label{Eq:DefinitionsofDC2}
\left(\kappa^{ij}, X^{IJ}, \theta^{iJ}\right)(\omega) &=& \left( \frac{1}{i\omega} \langle Q^{i} Q^{j} \rangle , i \omega \langle O^{I} O^{J} \rangle, \langle Q^{i} O^{J} \rangle \right) \; .
\end{eqnarray}
We have split them into two sets; the leading low frequency terms of the first set \eqref{Eq:DefinitionsofDC1} are fixed by symmetry \cite{amoretti:magnetothermaltransporta,amoretti:hydrodynamicmagnetotransport} and are the same for every system satisfying our general assumptions. The second set \eqref{Eq:DefinitionsofDC2} depend on the specific microscopic theory of the system. With that said, due to the ladder nature of the Ward identities \eqref{Eq:2ptWard}, the arbitrary frequency values of $\alpha^{ij}(\omega)$, $\kappa^{ij}(\omega)$ and $\theta^{iJ}(\omega)$ can all be derived from $\sigma^{ij}(\omega)$ and $\theta^{iJ}(\omega)$. The coefficient $X^{IJ}(\omega)$ stands on its own at the level of the spontaneous two-point Ward identities being not related to any of the others. Hence the independent conductivities are $\sigma^{ij}(\omega)$, $\gamma^{iJ}(\omega)$ and $X^{IJ}(\omega)$ and we shall give hydrodynamic expressions for these.}
{\ Since the analytic expressions for the conductivities are rather large, we employ a matrix notation similar to the one used in \cite{amoretti:hydrodynamicmagnetotransport}. We construct the following matrices of hydrodynamic transport coefficients (first line) and AC conductivities (second line)
\begin{align}
\label{Eq:SpontaneousTransportCoeffs}
(\hat\sigma,\hat\sigma_{\phi},\hat\gamma)&=(\sigma,\sigma^{\phi},\gamma)_{(\mathrm{L})}\mathds{1}_2-(\sigma,\sigma^{\phi},\gamma)_{(\mathrm{H})}F\;,\\
\label{Eq:DCtransportcoeffs}
(\hat\sigma,\hat\alpha,\hat\kappa,\hat\gamma,\hat X,\hat\theta)(\omega)&=(\sigma,\alpha,\kappa,\gamma,X,\theta)_{(\mathrm{L})}(\omega)\mathds{1}_2-(\sigma,\alpha,\kappa,\gamma,X,\theta)_{(\mathrm{H})}(\omega)F^{-1} \; .
\end{align}
We also define the following additional terms for notational convenience
\begin{align}
\hat\sigma'&=\hat\gamma^2+\hat\sigma\cdot\hat\sigma_\phi&\hat\rho&=2\hat\gamma+F\cdot\hat\sigma-n_f\mathds{1}_2 \; .
\end{align}
The coefficients of \eqref{Eq:SpontaneousTransportCoeffs} are the transport coefficients appearing in the constitutive relations \eqref{eqn:constitutive_relations}. In terms of them the three independent AC conductivities are
\begin{subequations}
\label{Eq:ACSpontaneousconductivities}
\begin{align}
\hat\sigma(\omega)&=\hat\Lambda^{-1}\cdot\left[\omega P_l\left(in_f\hat\rho-\omega w_f\hat\sigma\right)+n_f^2\hat\sigma_\phi-\left(n_fF+i\omega\chi_{\pi\pi}\mathds{1}_2\right)\hat\sigma'\right]\;,\\
\hat\gamma(\omega)&=\hat\Lambda^{-1}\cdot\left(i\omega w_f\hat\gamma-n_f\hat\sigma_\phi+F\cdot\hat\sigma'\right)\;,\\
\hat X(\omega)&=\hat\Lambda^{-1}\cdot\left(F\cdot\hat\rho+i\omega w_f\mathds{1}_2-\hat\sigma_\phi\right)\;,\\
\hat\Lambda&=\omega P_l\left(iF\cdot\hat\rho-\omega w_f\mathds{1}_2\right)+\hat\sigma_\phi(Fn_f-i\omega\chi_{\pi\pi}\mathds{1}_2)
-F^2\cdot\sigma'\;,
\end{align}
\end{subequations}
where we defined $w_f$ to be the fluid enthalpy density $w_f=\epsilon_f+P_f$. The above expressions are a result of hydrodynamics alone following from using our constitutive relations \eqref{eqn:constitutive_relations}.
{\ Undetermined in the above are the hydrodynamic transport coefficients: $\sigma_{IJ}$, $\sigma^{\phi}_{IJ}$ and $\gamma_{IJ}$. For the spontaneous case, we will write these coefficients as functions of $X_{iJ}(0)$, $\theta_{IJ}(0)$ and $\kappa_{ij}(0)$ by employing the Ward identities (as in \cite{amoretti:magnetothermaltransporta,amoretti:hydrodynamicmagnetotransport}). We find
\begin{subequations}
\label{Eq:ACSpontaneoustransportcoeffs}
\begin{align}
\hat\sigma&=\hat\Phi^{-1}\cdot\left(\hat\kappa(0)+2\chi_{\pi\pi}\hat\theta(0)-\chi_{\pi\pi}^2\hat X(0)-\mu^2n_fF^{-1}\right)+n_f F^{-1} \; , \\
\hat\sigma_{\phi}&=\hat\Phi^{-1}\cdot\left[F\cdot(P_l^2\hat X(0)-\hat\kappa(0)-2P_l\hat\theta(0))+\mu(\mu n_f-2w_f)\mathds{1}_2\right]\cdot F \; , \\
\begin{split}
\hat\gamma&=\hat\Phi^{-1}\cdot\left[F\cdot\left(P_l\chi_{\pi\pi}\hat X(0)+(w_f-2\chi_{\pi\pi})\hat\theta(0)-\hat\kappa(0)\right)\right.\\
&\qquad\left.+\mu(\mu n_f-w_f)\mathds{1}_2\right] \; ,
\end{split}\\
\hat\Phi&=\left(\mu\mathds{1}_2-F\cdot\hat\theta(0)\right)^2+\left(F\cdot\hat\kappa(0)-\mu(\mu n_f-2\chi_{\pi\pi})\mathds{1}_2\right)\cdot F\cdot\hat X(0) \; .
\end{align}
\end{subequations}
The expressions \eqref{Eq:ACSpontaneousconductivities} and \eqref{Eq:ACSpontaneoustransportcoeffs} correctly reduce to the expressions obtained in \cite{amoretti:hydrodynamicmagnetotransport} at $P_l=0$ once appropriate identifications are made between the transport coefficients to account for the differing ways that the constitutive relations were constructed.}
\subsection{Pseudo-spontaneous and explicit cases}
{\ We now consider the cases of pseudo-spontaneous and explicit breaking of translation invariance by the scalars. The distinction between these two cases is somewhat loose, but we remind the reader that we identify the former as satisfying $\varphi \ll |\partial_i \langle O_i \rangle|$ while the latter consists of all other situations. Again, we will employ the formalism of \cite{Armas:2020bmo}, but now we must account for a non-zero phase relaxation in addition to the existence of an external magnetic field.}
\subsubsection{Constitutive relation}
{\ In the explicit case the procedure for constructing the effective hydrodynamic theory is broadly the same. In particular, the constitutive relations \eqref{eqn:constitutive_relations} and charge conservation equations remain unchanged modulo the inclusion of an explicit mass term for the scalar fields in the evolution equation for the spatial momentum
\begin{equation}
\label{Eq:MomentumConservation}
\partial_t P^i+\partial_j T^{ij}=F^{i\mu}J_{\mu}-K_I^{\text{ext}}e^{Ii}+\omega_0^2\chi_{\pi\pi}O^I\delta^{Ii} \; ,
\end{equation}
where $\omega_0^2$ is the pinning frequency. This addition follows directly from the one-point Ward identities \eqref{Eq:ConservationEquations2} when the scalars are sourced by a homogeneous and isotropic term of the form $\Phi^{I} = \varphi x^{I}$. In particular, we identify $\varphi = \omega_{0}^2 \chi_{\pi \pi}$ as in \cite{amoretti:hydrodynamicmagnetotransport}.}
{\ While the conservation equations are mostly unchanged, to describe the evolution of the scalars accurately we must also account for the fact that they are no longer massless and will have a tendency to spread out in space-time. We can phenomenologically track this effect by adding a non-zero phase relaxation term $\Omega^{IJ} O_J$ to the Josephson relation \cite{amoretti:hydrodynamicmagnetotransport} i.e.~
\begin{eqnarray}
\label{eqn:configuration_equation_explicit}
\sigma^{\phi}_{IJ}u^{\mu}e^I_{\mu}+\gamma'_{JK}P^{K\mu}\left(T\partial_{\mu}\frac{\mu}{T}-E_{\mu}\right)+\nabla_{\mu}\left(r_{JK}e^{K\mu}\right)=\Omega^{IJ} O_J + K^{\text{ext}}_J \; .
\end{eqnarray}
The phase-relaxation tensor $\Omega^{IJ}$, like the other hydrodynamic transport coefficients discussed here, decomposes with respect to $SO(2)$ rotation invariance and microscopic parity invariance
\begin{equation}
\Omega^{IJ}=\Omega_{(\mathrm{L})}\delta^{IJ}+\Omega_{(\mathrm{H})}F^{IJ} \; .
\end{equation}
While in principle the new phase relaxation term in \eqref{eqn:configuration_equation_explicit} is independent of the other transport coefficients, it turns out that Onsager relations impose a constraint on its value, in particular we found that
\begin{equation}\label{eqn:onsager}
\Omega^{IJ} = \omega_0^2\chi_{\pi\pi}\delta^{IJ}\;.
\end{equation}
This may seem surprising given that in previous works \cite{amoretti:hydrodynamicmagnetotransport} the phase relaxation is an independent transport coefficient with a non-zero Hall term. What is missing here is that compared to the formalism of \cite{amoretti:hydrodynamicmagnetotransport}, the time evolution of the scalar field (i.e.~the first term of \eqref{eqn:configuration_equation_explicit}) is not normalised to the identity matrix. Consequently, in the present formalism, the inverse crystal diffusivity
$(\sigma^{\phi})^{-1}$
plays the same role as the phase-relaxation tensor of \cite{amoretti:hydrodynamicmagnetotransport}. To compare the results with \cite{amoretti:hydrodynamicmagnetotransport}, in the limit of vanishing lattice pressure, one must rescale \eqref{eqn:configuration_equation_explicit} with the inverse crystal diffusivity and identify the phase relaxation tensor of \cite{amoretti:hydrodynamicmagnetotransport} to be $\omega_{0}^2 \chi_{\pi \pi} (\sigma^{\phi})^{-1}$.}
\subsubsection{AC conductivities}
{\ To compute the AC conductivities and identify the hydrodynamic transport coefficients we proceed as for the spontaneous case. The global thermodynamic equilibrium configuration is unchanged and we can fluctuate again with \eqref{eqn:fluctuations} to find the linearized expressions. However, to express the results in a compact form we once more require some additional notation. To begin with we define two new AC transport terms:
\begin{eqnarray}
(\varpi^{iJ}, \zeta^{IJ})(\omega) = \frac{1}{i \omega} \left(\langle J^{i} O^{J} \rangle, \langle O^{I} O^{J} \rangle - \frac{1}{\varphi} \delta^{IJ} \right) \; .
\end{eqnarray}
Contrasted to \eqref{Eq:DefinitionsofDC1} and \eqref{Eq:DefinitionsofDC2} these contain differing overall powers of the frequency reflecting the differing behaviour of the explicit correlators at low frequencies. Subsequently we decompose our AC conductivities as new matrices
\begin{eqnarray}
\label{Eq:Thermalconductivities}
(\hat\sigma,\hat\alpha,\hat\kappa,\hat\varpi,\hat\zeta)(\omega)=(\sigma,\alpha,\kappa,\varpi,\zeta)_{(\mathrm{L})}(\omega)\mathds{1}_2+(\sigma,\alpha,\kappa,\varpi,\zeta)_{(\mathrm{H})}(\omega)F \; .
\end{eqnarray}
The use of $F$ in the explicit case, rather than $F^{-1}$ as we used in the spontaneous case \eqref{Eq:DCtransportcoeffs}, reflects the smoothness of these latter conductivities as $B \rightarrow 0$. We also introduce one additional new quantity,
\begin{align}\label{Gammafreq}
\Gamma&=\omega_0^2\chi_{\pi\pi}-\omega^2P_l \; ,
\end{align}
not to be confused with any explicit momentum loss tensor.}
{\ With these definitions to hand we find that the three independent AC correlators are then given by
\begin{subequations}
\label{Eq:ExplicitACconductivities}
\begin{align}
\hat\sigma(\omega)&=\hat\Xi^{-1}\cdot\left[\Gamma\omega w_f\hat\sigma+\omega n_f^2\hat\sigma_\phi-i(\omega^2-\omega_0^2)\chi_{\pi\pi}\hat\sigma'-n_f\left(i\Gamma\hat\rho+\omega F\cdot\hat\sigma'\right)\right]\;,\\
\hat\varpi(\omega)&=\hat\Xi^{-1}\cdot\left(\omega w_f\hat\gamma+i(n_f\hat\sigma_\phi-F\cdot\hat\sigma')\right)\;,\\
\hat\zeta(\omega)&=\frac{1}{\omega_0^2\chi_{\pi\pi}}\hat\Xi^{-1}\cdot\left(\omega\chi_{\pi\pi}\hat\sigma_\phi-\omega P_l\left(F\cdot\hat\rho+i\omega w_f\mathds{1}_2\right)+iF\cdot(n_f\hat\sigma_\phi-F\cdot\hat\sigma')\right)\;,\\
\hat\Xi&=\Gamma\left(\omega w_f\mathds{1}_2-iF\cdot\hat\rho\right)+\omega n_fF\cdot\hat\sigma_\phi-i(\omega^2-\omega_0^2)\chi_{\pi\pi}\hat\sigma_\phi-\omega F^2\cdot\hat\sigma'\;. \label{denominatore}
\end{align}
\end{subequations}
Consequently, by employing the Ward identities \eqref{Eq:2ptWard} the three hydrodynamic transport matrices $\hat\sigma$, $\hat\sigma_{\phi}$ and $\hat\gamma$ can be expressed in terms of the DC values of the electric, thermoelectric and thermal conductivity $\hat\sigma(0)$, $\hat\alpha(0)$ and $\hat\kappa(0)$ hydrodynamic transport coefficients:
\begin{subequations}
\label{Eq:ExplicitACtransportcoeffs}
\begin{align}
\hat\sigma&=-\hat\Psi^{-1}\cdot\hat\pi(0)\;,\\
\hat\sigma_\phi&=\hat\Psi^{-1}\cdot\left[w_f^2\mathds{1}_2
+\left(F\cdot\hat\pi(0)-2w_f(\hat\alpha(0)+\mu\hat\sigma(0))\right)\cdot F\right]+n_fF\;,\\
\hat\gamma&=\hat\Psi^{-1}\cdot\left[F\cdot\hat\pi(0)-w_f(\hat\alpha(0)+\mu\hat\sigma(0))\right]+n_f\mathds{1}_2\;,\\
\hat\Psi&=\mu^2\hat\sigma(0)+2\mu\hat\alpha(0)+\hat\kappa(0)\;,
\end{align}
\end{subequations}
where we have defined
\begin{equation}
\hat\pi(0)=\hat\alpha^2(0)-\hat\kappa(0)\cdot\hat\sigma(0)\;,
\end{equation}
i.e.~minus the determinant of the DC thermoelectric conductivity matrix.
Notice that, compared to the spontaneous case, in the explicit case the DC electric, thermo-electric and charge-scalar conductivities are not fixed by symmetries. This implies that the value of the transport coefficients \eqref{Eq:ExplicitACtransportcoeffs} can be in principle obtained in a real experiment by measuring only the DC electric thermo-electric and thermal conductivities.
Finally, we can again take the limit of $P_{l} \rightarrow 0$ and match these expressions, \eqref{Eq:ExplicitACconductivities} and \eqref{Eq:ExplicitACtransportcoeffs}, against those in \cite{amoretti:hydrodynamicmagnetotransport} with the appropriate identifications. The agreement is perfect.}
\section{Holographic model}
\label{sec:holomodel}
{\ To test our hydrodynamic theory in a precise scenario we now study the transport properties of a holographic model consisting of a bidimensional `Q-lattice'~\cite{Donos:2014uba} with action
\begin{eqnarray}
\label{Eq:HolographicAction}
S &=& \int d^{3+1}x \sqrt{-g}\biggl( R-V[\phi]-\frac{1}{2}(\partial\phi)^2-\frac{Z[\phi]}{4}F^2-\frac{1}{2}Y[\phi]\sum_{i=1,2}(\partial\psi_i)^2\biggr) \,.
\label{eq:holoact}
\end{eqnarray}
This model is suitable for describing the symmetry breaking pattern we considered in previous sections since it enjoys a shift symmetry $\psi_i\to \psi_i+c_i$. Indeed, we will impose that
\begin{equation}
\label{Eq:BackgroundPsi}
\psi_i= k x^i\,,\qquad x^i=\{x,y\}\,,
\end{equation}
which breaks spatial translations and the shift symmetry to a diagonal $U(1)$. This form for the background values of $\psi_{i}$ allow us to find solutions where the metric, $U(1)$ gauge field and scalar $\phi$ in \eqref{eq:holoact} depend only on the radial coordinate; translations are broken homogeneously \cite{Amoretti:2017axe,Amoretti:2017frz,Amoretti:2018tzw,Amoretti:2019buu,Amoretti:2019kuf,Amoretti:2020ica}. From this point onward, given our choice of background axion in \eqref{Eq:BackgroundPsi} we will now identify capital Latin indices $I,J,K,L,\ldots$ which labeled crystal directions with the equivalent spatial indices $i,j,k,l,\ldots$.}
{\ We take as an ansatz for our background solution to the equations of motion coming from \eqref{Eq:HolographicAction} the following:
\begin{equation}
\label{Eq:BackgroundAnsatz1}
ds^2 = \frac{1}{r^2}\left(-f(r)dt^2+ \frac{dr^2}{f(r)} + g(r)d\vec{x}^2 \right), \quad
A=a(r) dt-B\,y dx\,,\quad\phi=\phi(r)\,.
\end{equation}
The expressions in \eqref{Eq:BackgroundAnsatz1} can accommodate configurations which are asymptotic to AdS, provided we impose particular $\phi\to0$ asymptotics for the scalar couplings
\begin{eqnarray}\label{Eq:Asymptoticpotential}
V[\phi] = - 6 - \phi^2 + \mathcal{O}(\phi^3) \; , \qquad Z[\phi] = 1 + \mathcal{O}(\phi) \; , \qquad Y[\phi] = \frac{\Upsilon}{2} \phi^2 + \mathcal{O}(\phi^3) \,. \qquad
\end{eqnarray}
As has been thoroughly discussed in \cite{Amoretti:2017frz,Amoretti:2018tzw,Amoretti:2019kuf}, the asymptotic behavior of $\phi$ towards the boundary of AdS will determine whether translations are broken explicitly or spontaneously. Given the asymptotics of \eqref{Eq:Asymptoticpotential}, the scalar $\phi(r)$ behaves as
\begin{equation}
\label{Eq:BackgroundAnsatz2}
\phi(r)=\lambda r+\phi_v r^2+O(r^3) \; .
\end{equation}
Briefly, these asymptotic choices for the scalar and axions allow one to repackage the fields ($\phi, \psi_i$) into a pair of complex scalars $\Phi_i\sim\phi\exp(i\psi_i)$ as in \cite{Donos:2013eha}. Solutions with $\lambda=0$ correspond to a theory where the operators dual to $\psi_i$ break translations spontaneously, while backgrounds with $\lambda\neq0$ break them explicitly. At zero magnetic field, both spontaneous and explicit solutions with potentials satisfying \eqref{Eq:Asymptoticpotential} have been constructed in \cite{Amoretti:2017axe,Amoretti:2017frz,Amoretti:2018tzw,Amoretti:2019buu}. In this section
we will explore such solutions further, focusing on their transport properties in the presence of an external magnetic field, and for a choice of potentials where
\begin{eqnarray}\label{potential}
\label{eq:holopots}
V[\phi] = - 6 \cosh\left( \frac{\phi}{\sqrt{3}} \right) \, , \qquad Z[\phi] = \exp\left( - \frac{\phi}{\sqrt{3}} \right) \,, \qquad Y[\phi] = \left( 1 - e^{\phi} \right)^2 \; .
\end{eqnarray}
Consequently we set $\Upsilon=2$ in \eqref{Eq:Asymptoticpotential}.}
\subsection{Summary of the thermodynamics}
{\ To match the holographic model to our hydrodynamic theory, we first need to determine the thermodynamic quantities in the boundary field theory corresponding to our backgrounds \eqref{Eq:BackgroundAnsatz1}. On the boundary, the thermodynamic state of our theory is determined by the temperature $T$, the chemical potential $\mu$, the external magnetic field $B$ and in principle the wavelength $k$ of the crystal. In our models however we will treat $k$ as an external parameter and not minimize the free energy with respect to this quantity. That this approach gives, at the level of the transport properties, the same result as a more sophisticated model (see \textit{e.g.}~\cite{Andrade:2017cnc,Andrade:2020hpu}) where $k$ is fixed by minimizing the free energy, is discussed for example in \cite{Amoretti:2017frz,Amoretti:2020ica}.
Additionally, to specify the groundstate at the boundary we require either the vev of the scalar in the spontaneous case ($\phi_{v}$) or the boundary source ($\lambda$) in the explicit case.}
{\ As detailed in Appendix \ref{app:numerics} we construct numerical solutions corresponding to non-zero temperature and charge density states in an external magnetic field. In addition to \eqref{Eq:BackgroundAnsatz2}, one can show that the boundary behavior of the functions in our background ansatz \eqref{Eq:BackgroundAnsatz1} is of the form
\begin{subequations}
\label{Eq:UVexpansionbackgroundexp}
\begin{eqnarray}
\label{Eq:UVexpansionbackgroundexpf}
f(r) &=& 1 - \frac{\lambda^2}{4} r^2 - \frac{\epsilon_{f}}{6} r^3+O(r^4) \,, \\
g(r)&=& 1 - \frac{\lambda^2}{4} r^2 - \frac{\lambda \phi_{v}}{3} r^3 +O(r^4) \,, \\
\label{Eq:UVexpansionbackgroundexpgauge}
a(r) &=& \mu- n_{f} r+O(r^3) \,,
\end{eqnarray}
\end{subequations}
where in the gauge field expansion \eqref{Eq:UVexpansionbackgroundexpgauge} we recognise the chemical potential $\mu$, and the free electric charge density $n_{f}$. Similarly in \eqref{Eq:UVexpansionbackgroundexpf} we find the energy density $\epsilon_{f}$ appearing at some subleading order. Solutions dual to states at finite temperature present a horizon at a finite $r_h$ in the bulk. The asymptotic behavior of our background fields towards the horizon read
\begin{eqnarray}
\label{Eq:Nearhorizonfields}
ds^2 &=& -4 \pi T (r_{\mathrm{h}}-r) dt^2 + \frac{dr^2}{4 \pi T (r_{\mathrm{h}}-r)} + \frac{s_{f}}{4 \pi}(dx^2+dy^2) \;, \qquad \\
a(r) &=& a_{h,1} (r_{\mathrm{h}}-r)+...,
\qquad \phi = \phi_{\mathrm{h}}+... \,,
\end{eqnarray}
where $T$ and $s_{f}$ correspond respectively to the temperature and the free entropy density of the dual system.
{\ We can make further progress in determining the thermodynamics of the system in terms of the near-horizon asymptotics of our solutions by making use of two radially-conserved quantities that follow from the background equations of motion. First, the Maxwell equation implies the conservation of
\begin{equation}
\label{Eq:Maxwellradial}
- g(r) Z[\phi(r)] a'(r)\,,
\end{equation}
which asymptotes to the free electric charge density $n_{f}$ at the boundary (hence the overall sign choice). Consequently, the leading term in the near horizon expansion of the gauge field ($a_{h,1}$) can be given in terms of the boundary electric charge density $n$,
\begin{eqnarray}
a_{h,1} = - \frac{4 \pi n_{f}}{s_{f} Z_{\mathrm{h}}} \; , \qquad Z[\phi_{\mathrm{h}}] = Z_{\mathrm{h}} \; .
\end{eqnarray}
The second radially conserved quantity follows from the Einstein equations which imply
\begin{equation}
\label{Eq:Einsteinradial}
\left[n_{f} a(r)+ \frac{g^2(r)}{r^2} \left(\frac{f(r)}{g(r)}\right)' - k^2 I_{Y}(r) - B^2 I_{Z}(r) \right]' = 0 \, ,
\end{equation}
where we have introduced the integrals
\begin{equation}
\label{Eq:IYIZintegrals}
I_{Y}(r) = \int_{w=0}^{r} d w \; \frac{Y[\phi(w)]}{w^2} \,,\qquad I_{Z}(r) = \int_{w=0}^{r} d w \; \frac{Z[\phi(w)]}{g(w)} \,.
\end{equation}
The former of these represents the thermodynamic response of the theory to broken translation invariance which is captured in the ``lattice pressure''~\cite{Armas:2019sbe} $P_{l} = - k^2 I_{Y}(r_{\mathrm{h}})$, while the latter is related to the magnetisation density, $m = - B I_{Z}(r_{\mathrm{h}})$, as discussed in~\cite{Blake:2015ina,Lucas:2015pxa}. Importantly, at the horizon and boundary \eqref{Eq:Einsteinradial} takes the following values
\begin{eqnarray}
\mathrm{boundary} : \mu n_{f} + \lambda \phi_{v} - \frac{\epsilon}{2} \,, \qquad \mathrm{horizon} : - s_{f} T + m B + P_{l} \; .
\end{eqnarray}
Equating these expressions shows that these thermodynamic quantities satisfy a Smarr-type relation
\begin{eqnarray}
\epsilon_{f} = 2 \left( s_{f} T + \mu n_{f} - m B - P_{l} + \lambda \phi_{v} \right) \, .
\end{eqnarray}
}
\subsection{Conserved bulk radial currents at the fluctuation level}
\label{ssec:holoDCxp}
{\ It will be possible to compute analytically the zero frequency value of several correlators in the holographic model. These will include our input data for the hydrodynamic model (\textit{e.g.}~the DC conductivities\footnote{See also \cite{Donos:2018kkm,Donos:2019tmo,Donos:2019hpp,Gouteraux:2018wfe} for the derivation of holographic DC quantities in analogous models.}). To do this we shall turn on a constant background electric field at the level of fluctuations and compute the response of the theory. We therefore consider the following perturbations for the gauge field and metric
\begin{subequations}
\label{Eq:DiffusionAnsatz}
\begin{eqnarray}
\delta A_{x}(r) = a_{x}(r)-p_{x}(r) t \; , \quad \delta g_{tx}(r) = \frac{1}{r^2} \left( h_{x}(r)-\tilde{p}_{x} (r) t \right) \; , \quad \delta g_{rx}(r) = \frac{1}{r} \tilde{h}_{x}(r) \; , \qquad\;\;\\
\delta A_{y}(r) = a_{y}(r)-p_{y}(r) t \; , \quad \delta g_{ty}(r) = \frac{1}{r^2} \left( h_{y}(r)-\tilde{p}_{y} (r) t \right) \; , \quad \delta g_{ry}(r) = \frac{1}{r} \tilde{h}_{y}(r) \; , \qquad \;\;\;
\end{eqnarray}
\end{subequations}
where the radial functions $p_{x}(r)$, $p_{y}(r)$, $\tilde{p}_{x}(r)$ and $\tilde{p}_{y}(r)$ will correspond to turning on a small electric field at the fluctuation level. The difference between the spontaneous and explicit case will be encoded in the axion fields. In the former case they are taken to have the form
\begin{eqnarray}
\delta \psi_{x}(r) = \frac{\chi_{x}(r)}{r} - k \delta V_{x} t \; , \qquad \delta \psi_{y}(r) = \frac{\chi_{y}(r)}{r} - k \delta V_{y} t \; ,
\end{eqnarray}
where $\delta V_{x}$ and $\delta V_{y}$ are sliding modes which encode an ambiguity in the definition of the vev of the axions at the boundary \cite{Davison:2015taa,Donos:2018kkm,Amoretti:2017frz,Donos:2019hpp}. This ambiguity is fixed by conditions at the horizon. Meanwhile in the explicit case the axions are purely radial functions
\begin{eqnarray}
\delta \psi_{x}(r) = \chi_{x}(r) \; , \qquad \delta \psi_{y}(r) = \chi_{y}(r) \; ,
\end{eqnarray}
with no sliding mode ambiguity.}
{\ The explicit time dependence of the ans\"{a}tze \eqref{Eq:DiffusionAnsatz} will drop out from the linearised equations of motion provided that $p_{x}(r)$, $p_{y}(r)$, $\tilde{p}_{x}(r)$ and $\tilde{p}_{y}(r)$ take particular forms:
\begin{subequations}
\label{Eq:p1p2rsols}
\begin{eqnarray}
p_{x}(r)= p_{x}^{(0)} + n_{f} \bar{E}_{x} a(r) \ , \qquad \tilde{p}_{x}(r)= -n_{f} \bar{E}_{x} f(r) \ , \\
p_{y}(r)= p_{y}^{(0)} + n_{f} \bar{E}_{y} a(r) \ , \qquad \tilde{p}_{y}(r)= -n_{f} \bar{E}_{y} f(r) \ ,
\end{eqnarray}
\end{subequations}
where $p_{x}^{(0)}$, $p_{y}^{(0)}$, $\bar{E}_{x}$ and $\bar{E}_{y}$ are free constants. This perturbation represents a stationary state where the applied electric field is balanced against momentum loss at the fluctuation level. From this starting point one can construct Frobenius expansions for the fluctuation fields in the near horizon region and at the boundary. At the horizon we impose regularity conditions.}
\subsubsection{Spontaneous case}
{\ Given the perturbations we have switched on, it is relatively straightforward to massage the bulk equations of motion into sets of conservation equations for radial currents. For example, from the Maxwell equations for the gauge field perturbation one can identify the following radially conserved currents:
\begin{subequations}
\label{Eq:BulkChargeCurrents}
\begin{eqnarray}
\delta \mathcal{J}_{x}(r) &=& Z[\phi(r)] f(r) \left( a_{x}'(r) + \frac{r B}{g(r)} \tilde{h}_{y}(r) \right) - \frac{n_{f}}{g(r)} h_{x}(r) - n_{f} B \bar{E}_{y} I_{Z}(r) \; , \qquad \\
\delta \mathcal{J}_{y}(r) &=& Z[\phi(r)] f(r) \left( a_{y}'(r) - \frac{r B}{g(r)} \tilde{h}_{x}(r) \right) - \frac{n_{f}}{g(r)} h_{y}(r) + n_{f} B \bar{E}_{x} I_{Z}(r) \; . \qquad
\end{eqnarray}
\end{subequations}
At the boundary these currents encode electric charge conservation in the zero frequency limit and their form as expressed above does not depend on whether we are considering the spontaneous or explicit case. We can arrange for these bulk currents to tend to the vev of the electric charge current as $r \rightarrow 0$ i.e.~$\delta \mathcal{J}_{i}(0) =\lim_{r \rightarrow 0} \partial_r a_i(r) = \langle J^{i} \rangle$. We do not make any explicit magnetisation subtractions. This requires that we make the following identifications
\begin{subequations}
\label{Eq:p1p2identificationspontaneous}
\begin{eqnarray}
p_{x}^{(0)} &=& \left( s_{f} T - m B - k^2 I_{Y} \right) \bar{E}_{x} - \frac{B}{n_{f}} \langle J^{y} \rangle + \frac{\delta s_{x}}{n_{f}} \; , \\
p_{y}^{(0)} &=& \left( s_{f} T - m B - k^2 I_{Y} \right) \bar{E}_{y} + \frac{B}{n_{f}} \langle J^{x} \rangle + \frac{\delta s_{y}}{n_{f}} \; , \\
\label{Eq:p1p2identificationspontaneousvevs}
\delta s_{i} &:=& k \phi_{v}^2 \chi_{i}(0) \; ,
\end{eqnarray}
\end{subequations}
in the spontaneous case where $\langle J^{i} \rangle$ is the vev of the total spatial electric charge current at the boundary and $\delta s_{i}$ is the boundary source for the Goldstone field. As the values of $p_{x}^{(0)}$ and $p_{y}^{(0)}$ were free in \eqref{Eq:p1p2rsols} this presents no difficulty. That the identification of the phonon source is correct and unambiguous was addressed in \cite{Amoretti:2018tzw,Amoretti_2019}.}
{\ There are also a pair of conserved currents related to heat transport. These have the form
\begin{subequations}
\label{Eq:SpontaneousBulkHeat}
\begin{eqnarray}
\delta \mathcal{Q}_{x}(r) &=& - f(r) \left( \frac{h_{x}(r)}{r^2} \right)' + \left( \frac{f(r)}{r^2} \right)' h_{x}(r) + \left( a(r) - \frac{B^2}{n_{f}} I_{Z}(r) \right) \langle J^{x} \rangle \qquad \nonumber \\
&\;& + B \bar{E}_{y} \left( M_{Q}(r) - (s_{f} T + n_{f} a(r) - m B - k^2 I_{Y} ) I_{Z}(r) \right) \nonumber \\
&\;& - \frac{B}{n_{f}} \delta s_{y} I_{Z}(r) - k^2 \delta V_{x} I_{Y}(r) \; , \qquad \\
\delta \mathcal{Q}_{y}(r) &=& - f(r) \left( \frac{h_{y}(r)}{r^2} \right)' + \left( \frac{f(r)}{r^2} \right)' h_{y}(r) + \left( a(r) - \frac{B^2}{n_{f}} I_{Z}(r) \right) \langle J^{y} \rangle \qquad \nonumber \\
&\;& - B \bar{E}_{x} \left( M_{Q}(r) - (s_{f} T + n_{f} a(r) - m B - k^2 I_{Y} ) I_{Z}(r) \right) \qquad \nonumber \\
&\;& + \frac{B}{n_{f}} \delta s_{x} I_{Z}(r) - k^2 \delta V_{y} I_{Y}(r) \; , \qquad
\end{eqnarray}
\end{subequations}
where we have taken\footnote{There is an ambiguity in the definition of $M_{Q}(r)$ as it appears in the bulk heat current. In particular, redefining $M_{Q}(r)$ to be the linear combination
\begin{eqnarray}
M_{Q}(r) &=& - 2 n_{f} B \int_{0}^{r} \frac{dw}{g(w)} \; \left( (\alpha - 1) a(w) Z[\phi(w)] - (\alpha + 1) \frac{I_{Z}(w)}{Z[\phi(w)]} \right) \; ,
\end{eqnarray}
where $\alpha$ is an arbitrary constant, also leads to a conserved heat current that tends to the correct form at the boundary. However, it differs at the horizon and leads to a shift of the thermal Hall conductivity. We have chosen $M_{Q}$ such that $M_{Q}(r_\mathrm{h}) = M_{E} - \mu m$ where $M_{E}$ is the thermodynamic magnetisation energy.}
\begin{eqnarray}
M_{Q}(r) = - 2 n_{f} B \int_{0}^{r} dw \; \frac{Z[\phi(w)] a(w)}{g(w)} \; , \qquad
M_{Q} = M_{Q}(r_{\mathrm{h}}) \; . \qquad
\end{eqnarray}
It can be shown using the asymptotic expansions that these expressions \eqref{Eq:SpontaneousBulkHeat} tend to the canonical heat current at the boundary; again we have made no magnetisation subtractions.}
{\ Additionally there are radially conserved currents corresponding to fluctuations of the axions. These turn out to be linear combinations of the bulk heat and electric charge currents so we relegate their expressions to appendix \ref{appendix:scalarcurrent}.}
{\ Comparing the bulk electric charge current and heat currents at the horizon and boundary allows us to compute the DC conductivities in the standard manner. Firstly, it is possible to identify the boundary electric field ($E_{i}$) and thermal gradients ($\partial_{i} T/T$) from the Frobenius expansions
\begin{eqnarray}
\label{Eq:boundarysourceidentifications}
E_{i} = -\lim_{r\to0} \partial_{t} \left( \delta a_{i} + \frac{\mu r^2}{f(r)} \delta g_{ti} \right) \; , \qquad \frac{\partial_{i} T}{T} = \lim_{r\to0}\partial_{t} \left( \frac{r^2}{f(r)} \delta g_{ti} \right) \; . \qquad
\end{eqnarray}
Substituting asymptotic solutions for our field fluctuations into \eqref{Eq:boundarysourceidentifications} we find the following relations
\begin{eqnarray}
\label{Eq:boundaryidentifications}
\langle J^{i} \rangle &=& (F^{-1})^{ij} \left( n_{f} E_{j} + ( s_{f} T - P_{l} - m B) \frac{\partial_{j} T}{T} - \delta s_{j} \right) \; , \qquad \bar{E}_{i} = \frac{\partial_{i} T}{n_{f} T} \; . \qquad
\end{eqnarray}
Consequently in the spontaneous case one can immediately read off all DC values of the transport coefficients involving the electric charge current i.e.~
\begin{eqnarray}
\sigma_{(\mathrm{H})}(0) = - n_{f} \; , \; \;
\alpha_{(\mathrm{H})}(0) = - \left( s_{f} T - m B - P_{l} \right) \; , \; \;
\gamma_{(\mathrm{H})}(0) = - 1 \; ,
\end{eqnarray}
with all other conductivities involving the current operator being zero. We can derive these without reference to the near horizon values of the bulk current because they are the transport coefficients dictated by symmetry.}
{\ The other DC observables in the spontaneous case can be found by matching the near horizon and boundary values of the conserved radial heat currents. Unfixed thus far are the expectation value of the boundary stress tensor and the values of the sliding mode coefficients $\delta V_{i}$. Employing our expressions \eqref{Eq:boundaryidentifications}, one can write the bulk heat currents $\delta \mathcal{Q}_{i}$ in terms of $E_{i}$, $\partial_{i} T/T$ and $\delta s_{i}$. The resultant thermal conductivities obtained from this bulk current are then:
\begin{eqnarray}
\kappa_{(\mathrm{L})}(0) = \frac{1}{T} \left( \frac{Z_{\mathrm{h}} \left(s_{f} T- P_{l}\right)^2}{n_{f}^2 + B^2 Z_{\mathrm{h}}^2} + \frac{4 \pi P_{l}^2}{s_{f} Y_{\mathrm{h}}} \right) \; , \; \;
\kappa_{(\mathrm{H})}(0) = - \frac{n_{f} \left(s_{f} T - P_{l} \right)^2}{T \left(n_{f}^2 + B^2 Z_{\mathrm{h}}^2 \right)} - \frac{M_{Q}}{n_{f} T} \; , \qquad
\end{eqnarray}
and the thermal-Goldstone zero frequency terms are
\begin{eqnarray}
\theta_{(\mathrm{L})}(0) = \frac{4 \pi I_{Y}}{s Y_{\mathrm{h}}} - \frac{( s_{f} T - P_{l} ) Z_{h}}{n_{f}^2 + B^2 Z_{\mathrm{h}}^2} \; , \qquad
\theta_{(\mathrm{H})}(0) = \frac{n_{f} ( s_{f} T - P_{l} ) Z_{h}}{n_{f}^2 + B^2 Z_{\mathrm{h}}^2} \; .
\end{eqnarray}
Meanwhile, the DC term corresponding to the Goldstone-Goldstone correlator can be obtained from \eqref{eqn:configuration_equation}. To do this we work in equilibrium, vary both sides of equation \eqref{eqn:configuration_equation} with respect to $\delta s_{i}$ and then take the limit of $\omega \rightarrow 0$. In this limit we can identify $X^{ij}(0)$ with the residue of $\langle O_{i} O_{j} \rangle$ at $\omega=0$, see \eqref{Eq:ACSpontaneousconductivities}. Consequently we find
\begin{eqnarray}
X^{ij}(0) = - \frac{\delta V^{i}}{\delta s_{j}} \; , \qquad
\end{eqnarray}
As we have already determined the sliding mode $\delta V_{i}$ in terms of the boundary sources (the expression is too long to include here) we readily find
\begin{eqnarray}
X_{(\mathrm{L})}(0) = - \left( \frac{4 \pi}{k^2 s_{f} Y_{\mathrm{h}}} + \frac{Z_{\mathrm{h}}}{n_{f}^2 + B^2 Z_{\mathrm{h}}^2} \right) \; , \qquad
X_{(\mathrm{H})}(0) = \frac{n_{f}}{(n_{f}^2 + Z_{\mathrm{h}}^2 B^2)} \; . \qquad
\end{eqnarray}
}
\subsubsection{Explicit case}
{\ The expressions for the bulk electric charge currents \eqref{Eq:BulkChargeCurrents} are unchanged between the explicit and spontaneous case. However the identifications of the constants $p_{x}^{(0)}$ and $p_{y}^{(0)}$ of \eqref{Eq:p1p2rsols} are different
\begin{subequations}
\label{Eq:p1p2identificationexplicit}
\begin{eqnarray}
p_{x}^{(0)} &=& \left( s_{f} T - m B - k^2 I_{Y} \right) \bar{E}_{x} - \frac{B}{n_{f}} \langle J^{y} \rangle - \frac{\langle O^{x} \rangle}{n_{f}} \; , \\
p_{y}^{(0)} &=& \left( s_{f} T - m B - k^2 I_{Y} \right) \bar{E}_{y} + \frac{B}{n_{f}} \langle J^{x} \rangle - \frac{\langle O^{y} \rangle}{n_{f}} \; , \\
\label{Eq:p1p2identificationexplicitvevs}
\langle O^{i} \rangle &:=& k \lambda^2 \chi_{i}'(0) \; .
\end{eqnarray}
\end{subequations}
Consequently, in the explicit case, the expressions for the boundary current \eqref{Eq:boundaryidentifications} are modified to
\begin{eqnarray}
\label{Eq:boundaryidentificationsexplicit}
\langle J^{i} \rangle &=& (F^{-1})^{ij} \left( n_{f} E_{j} + ( s_{f} T - k^2 I_{Y} - m B) \frac{\partial_{j} T}{T} + \langle O_{j} \rangle \right) \; , \qquad
\end{eqnarray}
where again we have employed \eqref{Eq:boundarysourceidentifications}.}
{\ The bulk (radially conserved) heat currents in the explicit case are
\begin{eqnarray}
\delta \mathcal{Q}_{x}(r) &=& - f(r) \left( \frac{h_{x}(r)}{r^2} \right)' + \left( \frac{f(r)}{r^2} \right)' h_{x}(r) \nonumber \\
&\;& + \left( a(r) - \frac{B^2}{n_{f}} I_{Z}(r) \right) \langle J^{x} \rangle
+ \frac{B}{ n_{f}} \langle O_{y} \rangle I_{Z}(r) \qquad \nonumber \\
&\;& + B \bar{E}_{y} \left( M_{Q}(r) - (s_{f} T + n_{f} a(r) - m B - k^2 I_{Y} ) I_{Z}(r) \right) \; , \qquad \\
\delta \mathcal{Q}_{y}(r) &=& - f(r) \left( \frac{h_{y}(r)}{r^2} \right)' + \left( \frac{f(r)}{r^2} \right)' h_{y}(r) \nonumber \\
&\;& + \left( a(r) - \frac{B^2}{n_{f}} I_{Z}(r) \right) \langle J^{y} \rangle
- \frac{B}{ n_{f}} \langle O_{x} \rangle I_{Z}(r) \qquad \nonumber \\
&\;& - B \bar{E}_{x} \left( M_{Q}(r) - (s_{f} T + n_{f} a(r) - m B - k^2 I_{Y} ) I_{Z}(r) \right) \; ,
\end{eqnarray}
These differ from the spontaneous case through the dropping of terms dependent on the sliding modes $\delta V_{i}$ and the replacement of $\delta s_{i}$ by terms proportional to $\langle O_{i}\rangle$.}
{\ Once more, we use our identifications in \eqref{Eq:boundarysourceidentifications}, and the matching of the bulk currents at boundary and horizon, to express the subleading term in the boundary expansion of the bulk graviton and $ \langle O_{i} \rangle$ in terms of the boundary electric field $E_{i}$ and the temperature gradient $\partial_{i} T/T$ - there is no explicit axion source term in our expressions to worry about. Doing this fixes $\langle O_{i} \rangle$ appearing in \eqref{Eq:boundaryidentificationsexplicit} in terms of the boundary fluctuations of temperature and electric field. Consequently we can read off the various DC thermo-electric conductivities. For example, the electric charge conductivities are
\begin{eqnarray}
\sigma_{(\mathrm{L})}(0) &=& \frac{k^2 s_{f} Y_{\mathrm{h}} \left( k^2 s_{f} Y_{\mathrm{h}} Z_{\mathrm{h}} + 4 \pi ( n_{f}^2 + B^2 Z_{\mathrm{h}}^2 ) \right)}{(4 \pi n_{f} B)^2 + \left( k^2 s_{f} Y_{\mathrm{h}} + 4 \pi Z_{\mathrm{h}} B^2 \right)^2} \; , \\
\sigma_{(\mathrm{H})}(0) &=& - \frac{8 \pi n_{f} \left( k^2 s_{f} Y_{\mathrm{h}} Z_{\mathrm{h}} + 2 \pi ( n_{f}^2 + B^2 Z_{\mathrm{h}}^2 ) \right)}{(4 \pi n_{f} B)^2 + \left( k^2 s_{f} Y_{\mathrm{h}} + 4 \pi Z_{\mathrm{h}} B^2 \right)^2} \; .
\end{eqnarray}
\captionsetup[subfigure]{labelformat=empty}
\begin{figure}
\centering
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{kappaLspontaneousArean}
\end{subfigure} \hspace{.01\textwidth}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{thetaLspontaneousArean}
\end{subfigure} \hspace{.01\textwidth}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{XLspontaneousArean}
\end{subfigure} \hfill \\
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{kappaHspontaneousArean}
\end{subfigure} \hspace{.01\textwidth}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{thetaHspontaneousArean}
\end{subfigure} \hspace{.01\textwidth}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{XHspontaneousArean}
\end{subfigure} \hfill
\caption{Spontaneous case AC correlators at $k/\mu=10^{-1}$.
Grey dots are numerical data, solid lines are our analytic hydrodynamic expressions and the dashed grey line is the DC value of the coefficient. \textbf{Left column:} The thermal conductivities ($\kappa(\omega)$) at $T/\mu = 0.06$ and $B/\mu^2 \approx 4.4 \times 10^{-4}$. \textbf{Central column:} The heat-Goldstone correlators ($\theta(\omega)$) at $T/\mu=0.04$ and $B/\mu^2 \approx 3.9 \times 10^{-4}$. \textbf{Right column:} The Goldstone-Goldstone correlators ($X(\omega)$) at $T/\mu=0.02$ and $B/\mu^2 \approx 3.5 \times 10^{-4}$.}
\label{fig:spontaneousAreanACconductivities}
\end{figure}
The rest of the expressions are relegated to appendix \ref{appendix:formulae}. Our expressions involving the conserved currents agree with those computed in \cite{Blake:2015ina} and we have also determined the zero frequency limits for correlators involving the scalars. Finally, from the DC limit of the electric-axion correlator one can identify
\begin{eqnarray}
\zeta_{(\mathrm{L})}(0) &=& \frac{ n_{f} (k^2 s_{f} Y_{\mathrm{h}})^2}{(4 \pi n_{f} B)^2 + \left(k^2 s_{f} Y_{\mathrm{h}} + 4 \pi B^2 Z_{\mathrm{h}}\right)^2} \; , \qquad \\
\zeta_{(\mathrm{H})}(0) &=& \frac{ k^2 s_{f} B Y_{\mathrm{h}} \left(k^2 s_{f} Y_{\mathrm{h}} Z_{\mathrm{h}} + 4 \pi \left(B^2 Z_{\mathrm{h}}^2 + n_{f}^2 \right)\right)}
{(4 \pi n_{f} B)^2 + \left(k^2 s_{f} Y_{\mathrm{h}} + 4 \pi B^2 Z_{\mathrm{h}}\right)^2} \;
\end{eqnarray}
The remaining DC terms in the explicit case are listed in appendix \ref{appendix:formulae}.}
\subsection{AC correlators}
{\ With exact expressions for the DC values of the various correlators we can now employ our hydrodynamic expressions for the correlators and compare to the equivalent quantities obtained from holography. Naturally many of our observations in this section will be model dependent at the quantitative level; however certain features we expect to hold in general models. Moreover, we can put our hydrodynamic theory to a precision test.}
\subsubsection{Spontaneous case}
{\ There are twelve potential AC correlators to display including the transport coefficients, the current-Goldstone correlators and the Goldstone-Goldstone correlators. Given the difficulties in the past of matching the thermal DC conductivities to hydrodynamics we shall choose to display these and not the other conductivities as they tend to be quite robust to small errors. We shall also show the AC thermal-Goldstone correlator and the Goldstone-Goldstone correlators with a magnetic field as these types of correlators are novel in the literature.}
\captionsetup[subfigure]{labelformat=empty}
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{omega0flowwithlambda}
\caption{The pinning frequency against $\lambda/\mu$ at $k/\mu = 0.1$, $B/\mu^2 = 10^{-2}$ and $T/\mu = 5 \times 10^{-2}$. The solid purple line is a best fit to the data points with an expression proportional to $\sqrt{|\lambda|/\mu}$.}
\label{fig:omega0flowwithlambda}
\end{figure}
{\ As can be seen in fig.~\ref{fig:spontaneousAreanACconductivities} the matching between our analytic hydrodynamic expressions and the holographic model is excellent over a wide range of parameters. In fact, it is somewhat surprising that they work so well down to rather low temperatures and relatively high magnetic fields. There exists one peak at $\omega>0$ in the electric, thermo-electric and thermal conductivities corresponding to the cyclotron mode.}
{\ Somewhat new to the literature, but perhaps not unexpected, is the smoothing of the low frequency Goldstone-Goldstone correlators. It was observed in previous works \cite{Amoretti_2019} that these correlators have a double pole in frequency located at $\omega=0$. Taking the zero magnetic field limit of our longitudinal expression for the Goldstone-Goldstone correlator one again finds this double pole emerging. For finite $B$ however one of the degenerate poles is displaced and becomes the cyclotron modes; leaving an isolated pole at $\omega =0$. This can be seen from the lack of any $\omega \rightarrow 0$ divergence in $X_{(\mathrm{L})}(\omega)$ and $X_{(\mathrm{H})}(\omega)$ as displayed in fig.~\ref{fig:spontaneousAreanACconductivities}.}
\subsubsection{Explicit case}
{\ Hydrodynamics and the DC conductivities almost fix the transport coefficients appearing in our hydrodynamic expressions, \eqref{Eq:ExplicitACtransportcoeffs}, completely. There remains a single parameter that must be determined numerically: $\omega_{0}$. This is the pinning frequency of the phonon-like mode. There are a couple of methods by which this may be determined, and we have tested that both are consistent. Firstly, one may examine the quasinormal modes of the theory and solve for $\omega_{0}$ using their position in the complex plane. Alternatively, one may take any of the correlators at low frequency (so that we are in the hydrodynamic regime) and request that the analytic expression match the numerically determined one. Doing so allows one to solve for the pinning frequency.}
\captionsetup[subfigure]{labelformat=empty}
\begin{figure}
\begin{subfigure}{.47\textwidth}
\centering
\includegraphics[width=\linewidth]{kappaLexplicittwopeaksArean}
\end{subfigure} \hfill %
\begin{subfigure}{.47\textwidth}
\centering
\includegraphics[width=\linewidth]{peaksandtroughswithT}
\end{subfigure}
\caption{Conductivities in the (pseudo-) explicit breaking regime.
\textbf{Left:} The AC longitudinal thermal conductivity at $\lambda/\mu= - 10^{-5}$, $k/\mu =0.1$, $B/\mu^2 \approx 3 \times 10^{-4}$ and $T/\mu = 10^{-2}$. Notice
the two peaks, both displaced from $\omega = 0$. \textbf{Right:} The frequency of the maxima (red) and minima (blue) in the hydrodynamic longitudinal electric charge conductivity as a function of temperature at $\lambda/\mu=-10^{-5}$, $k/\mu=0.1$ and $B/\mu^2 = 10^{-3}$. At the lowest temperatures we have two peaks in the $\omega>0$ half-line and also a minimum at $\omega=0$. As the temperature increases the two pseudo-Goldstone modes join ($T/\mu \approx 0.083$) to become a single Drude-like peak at zero frequency. This zero frequency peak eventually drops out of the correlator, becoming a trough, at $T/\mu \approx 0.237$. }
\label{fig:Twopeaksexplicit}
\end{figure}
{\ Regarding the pinning frequency, for a range of $|\lambda/\mu| \in (10^{-5},10^{-2})$ we have found that the pinning frequency $\omega_{0}$ is proportional to $\sqrt{|\lambda|/\mu}$. This is in accordance with the behavior found in the same holographic model at zero magnetic field in \cite{Amoretti:2018tzw} and with more general quantum field theory arguments, as explained in \cite{Amoretti:2016bxs}. We display the flow of $\omega_{0}$ with $\lambda$ for a particular choice of temperature and magnetic field in fig.~\ref{fig:omega0flowwithlambda}. For increasing temperature the pinning frequency gets progressively smaller.}
{\ Now let us consider the behaviour of the suite of thermo-electric AC conductivities. In figures \ref{fig:Twopeaksexplicit} and \ref{fig:conductivities} we plot the electric and thermal conductivities showing an excellent agreement between our hydrodynamic expressions \eqref{Eq:ExplicitACconductivities} and the exact holographic data.
Notice that in the explicit case it is possible that these thermo-electric correlators have two peaks on the $\omega >0$ half-line, both displaced from $\omega=0$ to some finite value of $\omega$; i.e.~the point $\omega=0$ is a minimum.
An example of this phenomenon is displayed in the left hand plot of fig.~\ref{fig:Twopeaksexplicit} for the longitudinal thermal correlator. On the right hand side of the same figure we show the flow of the maxima and minima of the analytic hydrodynamic longitudinal conductivity as a function of $T/\mu$ for a particular choice of $\lambda/\mu$ and $B/\mu^2$.
One can identify a low temperature `phonon regime' where two peaks at nonzero
frequency are observed. At intermediate temperatures the correlator displays a Drude-like peak at the origin and a cyclotron peak at finite $\omega$. We denote
this temperature range as `Drude regime'.
These two peaks,
and the underlying quasinormal modes,
can be qualitatively interpreted
as the magnetophonon and mangnetoplasmon resonances expected in the
hydrodynamic regime of a weakly-pinned Wigner crystal~\cite{Delacretaz:2019wzh}.
At high temperatures
only a cyclotron peak at finite $\omega$ is observed and we expect the correlator to
be well described by magnetohydrodynamics with a decay rate (i.e.~see \cite{Sachdev} for further discussion).
The two inflection points marking the transition between the three regimes
occur when the following conditions are satisfied
\begin{eqnarray}
\zeta_{(\mathrm{L})}'(0) = 0 \, , \qquad
\zeta_{(\mathrm{L})}''(0) = 2 \left( 3 \mu^2 \sigma_{\mathrm{L}}(0) + 4 \mu \alpha_{(\mathrm{L})}(0) + \kappa_{(\mathrm{L})}(0) \right) \,,
\end{eqnarray}
which can be obtained from the Ward identity by requiring $\omega=0$ to be an inflection point.}
{\ In the Drude-like regime the peak at nonzero frequency is associated with the cyclotron mode and the corresponding pair of quasinormal modes.
The peak at $\omega=0$ is however a little unusual in that it is not associated with a single (imaginary) quasinormal mode, but instead with two complex modes. While we have termed this the `Drude regime' on account of the single peak at small frequency, one must be careful in interpreting this since, as explained in details in \cite{amoretti:hydrodynamicmagnetotransport}, it has nothing to do with any explicit coherent momentum decay rate in the hydrodynamic theory. If one looks at hydrodynamics in an external magnetic field with a non-zero momentum decay rate tensor $\Gamma^{ij}$, and no translation breaking scalars, one finds only two quasinormal modes in the diffusive sector which can be identified as displaced cyclotron modes. In particular there is no Drude-like peak in such a system.}
\captionsetup[subfigure]{labelformat=empty}
\begin{figure}
\begin{subfigure}{.47\textwidth}
\centering
\includegraphics[width=\linewidth]{sigmalpseudospontaneousarean.pdf}
\end{subfigure} \hfill %
\begin{subfigure}{.47\textwidth}
\centering
\includegraphics[width=\linewidth]{sigmahpseudospontaneousarean.pdf}
\end{subfigure} \\
\begin{subfigure}{.47\textwidth}
\centering
\includegraphics[width=\linewidth]{sigmaLstronglyexplicitArean.pdf}
\end{subfigure} \hfill %
\begin{subfigure}{.47\textwidth}
\centering
\includegraphics[width=\linewidth]{sigmaHstronglyexplicitArean.pdf}
\end{subfigure}
\caption{AC electric conductivities at $B/\mu^2=10^{-3}$, $T/\mu=10^{-1}$ and $k/\mu=10^{-1}$. Red lines are the analytic hydrodynamic expressions while grey dots are numerical data. \textbf{Left:} The longitudinal conductivities in two regimes - the pseudo-spontaneous (top) where $\lambda \mu/\phi_{v} \approx 0.004$ and a strongly explicit regime (bottom) where $\lambda \mu/\phi_{v} \approx 0.95$. In both cases our hydrodynamic expressions closely match the data. \textbf{Right:} The Hall conductivities in the same regimes.}
\label{fig:conductivities}
\end{figure}
\subsection{On the spurious pole}
{\ Our final observation concerns the number of poles implied by our hydrodynamic expressions. Curiously, the formalism of \cite{Armas:2019sbe,Armas:2020bmo} predicts the existence of an additional gapped pole with respect to the hydrodynamic approach of \cite{amoretti:hydrodynamicmagnetotransport}. The existence of this additional pole is related to the presence of the lattice pressure $P_l$ as one can see from the frequency dependent term in \eqref{Gammafreq}, which gives rise to an extra zero in the denominator of the correlators \eqref{denominatore}, not present in hydrodynamic approach of \cite{Delacretaz:2017zxd,Delacretaz:2019wzh,amoretti:hydrodynamicmagnetotransport}. For the systems we have investigated, this extra pole has always a very large imaginary part (which we checked numerically) and its effect on the diffusive correlators can be ignored.}
\captionsetup[subfigure]{labelformat=empty}
\begin{figure}
\centering
\includegraphics[width=0.47\linewidth]{sigmaLexplicitDrudeArean}
\caption{The longitudinal electric charge conductivity against frequency at small frequencies in the Drude-like regime with $\lambda/\mu = -10^{-5}$, $k/\mu=10^{-1}$, $T/\mu = 0.3$ and $B/\mu^2 \approx 0.066$.
The solid red line corresponds to the hydrodynamic expression, the grey dots are
numerical data, and the dashed Green line shows a pure Drude-like approximation $\sim 1/ (\omega - i \Gamma)$.}
\label{fig:Drudeemergence}
\end{figure}
{\ In the papers where this formalism was developed a small frequency expansion was taken to eliminate this additional pole from the diffusive sector \cite{privatecomm}. In our systems, we could not take a low frequency expansion without washing out all of our poles as the external magnetic field gaps the system. Instead, when comparing our expressions with data, we checked that this additional ``spurious'' pole must reside deep in the complex plane according to our hydrodynamic expressions and as such could be ignored.}
{\ We did check to determine whether there was a hint that this spurious pole existed within the system by examining the numerical correlators at complex frequency around the position predicted by our hydrodynamic expressions. Thus far we have found no evidence of its presence in the diffusive part of the spectrum. In fact, we found that other quasinormal modes become relevant before any hint of this additional pole appears.}
In order to check that this additional pole is not an artifact of the frame choice, we computed the Green functions for the spontaneous case using the method described in Sec~\ref{sec:2}, at zero magnetic field and in two distinct frames, and we found the same results as for the Landau frame. In particular we considered a pseudo-Eckart frame (eliminating all first derivative terms from the electric charge current \eqref{eqn:constitutive_relations} except for the $\gamma$ term) and the true Eckart frame where $J^{\mu} = q u^{\mu}$ to all orders in derivatives. Notice that the $\gamma$ term in \eqref{eqn:constitutive_relations} is naively order zero in derivatives until one substitutes for $u^\mu e^I_\mu$ using the configuration equation.
\section{Conclusions}
\label{sec:conclusions}
In this paper we have provided a complete hydrodynamic description of holographic Q-lattice models which present a spontaneous or a pseudo-spontaneous breaking of translations in the presence of an external magnetic field. To take into account the presence of a non-trivial lattice pressure term $P_l$ in the thermodynamics of these holographic models, we have generalized the hydrodynamic approach of \cite{Armas:2019sbe,Armas:2020bmo} in order to include both a small mass for the Goldstone boson related to translation symmetry breaking and an external magnetic field. Moreover, using the method of \cite{amoretti:magnetothermaltransporta,amoretti:hydrodynamicmagnetotransport}, we have been able to express all the hydrodynamic AC correlators in terms of their DC values and the pinning frequency. Since the DC holographic thermo-electric conductivities have a closed analytic form in terms of horizon data, combining the hydrodynamic result with holography we have provided an analytic form for the holographic correlators in terms of the horizon data of the model and one undetermined quantity, the pinning frequency, which we have obtained numerically. The correlators computed in this way match excellently the holographic numerical result, and the behavior of the pinning frequency agrees with the one reported previously for the same model in the absence of an external magnetic field \cite{Amoretti:2018tzw}.
Finally, the identification of a regime where the AC correlators feature a deep IR peak that can be identified with the magnetophonon collective mode opens the way
for a further exploration of the holographic Q-lattice models as avatars of
strongly coupled electronic phases of matter.
\section*{Acknowledgments}
We would like to thank Jay Armas and Akash Jain for private communications about the hydrodynamic model described in this paper. We also thank Blaise Gout\'eraux for a careful reading of a previous version of the present paper. The project has been partially supported by the INFN Scientific Initiative SFT: ``Statistical Field Theory, Low-Dimensional Systems, Integrable Models and Applications''. D. A. is supported by the `Atracci\'on de Talento' programme
(2017-T1/TIC-5258, Comunidad de Madrid) and through the grants SEV-2016-0597 and PGC2018-095976-B-C21.
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{"url":"https:\/\/judgegirl.csie.org\/problem\/0\/50116","text":"# 50116. Play with digits\n\n## I'm a slow walker, but I never walk backwards.\n\nWe have a huge decimal number N. Write a program to determine the followings:\n\n\u2022 The number of digits in N.\n\u2022 Is N an even number?\n\u2022 The number of zeros in it.\n\u2022 Is N a multiple of 11? Note that we can determine if N is a multiple of 11 by checking the difference between the sum of the odd positioned digits and the sum of the even positioned digits. For example, 82375 is not a multiple of 11 because the sum of the even positioned digits is 2 + 7 = 9, and the sum of the odd positioned digits is 8 + 3 + 5 = 16, and the difference between 9 and 16 is 7, which is not a multiple of 11.\n\nWe will give you the number one digit per line. For example, if you get digits \u20181\u2019, \u20182\u2019, \u20183\u2019, \u20194\u2019, \u20180\u2019 in order, then the number is 12340. The number will not start with 0.\n\n## Input Format\n\nThe input has several lines. Each line has a digit. EOF indicates the end of input.\n\n## Output Format\n\nOutput the four answers above line by line. If the number is even output a 1; otherwise a 0. If the number is a multiple of 11 output a 1; otherwise output a 0.\n\n\u2022 10 points: you can store the decimal number in an integer without overflow\n\u2022 10 points: the number of digits is no more than 32768, so you can store digits in an array\n\u2022 80 points: you will get MLE if you use array\n\n## Sample Input 1\n\n12340\n\n\n## Sample Output 1\n\n5110","date":"2022-05-23 18:00:00","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\": 0.3344021141529083, \"perplexity\": 359.8141561066406}, \"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\/1652662560022.71\/warc\/CC-MAIN-20220523163515-20220523193515-00074.warc.gz\"}"} | null | null |
Read Full Report
Date: Sept. 13, 2007
Contacts: Jennifer Walsh, Media Relations Officer
William Kearney, Director of Media Relations
Kimberly Berryman, Media Relations Assistant
202-334-2138; e-mail <news@nas.edu>
<?xml:namespace prefix = st1 ns = "urn:schemas-microsoft-com:office:smarttags" />
U.S. Climate Change Science Program Making Good Progress
In Documenting And Understanding Changes, But Study Of Impacts On Humans And Communication With Decision Makers Lag
WASHINGTON -- Climate change research directed by the federal government has made good progress in documenting and understanding temperature trends and related environmental changes on a global scale, says a new report from the National Research Council. The ability to predict future climate changes also has improved, but efforts to understand the impact of such changes on society and analyze mitigation and adaptation strategies are still relatively immature, added the committee that wrote the report. Moreover, the U.S. Climate Change Science Program (CCSP), which oversees federal research in this area, has made inadequate progress in supporting decision making, studying regional impacts, and communicating with a wider group of stakeholders.
"CCSP, an important initiative that has broadened our knowledge of climate change, needs to package more of that knowledge for policymakers from the national to local level, and place more emphasis on understanding how people will be affected by climate change and how they might react," said committee chair Veerabhadran Ramanathan, Distinguished Professor of Atmospheric and Climate Sciences at the Scripps Institution of Oceanography, University of California, San Diego.
Adjustments will have to be made in the balance between basic science and applications if CCSP is to achieve its vision of producing information that can be used to formulate strategies for preventing, mitigating, and adapting to the effects of climate change, the committee stated. It did not offer recommendations for how to sustain and improve the program's basic science while strengthening its applications, but this will be among the subjects considered in a follow-up report that the committee expects to issue early next year.
The report was requested by CCSP's former director, who asked the Research Council to develop a process for evaluating the program and to conduct a preliminary assessment of its progress. The committee's report is the first review of CCSP's progress since the program was established in 2002.
The committee developed a two-stage evaluation process. The first stage, presented in this report, assesses the strengths and weaknesses of the entire program, and identifies areas where progress has not met expectations and that should be subject to more detailed analysis during a second stage of evaluation. This second stage, to be completed by CCSP because it requires detailed budget and management information not readily available to the committee, would diagnose the reasons for weaknesses and identify strategies for improving the program.
In its review, the committee concluded that discovery science and understanding of the overall climate system are proceeding well. For example, knowledge of the nature and extent of atmospheric warming and other climate changes over the past few decades and the influence of human activities on these observed changes has advanced significantly. In addition, models that have demonstrated reasonable success in reproducing past climate conditions are improving confidence in future projections. Understanding of the water cycle has also improved, and good progress has been made in documenting land-use changes and estimating how carbon is distributed around the planet.
Uncertainties remain in other aspects of global climate change, particularly the role of man-made aerosols in masking greenhouse warming, the response of hurricanes and ice sheets to global warming, and how climate feedbacks -- the dynamics of water vapor and clouds, for example -- amplify or dampen the effects of greenhouse gases and other climate-change forces.
Overall, research into the social sciences, including human drivers of climate change such as energy consumption, the impact on human systems such as political institutions and economies, and mitigation and adaptation options, is much less developed than research on the natural climate system. One reason for the slow progress is that only $25 million to $30 million of CCSP's $1.7 billion annual budget is devoted to such research. In addition, few social scientists are in leadership positions at the participating federal agencies, making it difficult for CCSP to increase emphasis in this area or to establish links with the academic social science community.
Even where good scientific progress is being made, use of new knowledge to support decision making and risk analysis is proceeding slowly, according to the committee. For instance, although CCSP's temperature trends assessment was influential in this year's report by the Intergovernmental Panel on Climate Change, 19 other synthesis and assessment products that were scheduled for release by now are still in production.
One way CCSP could bridge the gap between science and decision making would be to more closely examine the impact of climate change at regional and local scales, the report says. More accurate models, better regional observations, and the development of impact scenarios will be required to improve predictions of how climate change will affect smaller spatial scales.
Better communication from CCSP also will be critical for confronting climate change at the local level. CCSP should build upon the two-way dialogue envisioned in its strategic plan by engaging state and local officials, nongovernmental organizations, industry, and the climate change technology community. This dialogue should go beyond communicating research results to asking what is needed from the program. The committee acknowledged that more resources will be needed to bolster such relationships.
A major hurdle to CCSP progress is the program director's lack of authority to allocate or prioritize funding across participating agencies, the committee said. Likewise, many of the members of CCSP's interagency working groups have little budgetary authority to implement the program's research agenda. As a result, progress tends to occur when the priorities of the 13 participating agencies coincide with CCSP's goals.
The committee emphasized that high-quality data from satellites have been crucial to the advancement of climate change science. However, a number of planned satellite missions have been cancelled or seriously delayed, presenting perhaps the single greatest threat to the future success of CCSP, according to the committee. Without these satellites, scientists' ability to monitor and predict climate change will decline, even as the urgency of doing so increases.
The committee is holding a workshop in Washington, D.C., Oct. 15-17, to discuss future priorities for CCSP research, which will be the focus of its follow-up report.
The study was sponsored by the U.S. Climate Change Science Program. The National Academy of Sciences, National Academy of Engineering, Institute of Medicine, and National Research Council make up the National Academies. They are private, nonprofit institutions that provide science, technology, and health policy advice under a congressional charter. The Research Council is the principal operating agency of the National Academy of Sciences and the National Academy of Engineering. A committee roster follows.
Copies of Evaluating Progress Of The U.S. Climate Change Science Program: Methods And Preliminary Results will be available from the National Academies Press; tel. 202-334-3313 or 1-800-624-6242 or on the Internet at http://www.nap.edu. Reporters may obtain a pre-publication copy from the Office of News and Public Information (contacts listed above).
[This news release and report are available at http://national-academies.org ]
Committee on Strategic Advice on the U.S. Climate Change Science Program
Veerabhadran Ramanathan (Chair)
Distinguished Professor of Atmospheric and Climate Sciences,
and Victor C. Alderson Professor of Applied Ocean Sciences
Scripps Institution of Oceanography
Christopher O. Justice (Vice Chair)
Director of Research and Professor
John B. Carberry
Director of Environmental Technology
E.I. du Pont de Nemours and Co.
Wilmington, Del.
Robert E. Dickinson
Eileen E. Hofmann
Department of Ocean, Earth, and Atmospheric Sciences, and
Center for Coastal Physical
Old Dominion University
James W. Hurrell
Climate and Global Dynamics Division
National Center for Atmospheric
Jeanine A. Jones
Interstate Resources Manager
Roger E. Kasperson
Research Professor and Distinguished Scientist
Charles D. Kolstad
Donald Bren Professor of Environmental Economics and Policy
Bren School of Environmental Science and Management, and
Maria Carmen Lemos
Associate Professor of Natural Resources and Environment
Ann Arbor, and
Senior Policy Analyst
Udall Center for Studies of Public Policy
Department of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology, and
Joint Program in Oceanography and Ocean Engineering of MIT and Woods Hole Oceanographic Institution
Ellen S. Mosley-Thompson
Department of Geography, and
Byrd Polar Research Center
Guido D. Salvucci
Professor and Chair
Department of Earth Sciences, and
Department of Geography and
Susan E. Trumbore
Department of Earth System Science, and
Institute for Geophysics and Planetary Physics
T. Stephen Wittrig
Director of Advanced Technologies
Naperville, Ill.
RESEARCH COUNCIL STAFF
Anne Linn | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,898 |
\section{Introduction}
The formation of stellar black holes (BH), representing the ultimate stage of dying stars with an initial mass $\gtrsim 15-20 ~{\rm M}_\odot$, is a process occurring on time-scales of a few to tens of Myr. Tens to thousands of such BHs are expected to form in dense stellar environments, such as globular (GCs) or nuclear clusters (NCs) \citep{kulkarni93,sigurdsson93}. At their birth, BHs may receive strong natal kicks that potentially can lead to their ejection from the parent cluster. However, the recoiling velocity amplitude is still a matter of debate.
A number of works suggested that BHs the kick distribution is similar to what expected for neutron stars \citep{repetto12,janka13,sippel13,mandel16}. However, it seems possible that massive stars can undergo direct collapse, turning into BHs without losing a large mass fraction and avoiding supernova explosion \citep{adams2017}. According to this picture, BHs might have masses quite larger than previously thought, and experience no or low natal kicks
\citep{fryer99,belckzinski10,fryer12,spera15}.
Small recoiling velocities imply that the BHs retention fraction in dense stellar clusters is much larger than previously thought. A larger retention fraction is also required to explain the ever-growing observational evidence of BHs signatures in Galactic GCs \citep{strader12,chomiuk13,miller15,bahramian17,giesers18}, whose presence seem largely supported by recent numerical works \citep{morscher13,wong14,morscher15,repetto15,peuten16}.
Retained BHs would undergo rapid mass segregation, populating the inner regions of their parent cluster and likely forming a subsystem on a core-collapse time-scale \citep{spitzer87,portegies00,zwart02,fregeau2004,zwart04,gaburov08,freitag06c,AS16}.
Three-body interactions and multiple scatterings can drive the formation of BH binaries, which act as a power supply for the cluster core. These binaries kick out the most massive BHs, depleting the global BH reservoir, and eventually kick out each other through super-elastic encounters \citep{banerjee10,downing10,rodriguez15,askar17}.
The single-binary and binary-binary continuous interactions lead BH binaries to get harder and harder, until they are ejected from the cluster core or merge in there releasing gravitational waves (GWs) \citep{portegies00,banerjee10,downing10,rodriguez15,rodriguez16,wang16,askar17}.
Stars interacting with retained BHs are pushed on wider orbits, causing the expansion of the GC core and delaying core-collapse \citep{merritt04,mackey08,gieles10,wang16}.
Currently, there is no general consensus on the definition of a BHS. Recently, \cite{breen13} investigated how a population of BHs behaves in idealized star cluster models, revisiting the pioneering work made by \cite{spitzer71} \cite[but see also][]{kulkarni93,sigurdsson93}.
In the case of a so-called ``Spitzer-instable'' system, the BHs lose energy to the other stars and segregate toward the GC centre, causing a progressive reduction of the BHS half-mass radius. This contraction slows down as soon as the BHS energy is transferred to the surrounding stars and is balanced by the ``thermal energy'' provided by the BHs binaries formed in the very inner portion of the GC. The complex interactions provide sufficient energy to avoid further collapses, although a stable configuration is hardly achievable \citep{spera16,bianchini16}.
Under the simple assumption of a two-mass population of objects and using theoretical arguments, \cite{breen13} have shown that the energy generated from the repeated scattering between the light and heavy objects, which flows through the GC half-mass radius, regulates the evolution of the heavier component, which settles into the GC centre.
Their results suggest that GCs having a sufficiently long half-mass relaxation time can retain a sizeable number of BHs. However, is quite hard to define a BHS since BHs are typically mixed with other stars in realistic GC models, making difficult to define the BHS size and structure.
In this paper, we propose a novel method to define the BHS radial extent. Our definition of BHS radius is similar to the ``influence radius'' defined for supermassive BHs that inhabit galactic nuclei \citep{peebles72,merritt06,merritt13}.
Using a large suite of GC models simulated in the context of the ``MOCCA-Survey Database I project'', we determined a set of scaling relations aimed at allowing us to infer the presence of a BHS, and its main properties, through the observational and structural features of its host cluster. We found that GCs having a BHS are distributed in a narrow region of the surface brightness - average surface luminosity plane, well detached from GCs having a central IMBH or that exhibit none of these ``dark'' features.
These relations represent a unique tool to unveil the presence of a BHS in the centre of GCs. In a companion paper, we use these correlations to select a sample of 29 Galactic GCs that may harbor a BHS in their centre \citep{askar18}.
The paper is organized as follows: in section \ref{label:gcm} we briefly describe the MOCCA numerical models used in this work; in section \ref{sec:bhsingcs} we introduce our definition of BHS, discussing the basic relations connecting the BHS main properties; section \ref{label:scali} is focused on the scaling relations connecting the BHS main parameters and the host cluster structural properties; section \ref{sec:obser} presents the fundamental relation that allows connecting the GC observational properties to the BHS density. Finally, in section \ref{sec:end} are summarised the conclusions of this work.
\section{Globular cluster models}
\label{label:gcm}
\subsection{The MOCCA SURVEY DATABASE I.}
In this paper, we use the results from the MOCCA-Survey Database I \citep{askar17} that comprises of about 2000 realizations of GCs with different initial masses, structural and orbital parameters. These models were simulated with the MOCCA code for star cluster simulations, which treats the relaxation process using the method described by \citet{henon71}, conveniently improved by \citet{stdo86}, and recently by \cite{giersz08} \citep[but see also ][ and reference therein]{giersz13,Giersz15}. MOCCA implements the SSE and BSE codes \citep{hurley00,hurley2002} for treating binary and stellar evolution, while strong binary-single and binary-binary interactions are handled by the \texttt{FEWBODY} code \citep{fregeau2004}.
The initial parameters of the models simulated in the MOCCA-Survey Database I can be found in Table 1 in \citep{askar17}.
In nearly half of the simulated models, supernovae natal kick velocities for neutron stars and BHs are assigned according to a Maxwellian distribution, assuming a dispersion of $265$ km s$^{-1}$ \citep{hobbs}. In the remaining cases, BH natal kicks were modified according to the mass fallback procedure described by \cite{belczynski02}. The model metallicities are selected between $Z = 0.0002, ~0.001, ~0.005, ~0.006$ or $0.02$.
All MOCCA models are characterized by a \cite{kroupa01} initial mass function, with a minimum and maximum initial stellar mass of $0.08~{\rm M}_\odot$ and $100 ~{\rm M}_\odot$, respectively.
The total number of objects sampled in our simulated GCs are $4\times 10^4$, $10^5$, $4\times 10^5$, $7\times 10^5$ and $1.2\times 10^6$, including both single stars and primordial binaries. All our GCs are described by \cite{King} models, with central concentration parameters values $W_0 = 3,~6$ and $9$. We assumed an initial tidal radius $R_t = 30,~60$ or $120$ pc, while the ratio between the tidal radius and the GC half-mass radius is $50$, $25$ or the model is tidally-filling. We allowed for four different values of the primordial binary fraction: $5\%$,
$10\%$, $30\%$ and $95\%$. In models characterised by an initial binary fraction equal to or lower than $30\%$, we selected the initial eccentricities of binary systems according to a thermal distribution \citep{jeans19}, the semi-major axes according to a flat logarithmic distribution, and the mass ratio according to a flat distribution. For models containing a larger binary fraction, instead, the initial binary properties are selected according to the distribution provided by \citet{kroupa95,kroupa11}.
The GCs are assumed to move on a circular orbit at Galactocentric distances between $1$ and $50$ kpc. The Galactic potential is modelled in the simple point-mass approximation, taking as central mass the value of the Galaxy mass enclosed within the GCs orbital radius.
As pointed out in \citet{askar17}, the initial conditions assumed to create the MOCCA-Survey Database I were not specifically selected to reproduce the Galactic GC population. Nevertheless, their observational parameters calculated at the present-day exhibit a remarkably good agreement with Milky Way GCs.
\subsection{Globular clusters hosting a black hole subsystem}
In order to focus on the BHS properties, we selected models retaining at least 10 BHs after 12 Gyr. Our subsample, comprised of $N_{\rm sub}=172$ out of the over 2000 simulated systems, contains GCs with different properties, spanning a wide range of initial masses, binary fraction, and initial metallicity.
\\
The corresponding initial mass distribution peaks at $M_{\rm GC} \sim 6.3\times 10^5~{\rm M}_\odot$, with $\sim 154$ models having masses in between $M_{\rm GC}\sim 3.2-10\times 10^5~{\rm M}_\odot$ while the remaining are smaller ($M_{\rm GC} \lesssim 2.2\times 10^5~{\rm M}_\odot$).
The GC inital core radii ($r_c$) are smaller than 1.5 pc in nearly $80\%$ of the models, with more than 50 models having $r_c = 0.8$ pc. Nearly $90$ models have an initial half-mass radius ($r_h$) smaller than 3 pc, 55 have $r_h = 4.5$ pc while the remaining are more extended tidally-filling models having $6<r_h<17$ pc.
The initial \cite{King} $W_0$ parameter is evenly distributed among small $W_0 = 3$ (65 models), intermediate $W_0 = 6$ (70 models), while a smaller number of cases have higher values $W_0 = 9$ (37 models).
The GCs initial relaxation time, $T_{\rm rel}$, varies in a wide range: $\lesssim 0.65 Gyr$ (19 models), $\simeq 1-2 Gyr$ (118 models), $\gtrsim 10 Gyr$ (35 models).
More than a half of the GCs in the sample (96) are characterised by metallicities around $ Z = 10^{-3}$, while a few models have $Z \lesssim 2.5\times 10^{-4}$ (9), and 42 models have sub-solar metallicities ( $Z \sim 6\times 10^{-3} $). In the remaining 25 models, instead, the initial metallicity assumes solar-values.
In all the models containing a BHS, the BH natal kicks was calculated taking into account the amount of matter that fallbacks after supernova explosion, according to \cite{belczynski02}.
Our sample consists of GCs having a low initial binary fraction ($f_{\rm bin}\leq 0.1$ for 88 models) or intermediate values ($0.1 < f_{\rm bin} \leq 0.3$ for 20 models), while a substantial fraction is ``binary-rich'' ($f_{\rm bin}=0.95$ for 64 models).
Overall, the sample seems quite heterogeneous and is characterised by quite different initial conditions, thus highlighting at a glance that BHS can be a common feature of GCs. In the next section we will show how it is possible to infer the BHS main parameters from the observational and structural properties of the parent cluster.
Interestingly, a substantial number of our selected models ($\sim 120$) have large galactocentric radii, $R_0 > 2$ kpc, although almost $1/3$ of them
orbits at smaller distances from the Galactic Centre. GCs moving at smaller distances have masses in between $4\times 10^5 ~{\rm M}_\odot$ and $1.1\times 10^6~{\rm M}_\odot$.
Clusters having sufficiently small apocentres can segregate toward the Galactic Centre due to the intense action of dynamical friction (df) \citep{Trem76,Dolc93}.
Figure \ref{F1} shows how the df time-scale $t_{\rm df}$ varies at varying GCs masses and Galactocentric distances for the MOCCA models containing either an intermediate mass BH (IMBH) with mass above $10^2~{\rm M}_\odot$ or at least 10 BHs after 12 Gyr.
The df time is calculated following \cite{ASCD14a} (but see also \cite{ASCD15He}), according to which $t_{\rm df}\propto M_{\rm GC}^{-0.67}R_0^{1.76}$.
To represent the Milky Way we used the model recently provided by \cite{kafle14}, consisting of a \cite{Her90} sphere with scale length $\sim 11$ kpc and total mass $6\times 10^{11}~{\rm M}_\odot$.
We see that a substantial number of MOCCA models with large masses and small Galactocentric distances will quickly diffuse to the Galactic center reducing substantially the number of models with (IMBH), but not strongly influencing the number of models with BHSs. To form an IMBH, clusters have to be initially very dense \citep{Giersz15},- massive with small tidal radius, but to sustain BHSs until the Hubble time clusters need to be initially not too dense \citep{breen13} - relatively large half-mass radius.
In order to provide a very preliminary investigation about whether the GCs global properties can be used also to infer the presence of an IMBH in their inner regions, we selected 470 MOCCA models hosting a central BH heavier than $150~{\rm M}_\odot$ at 12 Gyr. We stress here that this subject will be deeply discussed in a companion paper.
Note that the possibility that some GCs deliver their IMBH or BHS toward the galactic centre can have interesting implications for IMBH-SMBH pairing and coalescence events, as recently investigated by \citep[Arca Sedda and Gualandris, in prep.]{ASCD17b,fragione17}.
Orbitally segregated GCs can deposit into the hosting galactic centre a substantial population of BHs living in binary systems. For instance, the progenitor of low-mass X-ray binaries (LMXBs), containing either an NS or a stellar BH, can easily be transported into the galactic centre from inspiral clusters.
As a consequence, the population of LMXBs inhabiting the galactic inner regions might benefit from the GC infall process.
Recently, detailed observations of the Milky Way nuclear cluster (MWNC) revealed the presence of as many as 20000 BHs probably orbiting the SMBH surroundings \citep{hailey18}. As suggested by a number of works, most of the MWNC likely formed through repeated mergers of $\sim 10-20$ massive star clusters with masses above $\simeq 10^6~{\rm M}_\odot$ \citep{AMB,antonini14,ASCD14b,ASCD15He,ASK17}.
As we will show in detail in the following, BHS constitute nearly the $70\%$ of the GC total BH reservoir. Assuming that one BH form every 1000 stars, which is expected from standard stellar evolution, this means that infalling clusters might have brought to the MWNC $\sim 0.7\times 10^{-3} \times 10^6 \times 20 = 15000$ BHs, either as a single object or in a binary system. This number fits nicely with the values inferred recently by \citep{hailey18}.
In a subsequent paper, we will explore whether delivered BHs can lead to the formation of a number of LMXB containing a stellar BH consistent with the latest observations and modelling \citep{generozov18}.
\begin{figure}
\centering
\includegraphics[width=8cm]{tdf2.eps}
\caption{
MOCCA GCs initial Galactocentric radius $R$ (Y axis) and initial total mass $M$ (X axis).The color-coded map marks the retained number of BHs after 12 Gyr. The shaded regions identify $M-R$ couples characterized by $t_{\rm df} = 1$ (red region), $5$ (cyan region) and $12$ Gyr (grey region). The lower boundary of each region represents $t_{\rm df}$ for circular orbits, while the upper boundary marks the limit in which the GC moves on a nearly radial orbit. Filled circles represent GCs having at least 10 BHs at 12 Gyr, while crosses identify those hosting an IMBH.
}
\label{F1}
\end{figure}
\section{Black Hole Subsystems in globular clusters}
\label{sec:bhsingcs}
\subsection{A novel definition for BH subsystem}
As shown by \cite{breen13}, in the idealized case that a massive GC can be modeled as a two-mass population system, the energy exchange rate between the BHS and the surrounding stars depend on the energy flow through the GC half-mass radius $r_h$ and the corresponding half-mass relaxation time $t_{rh}$.
In particular, they suggest that the ratio between the BHS and the GC core radius scale as the ratio between the average BHS and GC mass and the ratio of their total masses
\begin{equation}
\frac{r_{{\rm BHS},h}}{r_{{\rm GC},h}} \propto \left(\frac{m_{\rm BHS}}{m_{\rm GC}}\right)^{2/5}\left(\frac{M_{\rm BHS}}{M_{\rm GC}}\right)^{3/5}.
\end{equation}
This implies that to sustain a BHS up to the Hubble time, the GC half-mass relaxation time has to be larger than about 1 Gyr \citep{breen13}.
We note here that, as long as this relation remains valid, it can have profound implications on the BHS lifetime.
The most massive BHs will be ejected in strong binary-binary and binary-single encounters Promptly after the BHS core-collapse, thus reducing the total BHS mass and its average mass as well. As a consequence, the outward flux energy generated by the BH-BH/BH-stars interactions decreases and the BHS contracts. This, in turn, drives a density increase and a consequent enhancement of the dynamical interactions rate until they can sustain the energy flow.
Hence, GCs having a large initial relaxation time should contain massive and extended BHS.
However, as long as new binaries form and multi-body processes occur efficiently, resulting in the depletion of BHs, the energy supply is insufficient and the BHS slowly dissolves into the sea of other stars.
During these complex stages, which last on time-scales comparable to the half-mass relaxation times, the BHS can be sufficiently dense to mimic the effect of an intermediate-mass black hole, exhibiting similar scaling relations with the host GC mass \cite{AS16}.
Since BHs are usually ``mixed'' with other stars, a natural definition of BHS radius would be the region where BHs play a dominant role in determining the dynamics. Following this idea, we define the BHS size as the sphere enclosing
$50\%$ of the cumulative mass in BHs and the remaining in other stars.
By definition, the radius of this sphere, $R_{\rm BHS}$, encloses
twice the total mass of the BHS, thus representing an analogous of the well-known ``influence radius'' calculated for an isothermal sphere \citep{merritt13}. In fact, $R_{\rm BHS}$ defined this way marks the region over which BHs affect significantly the host GC inner dynamics.
To investigate possible similarities between our definition of BHS size and \cite{breen13} theoretical predictions, we show in Fig. \ref{F2} how do they compare with the actual BH half-mass radius, calculated at 12 Gyr for all the MOCCA models hosting more than 10 BHs.
\begin{figure}
\centering\includegraphics[width=8cm]{BH13comparison.eps}
\caption{Our definition of BHS radius (filled red squares) and \citet{breen13} predicted values (open black circles) as a function of the actual BH half-mass radius as calculated for our MOCCA sample. The straight black line represents the equality between calculated and predicted values, i.e. $f(x) = x$.}
\label{F2}
\end{figure}
\cite{breen13} definition of BHS size seems to over-predict the actual BH half-mass radius, especially for values above 1 pc. Interestingly, our definition agrees pretty well with the real $r_{{\rm BH}{\rm,h}}$ value.
A rough explanation for the similarity between $r_{{\rm BH}{\rm,h}}$ and $R_{\rm BHS}$ can be developed following simple arguments. Let's assume that the BH mass distribution can be described by an isothermal sphere,
\begin{equation}
M_{\rm BH}(r) = \frac{\sigma_{\rm BH}^2}{(2\pi G)} r,
\end{equation}
being $\sigma_{\rm BH}$ the central velocity dispersion of the BHs population. Therefore, the resulting BH half-mass radius will be given by:
\begin{equation}
r_{{\rm BH}{\rm, h}} = \frac{\pi G}{\sigma_{\rm BH}^2} M_{\rm BH} .
\end{equation}
The corresponding GC mass enclosed within $r_{{\rm BH},{\rm h}}$ can be calculated as
\begin{equation}
M(r_{{\rm BH},{\rm h}}) = \frac{1}{2}\left(\frac{\sigma}{\sigma_{\rm BH}}\right)^2 M_{\rm BH}.
\end{equation}
Hence, under the hypothesis of equilibrium between stars and BHs, $\sigma_{\rm BH} \sim \sigma$,the BH half-mass radius contains the same amount of mass in stars and BHs, and this roughly corresponds to $50\%$ of the whole BH mass.
This is clearly an oversimplification of the whole picture, but provides a simple explanation for the similarity between the BH half-mass radius and our definition of BHS size. As we will discuss in the next section, the BHS defined here can contain up to $70\%$ of the BHs total mass, thus deviating from the half-mass radius. However, we will show that using $R_{\rm BHS}$ instead of $r_{{\rm BH}{\rm ,h}}$ allows us to provide a large set of tight scaling relations connecting the GC and the BHs properties.
\subsection{BHS basic properties}
\label{label:scali}
Following the aforementioned assumptions, we define the BHS mass ($M_{\rm BHS}$) as the mass in BHs enclosed within $R_{\rm BHS}$, while $N_{\rm BHS}$ is the number of BHs inside $R_{\rm BHS}$, $m_{\rm BHS} = M_{\rm BHS} / N_{\rm BHS}$ represents the BHS average mass\footnote{Note that this is the mean mass of BHs contained within $R_{\rm BHS}$.}, and $\rho_{\rm BHS} = M_{\rm BHS}/R_{\rm BHS}^3$ the BHS typical density.
\begin{figure*}
\includegraphics[width=8cm]{Rbh_Mbh.eps}
\includegraphics[width=8cm]{M_N_R_BHS.eps}\\
\includegraphics[width=8cm]{Rbh_mbh.eps}
\includegraphics[width=8cm]{Rbh_rhobh.eps}\\
\caption{BHS main correlations. Top left panel: $M_{\rm BHS} -R_{\rm BHS}$ relation, the color-coded map identifies the host cluster mass at 12 Gyr. Top right panel: number of BHs in the BHS as a function of the BHS mass, the coloured map marks the BHS radius. Bottom left panel: average mass of BHs in the subsystem as a function of $R_{\rm BHS}$, the color-coded map refers to the GC central density at 12 Gyr. Bottom right panel: BHS density as a function of its size, mapped on the GC mass at 12 Gyr.
}
\label{F3}
\end{figure*}
Figure \ref{F3} shows the basic correlations linking the BHS fundamental parameters.
These relations allow us to connect the BHS total mass, radius, and typical density each other. As we show below, the latter quantity can be directly connected with the GC observational properties, making scaling relations the most promising tools to explore the BHS-GC connections.
The $M_{\rm BHS}$ and $R_{\rm BHS}$ relation is well described by a simple power-law, whose best fitting is given by
\begin{equation}
{\rm Log} \left(\frac{M_{\rm BHS}}{~{\rm M}_\odot}\right) = \alpha {\rm Log} \left(\frac{R_{\rm BHS}}{{\rm pc}}\right) + \beta,
\label{MRBHS}
\end{equation}
with $\alpha = 0.77\pm0.07$ and $\beta = 3.05\pm0.03$, while the BHS density $\rho_{\rm BHS}$ is linked to the BHS size through \begin{equation}
{\rm Log} \left(\frac{\rho_{\rm BHS}}{~{\rm M}_\odot ~{\rm pc}^{-3}}\right) = \alpha {\rm Log} \left(\frac{R_{\rm BHS}}{{\rm pc}}\right) + \beta,
\label{RhoRBHS}
\end{equation}
with $\alpha = -2.11\pm 0.07$ and $\beta = 2.86\pm0.03$.
A closer look at the top left panel of Figure \ref{F3} reveals an interesting connection between the BHS and its host cluster. Indeed, it suggests that more massive clusters harbor heavier BHS at fixed $R_{\rm BHS}$ values.
The number of BHs in the BHS correlates very tightly with the BHS mass, as shown in Figure \ref{F3}, through a power-law
\begin{equation}
{\rm Log} N_{\rm BHS} = \alpha{\rm Log} \left(\frac{M_{\rm BHS}}{~{\rm M}_\odot}\right)+ \beta
\end{equation}
with slope $\alpha = 0.903\pm0.008$ and intercept $\beta = -0.79\pm0.02$.
This implies a slow increase of $m_{\rm BHS}$ at increasing values of the BHS total mass, being $m_{\rm BHS} \propto M_{\rm BHS}^{0.1}$. This is in a good agreement with the BHS evolution picture presented at the beginning of this Section.
Note that the correlation becomes tighter at $N_{\rm BHS} > 20$, while below this threshold the data points are much more dispersed. Due to this, in the following we will take into account only subsystems containing at least 20 BHs.
Top right panel in Figure \ref{F3} makes evident that at a fixed $N_{\rm BHS}$ value, larger BHS masses correspond to larger BHS sizes. On the other hand, for fixed BHS mass and $N_{\rm BHS} > 20$, a lower number of BHs corresponds to a larger BHS size thus suggesting that the larger the BHS average mass, the larger its size.
The BHS mean mass correlates with $R_{\rm BHS}$
\begin{equation}
{\rm Log}\left(\frac{m_{\rm BHS}}{~{\rm M}_\odot}\right) = \alpha{\rm Log} \left(\frac{R_{\rm BHS}}{{\rm pc}}\right)+\beta,
\label{mvsR}
\end{equation}
with $\alpha = 0.13\pm0.01 $, and $\beta = 1.083\pm 0.005$ the best fitting values.
Moreover, it turns out that the BHS structure depends on the GC central density at 12 Gyr, $\rho_{12}$. Indeed, our analysis suggests that low-density GCs seem to host BHS characterised by larger $R_{\rm BHS}$ values and comprised of heavier BHs than denser GCs, on average.
\subsection{Dynamical consequences of BHS in GCs: phenomenological relations}
In this section, we will investigate whether our BHS definition can be used to connect the GC dynamical status with the retained BH population. Indeed, the presence of a conspicuous number of BHs surviving into the host cluster core up to 12 Gyr is expected to shape significantly the GC properties.
\\
For instance, the top panel of Figure \ref{F4} shows how the BHS and GC densities vary at varying the number of stars within $R_{\rm BHS}$ and the BHS average mass.
This relation can hide some information about the dynamical status of the host GC which is rather difficult to see.
We can schematize the GC-BHs common evolution and ``dynamical feedback'' as follows:
\begin{enumerate}
\item massive stars evolve and become BHs while rapidly segregating to the GC core, leading to the formation of a massive BHS;
\item the BHS injects energy in the surroundings, losing energy to other stars and causing the GC core expansion, thus leading to a lower GC central density;
\item the formation of massive BH-BH binaries in the BHS provides a sufficient energy supply to sustain the GC core, leading eventually to its expansion;
\item repeated strong single and binary encounters occurring inside the BHS drive the ejection of the most massive BHs and stars, causing the BHS contraction due to the loss of the energy supply. Consequently, the mean BHS mass and size decrease while its density increases.
\end{enumerate}
\begin{figure}
\centering
\includegraphics[width=8cm]{Not_rhoBHOver12_meanBH.eps}
\includegraphics[width=8cm]{mbh_moth_Rbh.eps}
\includegraphics[width=8cm]{Mbh_Nbin_R.eps}
\caption{Top panel: Central BHS density, normalized to that of the host cluster at 12 Gyr, namely $\rho_{12}$, as a function of the number of stars moving inside $R_{\rm BHS}$. The coloured map highlights the BHS average mass.
Central panel: Mean mass of stars contained within $R_{\rm BHS}$, as a function of the BHS average mass. The color-coded map represents the BHS size.
Bottom panel: Ratio between the fraction of binaries containing at least one BH and the number of BHs in the BHS, as a function of the BHS mass. The coloured map represents the BHS size.
}
\label{F4}
\end{figure}
The scaling relation presented here seem to be compatible with the above scheme, allowing us distinguishing between ``dynamically young'' massive and relatively loose BHS that inhabit dense GCs, and ``dynamically old'' BHS, lighter, denser and inhabiting GCs characterized by smaller central densities.
Hence, it appears evident a correlation between the BHS-GC density ratio and the number of stars mixed with BHs inside the BHS radius. This implies that there is a relation between the potentially observable stellar properties and the BHs composing the BHS.
Indeed, the BHS average mass is tightly connected with the average mass ($m_{\rm oth}$) of stars enclosed within $R_{\rm BHS}$, as shown in the central panel of Figure \ref{F4}, through the relation
\begin{equation}
\left ( \frac{m_{\rm oth}}{~{\rm M}_\odot} \right) = \alpha \frac{1 + \beta\left (m_{\rm BHS}/~{\rm M}_\odot \right)}{1 + \gamma\left(m_{\rm BHS}/~{\rm M}_\odot \right)},
\label{Eq1}
\end{equation}
with $\alpha = 0.2 \pm 0.1$, and $\beta = -0.19\pm0.09$ and $\gamma = -0.14\pm0.02$.
More interestingly, the central panel in Figure \ref{F4} illustrates that at increasing $R_{\rm BHS}$ values, BHS host heavier BHs and lighter stars.
This implies that the larger the BHS average mass, the larger the number of stars ``mixed'' with the BHs in the subsystem, since $N_{\rm oth} m_{\rm oth} = N_{\rm BHS} m_{\rm BHS}$ by definition.
The relations found between ordinary stars and BHs in the BHS, together with the relation between the GC and the BHS central density, suggest that BHS hosting heavy stellar BHs have, on average, a low density concentration.
In dynamically young GCs, the BHS is large and sparse and its BHs have large masses. In these ``active systems'', BHs did not have enough time to contract sufficiently and form a dense BHS, while their self-interactions, which are the main engine for the ejection of massive BHs, did not become effective yet. Consequently, a large population of heavy BHs move inside the GC after 12 Gyr of evolution.
Hence, to provide a sufficient energy flow at the half-mass radius, only a small number of BH binaries is needed, being these extremely efficient energy sources that can lead to large GC half-mass radii.
At a fixed value of the semi-major axis, the heavier the binary the larger the binding energy. Under the general assumption that binaries binding energy undergoes a nearly constant variation ($\Delta(Eb)/Eb = -0.4$, \cite{heggie75} $\Delta(Eb)/Eb = -0.2$, \cite{spitzer87}), the heaviest binary BHs represent the most effective energy source in the cluster.
However, decreasing the binary mass from $m_{b1}$ to $m_{b2}$ implies that
the number of interactions needed to produce the same amount of energy must increase by a factor ($m_{b1}/m_{b2})^{2}$, which in turn implies larger densities. Ejection of the most massive BH binaries due to dynamical interactions leads to the contraction of the BHS in order to increase the energy generation by lower mass BH binaries. The more compact and dense the BHS is, the higher the number of interactions that are needed to sustain the energy flow through the half-mass radius. As a consequence, for dynamically older systems, BHS are denser, more compact and with smaller mass BHs.
Figure \ref{F4} demonstrates that GCs with more massive BHS have fewer number of their BHs in binary systems. Moreover, decreasing values of the $N_{\rm BHB}/N_{\rm BHS}$ ratio correspond to an increase of the BHS size. This is supported by the bottom panel of Fig. \ref{F4}, which shows how the ratio between the number of BHs in binary system and those in the BHS varies at varying the BHS mass and its radius. Indeed, heavier and larger BHS are characterised by a lower fraction of binary systems.
Our results compares very well with \cite{breen13} predictions. Clusters with larger half-mass relaxation times can sustain long living and more massive BHS than clusters with smaller half-mass relaxation times, for which the BHS contracts much faster and ``burns'' the more massive BHs that are needed to generate the required energy to support the host cluster.
Figure \ref{F7} shows how the GC relaxation time at 12 Gyr ($t_{\rm rel}$) varies with the ratio between the BHS and mixed stars average masses. Here, we used the standard definition of half-mass relaxation time-scale defined so far by \cite{spitzer87}.
The three panels in Figure \ref{F7} outlines that the densest BHS, characterised by a smaller ratio between $m_{\rm BHS}$ and $m_{\rm oth}$ on average, are found in GCs characterized by low $t_{\rm rel}$, thus dynamically old at 12 Gyr. On another hand, dynamically younger systems host low-density BHS, containing a significant fraction of massive stellar BHs and in general, a larger number of BHs, as shown in the bottom panel of Figure \ref{F7}.
\begin{figure}
\centering
\includegraphics[width=8cm]{Mmratio_trel_dens.eps}\\
\includegraphics[width=8cm]{Mmratio_trel_num.eps}\\
\includegraphics[width=8cm]{mbhs_trel_nbh.eps}
\caption{Top panel: Host cluster relaxation time-scale at 12 Gyr as a function of the ratio between stars and BHs averaged mass, calculated inside $R_{\rm BHS}$. The coloured map labels the BHS central density. Central panel: the same as in the top panel, but here the color-coded map represents the number of BHs in the subsystem. Bottom panel: the same as in the top panel, but on the X-axis is shown the BHS mean mass.}
\label{F7}
\end{figure}
Surprisingly, we found that our definition of BHS has crucial implication on the relation between the BHS and the whole BH population in a GC. Figure \ref{F8} shows the ratio between the BHS mass and the total mass of retained BHs after 12 Gyr as a function of the number of BHs in the subsystem. As long as the number of BHs in the subsystem remains below $\sim 100$, we found that the BHS contains up to $70\%$ of the whole BHs population. For subsystems containing a larger number of BHs, instead, this percentage oscillates betwee $70-85\%$.
The $M_{\rm BHS}-M_{\rm tBH}$ is a simple power-law
\begin{equation}
{\rm Log} \left(\frac{M_{\rm BHS}}{~{\rm M}_\odot}\right) = \alpha {\rm Log} \left(\frac{M_{\rm tBH}}{~{\rm M}_\odot}\right)+ \beta,
\end{equation}
where in this case $\alpha = 1.14\pm 0.02$ and $\beta = -0.62 \pm 0.06$.
Hence, our procedure allows calculating the mass of a central BH subsystem from the knowledge of the whole population of BHs present in the cluster at that time.
\begin{figure}
\centering
\includegraphics[width=8cm]{nbh_MbhOverBH_mmeanBH}\\
\caption{BHS mass, normalized to the mass of the whole population of BHs in the cluster as a function of $N_{\rm BHS}$. The color-coded map highlights the BHS average mass. }
\label{F8}
\end{figure}
\subsection{BHS observational scaling relations}
\label{sec:obser}
A challenging quest is to determine the presence of a BHS in the interior parts of GCs. For this purpose, we extracted from our MOCCA models the GC central velocity dispersion, $\sigma$, total luminosity $L$, observational half-mass radius $r_{\rm h,obs}$, and central surface brightness $\Sigma$.
Our aim is to provide a set of scaling relations that can be used to infer the presence of a BHS in any given GC for which these global observational properties are known.
In the following expressions, we will infer the BHS-GC observational correlations with the general expression (if not specified otherwise),
\begin{equation}
{\rm Log} \rho_{\rm BHS} = A {\rm Log} {\rm X} + B,
\end{equation}
where $X$ is the observational parameter considered. We use letters $A$ and $B$ instead of $\alpha$ and $\beta$ to better highlight the difference between observational and structural, or ``dynamical'', correlations.
Also, we will only consider models having at least 20 BHs within $R_{\rm BHS}$ as done in the previous sections (unless specified differently).
The BHS mean density seems to weakly correlate with the $r_{\rm h,obs}$, as shown in Figure \ref{radii} through a power-law
\begin{equation}
{\rm Log} \left(\frac{\rho_{\rm BHS}}{~{\rm M}_\odot {\rm pc}^{-3}}\right) = A {\rm Log} \left(\frac{r_{\rm h,obs}}{{\rm pc}}\right) + B,
\end{equation}
with intercept $B = 5.5\pm 0.1$ and $A = -3.4\pm 0.2$.
\begin{figure}
\centering
\includegraphics[width=8cm]{rho_rh_obs_mean.eps}\\
\caption{BHS density as a function of the GC observational half-mass radius. The color coded map represents the BHS average mass.}
\label{radii}
\end{figure}
In general, the correlation between the BHS density and the global GC properties are not very tight. From Figure \ref{glob}, it appears evident that neither the total magnitude in the B-band or the GC velocity dispersions are good indicators for extracting information about the BHS density.
\begin{figure}
\centering
\includegraphics[width=8cm]{rho_censurbri_mean.eps}\\
\includegraphics[width=8cm]{rhoBH_vc12.eps}\\
\includegraphics[width=8cm]{magni_rhoBHS.eps}
\caption{BHS scale density as a function as a function of the GC central surface brightness (top panel), central velocity dispersion (central panel) and magnitude (bottom panel). The color-coded map identifies the number of BHs in the BHS.}
\label{glob}
\end{figure}
In an attempt to define a set of correlations capable to link the BHS properties and several GC observables, we find a ``fundamental plane'' for BH subsystems that allows us to connect the BHS density with the GC average surface luminosity, namely $L/r_{\rm h,obs}^2$, and its velocity dispersion, $\sigma$.
This correlation, shown in the top panel of Figure \ref{fund}, suggests that the lower the GC average luminosity density (the term $L/r_{\rm h,obs}^3$) and its mean kinetic energy ($\sigma^2$) the lower the BHS density. Note that on average, low-density BHS have larger mean masses.
A much tighter relation, as shown in the bottom panel of Figure \ref{fund}, can be used simply combining $\rho_{\rm BHS}$ with the GC average surface luminosity $L/r_{\rm h,obs}^2$.
In this case, the relation can be written as
\begin{equation}
{\rm Log} \left(\frac{\rho_{\rm BHS}}{~{\rm M}_\odot {\rm pc}^{-3}}\right) = A\left[{\rm Log} \left(\frac{L}{{\rm L}_\odot}\right)-2{\rm Log} \left(\frac{r_{\rm h,obs}}{{\rm pc}}\right)\right]+B,
\label{fun}
\end{equation}
with $A = 1.34\pm 0.05$ and $B = -1.87\pm0.18$. This very simple relation implies that the total GC luminosity and its observational half-mass radius can be used to obtain the BHS density. Once $\rho_{\rm BHS}$ is obtained, the BHS size and mass can be obtained using
Equation \ref{RhoRBHS} and \ref{MRBHS}, respectively.
\begin{figure}
\centering
\includegraphics[width=8cm]{Fundamentalplane_SigmLrH.eps}
\includegraphics[width=8cm]{Fundamentalplane_LrH.eps}
\caption{Fundamental plane (top panel) and reduced fundamental plane (bottom panel)
for BHSs. The color-coded map refers to the BHs mean mass in the BHS.}
\label{fund}
\end{figure}
\section{A fundamental plane for IMBHs}
The procedure described above allowed us to define a handful of scaling relations connecting the GCs observational properties and their BHs population. In order to determine whether our treatment can be used also for IMBHs, we grouped all the MOCCA models harboring a central BH with a mass above $150~{\rm M}_\odot$ at 12 Gyr.
In this case, to define the typical ``IMBH size'', we made use of the widely known concept of influence radius, $R_{\rm IMBH}$, which is the region where the IMBH dominate the dynamics \citep{merritt04b}.
Similarly to BHS, the IMBH mass, $M_{\rm IMBH}$, is connected to $R_{\rm IMBH}$ through a power-law
\begin{equation}
{\rm Log} M_{\rm IMBH} = \alpha{\rm Log} R_{\rm IMBH} + \beta,
\label{Eibh1}
\end{equation}
with $\alpha = 0.81\pm0.06$ and $\beta = 3.68\pm0.02$, close to the values obtained for BHS. The corresponding relation is shown in Figure \ref{RMibh}.
\begin{figure}
\centering
\includegraphics[width=8cm]{Ribh_Mibh.eps}
\caption{IMBH mass as a function of the influence radius. The color-coded map identifies the host GC final mass. }
\label{RMibh}
\end{figure}
The similarity between Equation \ref{Eibh1} and \ref{MRBHS} suggests that BHS acts like a central point-like mass, shaping significantly the mass distribution in the inner regions of the parent cluster.
The IMBH mass and radius can be combined to define a typical density
\begin{equation}
\rho_{\rm IMBH} = M_{\rm IMBH} / R_{\rm IMBH}^3,
\end{equation}
which can be used to connect the GC ``dark'' properties and its observational parameters.
Even for IMBHs, is possible to define a fundamental plane, defined by $\rho_{\rm IMBH}$ and the GC typical surface luminosity, defined as the ratio between the total bolometric luminosity and the square of the half-light radius. Analogously to Eq. \ref{fun}, the fundamental plane is well described by a power-law.
\begin{equation}
{\rm Log} \left(\frac{\rho_{\rm IMBH}}{~{\rm M}_\odot {\rm pc}^{-3}}\right) = A\left[{\rm Log} \left(\frac{L}{{\rm L}_\odot}\right)-2{\rm Log} \left(\frac{r_{\rm h,obs}}{{\rm pc}}\right)\right]+B,
\label{fun}
\end{equation}
with
\begin{align}
A &= 1.35\pm0.04\\
B &= -2.3\pm0.2 ~.
\end{align}
Therefore, it seems that a strategy similar to the one applied to BHS could successfully be used to infer the basic properties of IMBHs and their environments on the basis of the host cluster observational parameters.
A deeper and more detailed analysis of how using our model to target GCs potentially harbouring an IMBH will be provided and discussed in our forthcoming paper.
\begin{figure}
\centering
\includegraphics[width=8cm]{Fundamentalplane_ibhLrH.eps}
\caption{
Fundamental plane for IMBHs heavier than $8\times 10^3~{\rm M}_\odot$ (filled triangles) and lighter than this limiting value (filled circles). The color-coded map identifies the average stellar mass enclosed within $R_{\rm IMBH}$.
}
\label{ibhfun}
\end{figure}
\section{Discerning BHSs, IMBHs and ordinary stars in GC centre}
\label{sec:disce}
The results discussed above allow us to obtain a simple set of scaling relations connecting GCs observable quantities and the main structural parameters characterising the population of BHs confined deeply into the host cluster.
In particular, we have shown that the cluster total luminosity and observed half-light radius represent the parameters that mostly constrain the BHS typical density, which in turn can be used to calculate the BHS mass, radius and average mass.
In this section, we investigate whether the quantity $L/r_{\rm h,ob}^2$ and the presence of a BHS can be connected uniquely. Moreover, we try to understand whether is possible to place any constrain on the putative presence of a BHS in Milky Way GCs.
With this purpose, we show in top panel of Figure \ref{comp} the central surface brightness $\Sigma$ and the average surface luminosity, defined above as $L/r_{\rm h,ob}^2$, for all the MOCCA models and for Milky Way GCs taken from the Harris catalogue \citep{harris96,harris10}.
Morever, in the bottom panel we compare the observational core radius ($r_{\rm c,ob}$) and half-light radius ($r_{\rm h,ob}$) of MOCCA models and actual GCs.
The $\Sigma - L/r_{\rm h,ob}^2$ plane is well divided in three different regions:
\begin{itemize}
\item one dominated by GCs with at most a few BHs after 12 Gyr;
\item one dominated by GCs containining an IMBH;
\item GCs containing a BHS.
\end{itemize}
BHS-dominated clusters have average surface luminosities in the range $10^2-10^{4.5}$ L$_\odot$ pc$^{-2}$ and surface brightness in between $10-10^4$ mag pc$^{-2}$.
However, it is quite evident that the three regions are not uniquely defined and overlap at their boundaries. For instance, both
systems hosting at least 10 BHs and those having up to 10 BHs gather in the same region of the plane. As expected, at a fixed value of the average surface luminosity, BHS dominate GCs having lower surface brightness.
More interestingly, BHS are found in clusters having a $\Sigma$ value smaller than IMBHs. Conversely, IMBHs are grouped in a well defined region of the plane, with average surface luminosities above $3 \times 10^3$ L$_\odot$ pc$^{-2}$ and $\Sigma > 10^3$ mag pc$^{-2}$, although some models encroach upon smaller values.
Overlapping the distribution of MOCCA data with observed GCs from the Harris catalogue \citep[][2010 version]{harris10} shows immediately that a noticeable number of Galactic globulars may harbor an IMBH or a massive BHS,
while in some others the BH population has been almost completely depleted due to high natal kicks and dynamical interactions.
Note that many Galactic GCs are expected to lie in the region characterized by ${\rm Log} ~L/r_{\rm h,ob}^2 = 2-4$. This preliminary comparison shows at a glance that many MW GCs can potentially host BHS characterised by relatively low-density and quite massive BHs, with average mass in between $14-22~{\rm M}_\odot$.
Some interesting hints about the BHS-dominated GCs are also provided by the relation between the observational half-light and core radii.
Indeed, nearly all the MOCCA models hosting a BHS with more than 10 BHs have observational core radius larger than $\sim 0.3$ pc and half-light radius larger than 1 pc. Interestingly, observed GCs follow the same trend of our MOCCA subsample in the $r_{\rm h,ob}-r_{\rm c,ob}$ plane. Note that IMBH-dominated models gather in a small portion of the plane, being characterized by relatively low core radii $r_{\rm c,ob}\lesssim 1$ pc \footnote{
We note here that $r_{\rm c,ob}$ might be ill-defined in models with a central IMBH, because of the presence of a very massive object in the cluster centre. In our forthcoming work, mostly focused on IMBHs, we will compute this quantity carefully and compare with the preliminary estimates provided here.}, and nearly constant half-light radii $r_{\rm h,ob}\simeq 1-4$ pc.
It appears evident that models with a small content in BHs overlap to both IMBHs and BHS systems, making difficult to remove the degeneracy between all the three possibilities.
In a companion paper, we identified a sample of 29 Galactic globulars that may host a central BHS. Using the correlations presented in this paper, we calculated for all these clusters the BHS main properties (mass, size, average mass) and provide an estimate of the number of retained BHs, either single or as the component of a binary system \citep{askar18}.
Interestingly, a number of these targeted GCs are already known in literature for being host of several phenomena related to BH physics.
One of the GCs falling in our selection is NGC 3201, which recently made the headlines thanks to the discovery of a BH in a detached binary \citep{giesers18}. As discussed in more detail in our companion paper, our estimate for this cluster are $\sim 20$ BHs as components of a binary system, and $\sim 10^2$ single BHs.
\begin{figure*}
\centering
\includegraphics[width=12cm]{observations.eps}\\
\includegraphics[width=12cm]{radii.eps}
\caption{Top panel: Central surface brightness as a function of the average surface luminosity, for all the MOCCA models at 12 Gyr and for MW GCs. Bottom panel: as above, but here we show the observational half-light radius as a function of the observational core radius. In both panels, we distinguish between GCs hosting an IMBH with mass above $150~{\rm M}_\odot$ (blue open triangle) and hosting a BHS containing at least 10 BHs (filled points).The colour-coded map identifies the number of BHs in the BHS. The purple open diamonds represent the observed population of MW GCs \citep{harris10}. }
\label{comp}
\end{figure*}
\section{Conclusions}
\label{sec:end}
In this paper, we used results from hundreds of GC models that were evolved as part of the MOCCA-Survey Database I to find correlations that can be used to infer the presence of a BHS in GCs using observational parameters.
Our main results can be summarized as follows:
\begin{itemize}
\item[i.] we provide a novel definition for a BHS and its boundaries in GC, according to which a BHS is the ensemble of BHs enclosed within the typical radius, $R_{\rm BHS}$, containing half the mass in BHs and half the mass in stars. The idea beyond $R_{\rm BHS}$ is conceptually similar to the definition of influence radius, i.e. the length scale over which a supermassive BH dominate the dynamics in a galactic nucleus;
\item[ii.] we define five main structural parameters for the BHS: radius, total mass, number of BHs, typical density, average mass and binarity. We found several correlations linking the quantities with each other. In particular, we found that the number of BHs increases smoothly with the total BHS mass, thus implying that heavier BHS are composed of heavier BHs, on average. At the same time, we found that heavier BHS have larger sizes and lower densities;
\item[iii.] comparing the BHS typical density and the host cluster central density, we found that at fixed GC density, heavier BHS are characterised by lower densities. On the other hand, in general the denser the GC the denser the BHS;
\item[iv.] the average mass of the BHs populating a BHS depends strongly on the BHS size and, intriguingly, it depends also on the average mass of the stars mixed within $R_{\rm BHS}$. In fact, higher star masses corresponds to lower BHs masses and vice-versa. In general, we found that heavier BHs are more likely to reside in the most extended BHS;
\item[v.] the BHS structural properties contains information about the dynamical age of the host cluster: high-density BHS, containing heavier BHs, reside in dynamically old GCs, while heavier BHS, with much lower densities are found in dynamically young GCs with long relaxation times;
\item[vi.] the relation between the GC dynamical age and the BHS properties is due to the nature of relaxation process. If a GC is dynamically young, its population of heavy BHs did not undergo yet ejection in strong dynamical interactions. Since the heavier the BHs, the harder the binary in which they bind and consequently, the larger the energy budget that they can exchange with the environment thus heavy BHs lead to sparser and larger BHSs. On the other hand, in a dynamically old GC, the most massive BHs underwent segregation and core-collapse, with consequent formation and ejection of massive BHs and BH binaries. The resulting BHS will have lower mass and higher concentration;
\item[vii.] the BHS properties also reflect the number of binaries containing at least one BH in the GC. In particular, we found that larger BHSs correspond to lower number of binaries, normalized to the total BHs in the BHS itself. Moreover, the fraction of binaries is lower at larger BHS sizes. This can again be related to the GC dynamical status, as dynamically young GCs (large and heavy BHSs) will have experienced no or little binary formation involving single or double BHs;
\item[viii.] the BHS properties are inherited by the initial GC BHs population. Indeed, we found that the BHS mass is about $70-80\%$ of the total BH mass if its number of BHs is $N_{\rm BHS} \gtrsim 100$, while it is below $70\%$ for smaller values of $N_{\rm BHS}$;
\item[ix.] we found a tight correlation in what we call ``the fundamental plane for BHSs'', defined by the GC average surface luminosity $L/r_{\rm h,ob}^2$ and the BHS density. This, combined with the $\rho_{\rm BHS} - R_{\rm BHS}$ and the $R_{\rm BHS} -M_{\rm BHS}$ correlations allows us to fully characterize the BHS properties from two observational quantities;
\item[x.] we found that BHS distribute in a well defined region of the plane delimited by the GC central surface brightness $\Sigma$ and its average surface luminosity $L/r_{\rm h,ob}^2$, quite detached by GCs with no BHs at 12 Gyr and from GCs hosting an IMBH. Comparing our models with observed GCs provided by the updated \cite{harris10} catalogue, we found that many Galactic GCs likely host a BHS with average masses in the range $14-22~{\rm M}_\odot$;
\item[xi.] we are also able to apply a similar procedure to MOCCA models hosting an IMBH, defining also in this case a typical radius $R_{\rm IMBH}$ and a typical density $\rho_{\rm IMBH}$ that is correlated with the IMBH mass and the GCs average surface luminosity density. A more detailed investigation will be carried out in a future paper.
\end{itemize}
The approach presented here aims at providing a simple and rapid treatment that serves to select in an easy way potentially interesting candidates for detailed numerical studies or dedicated observations. Clearly, while we focused the attention on Galactic GCs, the relations described in the paper can be, in principle, used to identify BHS or IMBHs in extragalactic GCs.
In our companion paper, we show how the information obtained through our simple scaling relation can be used to provide detailed information about the dark content of 29 Galactic GCs.
\section{Acknowledgement}
We would like to thank the anonymous referee whose comments and suggestions allowed us to significantly improve the contents of this manuscript. MAS acknowledges the Sonderforschungsbereich SFB 881 "The Milky Way System" (subproject Z2) of the German Research Foundation (DFG) for the financial support provided and the Nicolaus Copernicus Astronomical Center for the hospitality given during the development of this work. MG and AA were partially supported by the National Science Center (NCN), Poland, through the grant UMO-2016/23/B/ST9/02732. AA was also supported by NCN, Poland, through the grant UMO-2015/17/N/ST9/02573.
\clearpage
\footnotesize{
\bibliographystyle{mnras}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,412 |
package com.xrtb.tools.acct;
import java.text.DateFormat;
import java.text.SimpleDateFormat;
import java.util.ArrayList;
import java.util.Date;
import java.util.HashMap;
import java.util.List;
import java.util.Locale;
import java.util.Map;
import java.util.TimeZone;
import java.util.concurrent.ConcurrentHashMap;
import java.util.concurrent.atomic.AtomicLong;
import java.util.concurrent.atomic.DoubleAdder;
import java.util.concurrent.atomic.LongAdder;
import com.fasterxml.jackson.annotation.JsonIgnore;
import com.xrtb.pojo.BidRequest;
import com.xrtb.pojo.BidResponse;
import com.xrtb.pojo.WinObject;
import com.xrtb.tools.logmaster.Slice;
/**
* Implements the Spark accounting record
* @author Ben M. Faul
*
*/
public class Record {
public static int YEAR = 0;
public static int MONTH = 1;
public static int DATE = 2;
public static int HOUR = 3;
public LongAdder requests = new LongAdder();
public LongAdder pixels = new LongAdder();
public LongAdder bids = new LongAdder();
public LongAdder wins = new LongAdder();
public LongAdder clicks = new LongAdder();
public DoubleAdder bidPrice = new DoubleAdder();
public DoubleAdder winPrice = new DoubleAdder();
public String name;
public String campaignName;
public String accountName;
public long time;
static DateFormat format = new SimpleDateFormat("yyyy-MM-dd'T'HH:mm",
Locale.US);
public static Map<String, Record> records = new ConcurrentHashMap();
static {
format.setTimeZone(TimeZone.getTimeZone("Etc/UTC"));
}
String key;
@JsonIgnore
transient List<Integer> footprint = new ArrayList();
public Record(Object x) {
if (x instanceof Map) {
Map r = (Map)x;
r = (Map)r.get("ext");
time = (long)r.get("timestamp");
} else
if (x instanceof BidResponse) {
BidResponse r = (BidResponse)x;
time = r.utc;
} else
if (x instanceof WinObject) {
WinObject r = (WinObject) x;
time = r.utc;
}
process();
}
public static synchronized Record getInstance(Object x) {
long time = 0;
long wins = 0;
long bids = 0;
long requests = 0;
double bidPrice = 0;
double winCost = 0;
if (x instanceof Map) {
Map r = (Map)x;
r = (Map)r.get("ext");
time = (long)r.get("timestamp");
requests++;
} else
if (x instanceof BidResponse) {
BidResponse r = (BidResponse)x;
time = r.utc;
bids++;
bidPrice = r.cost;
} else
if (x instanceof WinObject) {
WinObject r = (WinObject) x;
time = r.utc;
wins++;
winCost = Double.parseDouble(r.price);
}
Date date = new Date(time);
String key = format.format(date);
Record r = records.get(key);
if (r == null) {
r = new Record(x);
records.put(key, r);
}
r.bidPrice.add(bidPrice);
r.winPrice.add(winCost);
r.bids.add(bids);
r.wins.add(wins);
r.requests.add(requests);
return r;
}
public void process() {
Date date = new Date(time);
key = format.format(date);
String [] parts = key.split("-");
// Year
footprint.add(Integer.parseInt(parts[0]));
parts[1] = parts[1].replaceFirst("^0+(?!$)", "");
// Month
footprint.add(Integer.parseInt(parts[1]));
String [] nparts = parts[2].split("T");
nparts[0] = nparts[0].replaceFirst("^0+(?!$)", "");
// Day
footprint.add(Integer.parseInt(nparts[0]));
nparts = nparts[1].split(":");
nparts[0] = nparts[0].replaceFirst("^0+(?!$)", "");
// Hour
footprint.add(Integer.parseInt(nparts[0]));
}
public String getKey() {
return key;
}
public String toString() {
return key + " = " + bids;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,764 |
529-0003-00L Advanced Quantum Chemistry
Dozierende M. Reiher, S. Knecht
Kurzbeschreibung Advanced, but fundamental topics central to the understanding of theory in chemistry and for solving actual chemical problems with a computer.
Examples are:
* Operators derived from principles of relativistic quantum mechanics
* Relativistic effects + methods of relativistic quantum chemistry
* Open-shell molecules + spin-density functional theory
* New electron-correlation theories
Lernziel The aim of the course is to provide an in-depth knowledge of theory and method development in theoretical chemistry. It will be shown that this is necessary in order to be able to solve actual chemical problems on a computer with quantum chemical methods.
The relativistic re-derivation of all concepts known from (nonrelativistic) quantum mechanics and quantum-chemistry lectures will finally explain the form of all operators in the molecular Hamiltonian - usually postulated rather than deduced. From this, we derive operators needed for molecular spectroscopy (like those required by magnetic resonance spectroscopy). Implications of other assumptions in standard non-relativistic quantum chemistry shall be analyzed and understood, too. Examples are the Born-Oppenheimer approximation and the expansion of the electronic wave function in a set of pre-defined many-electron basis functions (Slater determinants). Overcoming these concepts, which are so natural to the theory of chemistry, will provide deeper insights into many-particle quantum mechanics. Also revisiting the workhorse of quantum chemistry, namely density functional theory, with an emphasis on open-shell electronic structures (radicals, transition-metal complexes) will contribute to this endeavor. It will be shown how these insights allow us to make more accurate predictions in chemistry in practice - at the frontier of research in theoretical chemistry.
Inhalt 1) Introductory lecture: basics of quantum mechanics and quantum chemistry
2) Einstein's special theory of relativity and the (classical) electromagnetic interaction of two charged particles
3) Klein-Gordon and Dirac equation; the Dirac hydrogen atom
4) Numerical methods based on the Dirac-Fock-Coulomb Hamiltonian, two-component and scalar relativistic Hamiltonians
5) Response theory and molecular properties, derivation of property operators, Breit-Pauli-Hamiltonian
6) Relativistic effects in chemistry and the emergence of spin
7) Spin in density functional theory
8) New electron-correlation theories: Tensor network and matrix product states, the density matrix renormalization group
9) Quantum chemistry without the Born-Oppenheimer approximation
Skript A set of detailed lecture notes will be provided, which will cover the whole course.
Literatur 1) M. Reiher, A. Wolf, Relativistic Quantum Chemistry, Wiley-VCH, 2014, 2nd edition
2) F. Schwabl: Quantenmechanik für Fortgeschrittene (QM II), Springer-Verlag, 1997
[english version available: F. Schwabl, Advanced Quantum Mechanics]
3) R. McWeeny: Methods of Molecular Quantum Mechanics, Academic Press, 1992
4) C. R. Jacob, M. Reiher, Spin in Density-Functional Theory, Int. J. Quantum Chem. 112 (2012) 3661
http://onlinelibrary.wiley.com/doi/10.1002/qua.24309/abstract
5) K. H. Marti, M. Reiher, New Electron Correlation Theories for Transition Metal Chemistry, Phys. Chem. Chem. Phys. 13 (2011) 6750
http://pubs.rsc.org/en/Content/ArticleLanding/2011/CP/c0cp01883j
6) K.H. Marti, M. Reiher, The Density Matrix Renormalization Group Algorithm in Quantum Chemistry, Z. Phys. Chem. 224 (2010) 583
http://www.oldenbourg-link.com/doi/abs/10.1524/zpch.2010.6125
7) E. Mátyus, J. Hutter, U. Müller-Herold, M. Reiher, On the emergence of molecular structure, Phys. Rev. A 83 2011, 052512
http://pra.aps.org/abstract/PRA/v83/i5/e052512
Note also the standard textbooks:
A) A. Szabo, N.S. Ostlund. Verlag, Dover Publications
B) I. N. Levine, Quantum Chemistry, Pearson
C) T. Helgaker, P. Jorgensen, J. Olsen: Molecular Electronic-Structure Theory, Wiley, 2000
D) R.G. Parr, W. Yang: Density-Functional Theory of Atoms and Molecules, Oxford University Press, 1994
E) R.M. Dreizler, E.K.U. Gross: Density Functional Theory, Springer-Verlag, 1990
Voraussetzungen / Besonderes Strongly recommended (preparatory) courses are: quantum mechanics and quantum chemistry | {
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Вуэльта Испании 2004 — 59-я по счёту гонка Вуэльты Испании. Соревнование началось 4 сентября в Леоне, а закончилось 26 сентября 2004 года в Мадриде. За 23 дня гонщики преодолели 3034 километра. Титул победителя защитил Роберто Эрас из , вторым, отстав на 30 секунд, финишировал Сантьяго Перес из , третьим — Франсиско Мансебо из .
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-h-16t-2-50t-4-using-the-quadratic-formula","text":"# How do you solve h = -16t^2 + 50t + 4 using the quadratic formula?\n\nApr 27, 2017\n\n$h = \\frac{25 + \\sqrt{689}}{16}$\n\n$h = \\frac{25 - \\sqrt{689}}{16}$\n\n#### Explanation:\n\n$x = \\frac{- b \\pm \\sqrt{{b}^{2} - 4 a c}}{2 a}$\n\nwhere $a {x}^{2} + b x + c = 0$\n\n$h = - 16 {t}^{2} + 50 t + 4$\n\nLooking at $a {x}^{2} + b x + c = 0$\nI can see that your values are...\n\n$a = - 16$\n$b = 50$\n$c = 4$\n\nNow just put those values into the quadratic formula\n\n$h = \\frac{- b \\pm \\sqrt{{b}^{2} - 4 a c}}{2 a}$\n\n$h = \\frac{- \\left(50\\right) \\pm \\sqrt{{\\left(50\\right)}^{2} - 4 \\left(- 16\\right) \\left(4\\right)}}{2 \\left(- 16\\right)}$\n\n$h = \\frac{- 50 \\pm \\sqrt{2500 + 256}}{- 32}$\n\nNegative divided by negative makes our numerator and denominator positive\n\n$h = \\frac{50 \\pm \\sqrt{2756}}{32}$\n\nYou might think we are finished here but remember to always check if you can simplify the square root. In this case $2756$ can be divided by $4$, which will give us a $2$ outside of the square root.\n\n$h = \\frac{50 \\pm \\sqrt{4 \\cdot 689}}{32}$\n\n$h = \\frac{50 \\pm 2 \\sqrt{689}}{32}$\n\nNow notice how we can factor $2$ out of our problem, thanks to simplifying the square root\n\n$h = \\frac{2 \\left(25 \\pm \\sqrt{689}\\right)}{2 \\left(16\\right)}$\n\nCancel the common factors\n\n$h = \\frac{\\cancel{2} \\left(25 \\pm \\sqrt{689}\\right)}{\\cancel{2} \\left(16\\right)}$\n\n$h = \\frac{25 \\pm \\sqrt{689}}{16}$\n\n$\\textcolor{g r e e n}{h = \\frac{25 + \\sqrt{689}}{16}}$\n$\\textcolor{g r e e n}{h = \\frac{25 - \\sqrt{689}}{16}}$","date":"2020-09-21 13:51:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 24, \"mathjax_inline_tex\": 1, \"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\": 0.8867692947387695, \"perplexity\": 388.04558399824276}, \"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-2020-40\/segments\/1600400201699.38\/warc\/CC-MAIN-20200921112601-20200921142601-00243.warc.gz\"}"} | null | null |
10 siti di sfondi per poter personalizzare la vostra Sony Playstation Portable, ce ne sono per tutti i gusti e sono tutti gratis al 100%!
Hey! I just wanted to post and say thanks for listing me in your top 10! (I'm sonypspwalls.com) So ya, I worked on a better site and tried to really rework the way people post comments on the wall paper. Just inviting everyone to come check it out. And to say thanks… so … thanks!
dove posso trovare sfondi per psp e temi gratuiti? | {
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\section{Introduction}
\label{sec:intro}
\input{introduction}
\section{Related Work.}
\label{sec:relatedwork}
Fair resource allocation is well studied in the literature across various fairness and efficient notions \cite{endriss2018lecture,bouveret_chevaleyre_maudet_moulin_2016, MarkakisTRENDS2017,procaccia_moulin_2016,inbook}. When a definition of fairness is too strong or may not exist, we always look for its relaxation/approximation; researchers have also studied how likely it is that a fairness notion will not exist.
In this paper, we are majorly concerned with EF1 and USW.
EF1 allocations always exist and can be found in polynomial time. When agents have additive valuations, the round-robin algorithm always guarantees EF1 for (pure) goods or chores and the double round-robin algorithm for the combination of goods and chores. \cite{Caragiannis2018,Caragiannis2016}. When agents have general valuations, we can find EF1 allocations in $O(mn^3)$ using a cycle-elimination algorithm. \cite{Lipton2004}. Finding MUW allocations is also polynomial-time solvable for additive valuations, i.e., we iterate over items, assign the item to the agent who values it the most. However, finding MUW allocation amongst EF1 allocations is NP-hard even for two agents with additive valuations \cite{aziz2019constrained,Barman2019Nphard,ef1inumHaris,Aziz2016AAMAS,Caragiannis2018,KeijzerBKZ09}.
Authors in \cite{Highmultiplicitypaper2019} present a framework to compute $\epsilon$-Efficient and $\mathcal{F}$-Fair allocation, using parametric integer linear programming, which is double exponential in terms of the number of agents and items. In \cite{Highmultiplicitypaper2019}, they explore group Pareto Efficiency, which is equivalent to USW. Authors in \cite{ef1inumHaris} provide a pseudopolynomial-time algorithm for finding MUW within EF1 for any fixed number of agents for goods, which is exponential in the number of agents. In the paper, \cite{Iwillhaveorder}, the authors present an approximately optimal round-robin order that gives highly efficient (USW) EF1 allocations in the Reviewer Assignment setting; however, the setting is quite different from ours, as we are not concerned with the multiplicity of items.
In further related work, papers \cite{barman2018finding,Caragiannis2018} explore, PE and EF1 allocations, \cite{EQ1freemangoods,EQ1ChoresFreeman} explore PE and EQ1 allocations, \cite{azizPRop1andfpo} explore PE and Prop1 allocations for various items (goods or/and chores). There will always be a tradeoff between fairness and efficiency, corresponding to the study of the price of fairness \cite{barmanpriceoffairness,priceoffairnessbeiijcai19}.
Alongside, Researchers have also studied how likely it is that a fairness notion will not exist \cite{closinggaps,dickersonThecomputationalriseandfalloffairness,whendoenvyfreeallocationexist}. In \cite{closinggaps}, the authors show that Round Robin allocation is envy-free when $m \ge \Omega(n \log n / \log \log n)$.
Recently the EconCS community has been interested in learning mechanisms/algorithms using neural networks, esp. in a setting of theoretical limitations.
For, e.g., In paper \cite{Noah2018}, the authors provide a strategy-proof, multi-facility mechanism that minimizes expected social cost via NN. Authors in \cite{ICAByML}, the authors integrate machine learning in the combinatorial auction for preference elicitation. Further, in \cite{ICAByDL}, authors use a neural network to improve it and reformulate WDP into a mixed-integer program. Authors in \cite{manisha2018learning,TacchettiDeepmind} learn optimal redistribution mechanisms through NNs. Another line of work is Reinforcement Mechanism Design, such as learning dynamic price in sponsored search auctions ~\cite{Reinforcement2,Reinforcement1}. In \cite{PublicprojectNN}, the authors use NN to maximize the expected number of consumers and the expected social welfare for public projects.
\section{Preliminaries}
\label{sec:prelim}
We consider the problem of allocating $M=[m]$ indivisible items among
$N=[n]$ interested agents. Each agent $i \in N$ has a valuation function $v_i : 2^{M} \rightarrow \mathbb{R}$ and $v_i(S)$ is its valuation for a $S \subseteq M$ s.t. $v_i(\phi) = 0$. We consider three settings - pure goods, pure chores, and a combination of goods and chores. In combination, an item may be good for one agent and a chore for another. For an agent $i$, an item $j \in M$ is a \emph{good} if, $v_i(\{j\})\geq 0$, and a \emph{chore} if, $v_i(\{j\}) < 0$. We represent valuation profile $v=(v_1,v_2,\ldots,v_n)$. We consider additive valuations. The valuation of an agent $i \in N$ for bundle $A_i$ is $v_i(A_i) = \sum_{j\in A_i} v_i(\{j\}) $. Utilitarian Social Welfare (USW) is defined as $sw(A,v) = \sum_{i\in N} v_i(A_i)$.
We assume $\mathcal{F} = F_1 \times F_2, \ldots, \times F_n$ to be a known prior distribution over agents' valuations. We randomly draw $v_i \sim F_i$.
An allocation $A \in \{0, 1\}^{n\times m}$ is an $n$ way partition $(A_1, \ldots A_n)$ of $M$. Here, $A_i \in [m]$ is the bundle assigned to the agent $i$ and $A_i \cap A_k = \phi, \forall i, k \in N$ and $i \neq k$.
We consider a complete allocation of items, i.e., $\cup_i A_i = M$. We use the notation $n \times m$, for a problem setting with $n$ agents and $m$ items.
Given a valuation profile of agents $v=(v_1,v_2,\ldots,v_n)$, our goal is to find a fair and efficient allocation. First, we define fairness and efficiency properties.
\begin{definition}[Envy-free (EF)]
\label{def:ef}
An allocation $A$ is said to be EF, if no agent envies another agent, i.e., $\forall i,j \in N , v_i(A_i) \ge v_i(A_j)$.
\end{definition}
As EF allocation may not always exist for indivisible goods, we consider a generalized version of relaxation of the EF defined by Budish~\cite{Budish11}.
\begin{definition}[Envy-free up to one item (EF1)]
\label{def:ef1}
An allocation $A$ is said to be EF1 if envy of any agent can be eliminated by either removing any good from the envied agent's allocation or removing any chore from the agent's allocation. i.e., when either of the following is true
$\forall i,k \in N$,
\begin{enumerate}
\item $ \exists j \in A_k \ \mbox{s.t.}\ v_i(A_i) \ge v_i(A_k \setminus \{j\})$
\item $\exists j \in A_i \ \mbox{s.t.}\ v_i(A_i \setminus \{ j\}) \ge v_i(A_k)$
\end{enumerate}
\end{definition}
\begin{definition}[Maximum Utilitarian Welfare (MUW)]
An allocation $A^*$ is said to be \emph{efficient} or MUW if it maximizes the USW, i.e.\\ $$A^* \in \underset{A \in \{0,1 \}^{n\times m}}{arg\,max} sw(A,v) $$
\end{definition}
\begin{definition}[EEF1 Allocation]
We say an allocation is \emph{EEF1} if it satisfies EF1 fairness and maximizes USW amongst EF1 allocations.
\end{definition}
\section{Our Approach: EEF1-NN}
\balance
\label{sec:eef1-nn}
EEF1-NN\ represents a mapping from valuation space to allocation space, i.e., $\mathcal{A}^w :\mathbb{R}^{\{n \times m\}} \rightarrow \{0,1\}^{n\times m}$, where $w$ represents the network's weights. To learn the network parameters, we formulate our problem to optimize social welfare w.r.t. to fairness constraints in Section~\ref{ssec:loss_fn}. We construct the Langrangian Loss function of this optimization problem for the training of EEF1-NN.
We explain our architecture in Section ~\ref{ssec:networkdetails} and training details in Section~\ref{ssec:training}. Note that we represent EEF1-NN\ by $\mathcal{A}^w$ and an allocation by $A$.
\smallskip
\subsection{Optimization Problem.}
\label{ssec:loss_fn}
Consider $n$ interested agents and $m$ indivisible items; it can be good $v_i(\{j\}) \ge 0$, or chore $v_i(\{j\}) < 0$. We are given a set of valuation profile $v=(v_1,v_2,\ldots,v_n)$, where $v_i$ is drawn randomly from a distribution $\mathcal{F}_i$.
Among all possible allocations $A \in \{0,1\}^{n \times m}$, we need to find an optimal $A^*$ that maximizes utilitarian social welfare $sw(A, v)$ and satisfies a fairness constraint.
We formulate two fairness constraints - EF and EF1.
In Definition \ref{def:ef}, the envy of an agent $i \in N$ according to EF is as follows,
\begin{align}
\label{eq:EFenvy}
envy_i(A,v) =& \sum_{k \in N}\max
\Bigg\{ 0 , (v_i(A_k) - v_i(A_i)) \Bigg\}
\end{align}
In Definition. ~\ref{def:ef1}, the $ef1_i$ of an agent $i \in N$, according to EF1 is as follows:
\begin{align}
\label{eq:E1envy}
ef1_i(A,v) =& \sum_{k \in N}\max
\Bigg\{ 0 , (v_i(A_k) - v_i(A_i))
+ \nonumber \\ &\min \left\{ -\max_{j \in A_k} v_i(\{j\}) , \min_{j \in A_i} v_i(\{j\}) \right\} \Bigg\}
\end{align}
The above constraints are generalized formulations for both goods and chores. Given a set of valuation profiles, our goal is to maximize the expected welfare w.r.t. to the expected fairness.
\smallskip
\begin{center}
\fbox{\begin{minipage}{0.8\columnwidth}
\begin{equation}
\label{eq:our_prb}
\begin{aligned}
\mbox{minimize} & \; \; - \mathbb{E}_v\left[sw(A,v)\right] = \mathbb{E}_v[\sum_{i\in N} v_i(A_i)]
\\
\mbox{subject to} & \; \; \mathbb{E}_v\left[\sum_{i \in N} envy_i(A,v)\right] = 0 \quad \mbox{or,} \\
&\; \; \mathbb{E}_v\left[\sum_{i \in N} ef1_i(A,v)\right] = 0
\end{aligned}
\end{equation}\end{minipage}}
\end{center}
\smallskip
In the above optimization problem, we have 'OR' among fairness constraints, which we will elaborate on in the Ablation Study in Section ~\ref{subsec:ablaS}.
\smallskip
\noindent\textbf{EEF1-NN: Lagrangian Loss Function.}
We now formulate the objective function given by Eq. \ref{eq:our_prb} using the Lagrangian multiplier method. We use the Lagrangian multiplier $\lambda \in \mathbb{R}_{\ge 0}$ to combine the objective and constraints. Given $\mathcal{L}$ samples of valuation profiles $(v^1, \ldots, v^{\mathcal{L}})$ drawn from $\mathcal{F}$, we have the corresponding input $I_v^l$ and the loss for each sample is given by,
\begin{equation}
Loss(I_v^l, w, \lambda) = \frac{1}{n\times m}
\Bigg[ -sw(\mathcal{A}^w(I_v^l),v^l) +
\lambda\frac{ \sum_{i \in N} envy_i(\mathcal{A}^w(I_v^l),v^l)}{n} \Bigg]
\label{eq:loss_per_sample}
\end{equation}
We minimize the following loss w.r.t $w$,
\begin{equation}
\label{eq:loss}
\mathbf{L}_{EEF1}(I_v^l, w, \lambda)= \frac{1}{\mathcal{L}}\sum_l Loss(v^l,w)
\end{equation}
\subsection{Network Details}
\label{ssec:networkdetails}
We describe EEF1-NN\'s various components, including the input, architecture, and other training details in this section. EEF1-NN\ is a fully convolutional network (FCN) and processes input of varied sizes (i.e., height $\times$ width). Because of using an FCN, EEF1-NN\ runs independently of $n$ and $m$.
\smallskip
\noindent\textbf{EEF1-NN: Input.}
We construct an input tensor of size $n \times m \times 6$, i.e., the input to the network is a six channeled input $I_v \in \mathbb{R}^{n\times m \times 6}$. The first channel is an $n\times m$ matrix of given valuations, i.e., $\forall i,j ; I_v[i, j, 1] = v_i(\{j\})$. Note that we sample the valuation from a distribution $\mathcal{F}$.
We take a matrix $X$ of size $n \times m$ that contains valuation for items only corresponding to the agent who values it the most, and the rest elements are zeros, i.e.,
It takes a value $v_i(\{j\})$ for each item (column) for the agent (row) having maximum value for it, i.e,
\begin{align*}
\forall j \in M; \; X[i.j,1] &=
\begin{cases}
v_i(\{j\}) & \text{if } i \in argmax_i v_i(\{j\}) \\
0 & \text{otherwise}
\end{cases}
\end{align*}
We break ties arbitrarily. We expand this $n \times m$ matrix into five channels, such that the first one will contain information about items indexed as $0,5,10,\ldots, \floor{m/5}$, i.e.,
\begin{align*}
I[i.j,2] &=
\begin{cases}
X[i,j,1] & \text{if } j \in \{0,5,10,\ldots,\floor{m/5}\} \\
0 & \text{otherwise}
\end{cases}
\end{align*}
The next channel will have data from the previous channel and along with that about items indexed as $1,6,11,\ldots, 1+\floor{m/5}$.
\begin{align*}
I[i.j,3] &= I[i,j,2] +
\begin{cases}
X[i,j,1] & \text{if } j \in \{1,6,11,\ldots,1+\floor{m/5}\} \\
0 & \text{otherwise}
\end{cases}
\end{align*}
And so on. The last channel, $I_v[i,j,6]$ will be equal to $X$
We observe that giving single channeled input of only valuations, i.e., tensor of size $(n \times m \times 1)$, performs sub-optimal as opposed to the six-channel. We evaluate the performance of six channeled input across various other inputs in Section ~\ref{subsec:ablaS}.
With this representation, the network learns better.
\smallskip
\noindent\textbf{EEF1-NN: Architecture.}
Our architecture is inspired by U-Net architecture \cite{unetpaper}. U-Net is a fully convolutional network built to segment bio-medial images; it also requires assigning labels to image patches and not just classifying the image as a whole. Traditionally, a fully convolutional network is used for image segmentation. While we are working on valuation profiles rather than images, one of the primary motivations to use U-Net is to process arbitrary size images. If we use a feed-forward fully functional neural network to learn fair and efficient allocations, we need a different network for each $n \times m$. Moreover, just using a feed-forward functional network (multi-layer perceptron) learns EEF1 allocations for smaller values of $n$, cannot learn as $n$ increases; we will briefly mention this in our Ablation Study in Section ~\ref{subsec:ablaS}.
EEF1-NN\ contains series of convolution (contracting) and up-convolution (expanding) layers, as given by Fig. \ref{fig:EEF1-NNArchitecture}. EEF1-NN\ has three series of Conv-UpConv layers.
The convolutional layers consist of 4 repeated 3x3 convolution, each followed by a non-linear activation function, i.e., tanh, which is applied element-wise.
The up-convolution layers consist of 4 repeated 3x3 up-convolution, each followed by a tanh activation.
Note that we are not using maxpool or skip connections. We found that using a pooling layer or skip connections degraded the network performance. The final output represents the probability with which agent $i$ will receive item $j$. We apply softmax activation function across all agents for every item to ensure each item is allocated exactly once ,i.e., $ \forall j \in M \; \sum_{i \in N} \mathcal{A}^w_{i}(\{j\})=1 $. In total, we have 24 layers (convolution + up-convolution).
Using an FCN structure, we have a generalized network for $n \times m$; however, learning EEF1 allocations is not easy. We need to learn discrete variables, while neural networks are known for learning continuous output. We will describe these challenges in the next Section.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{figures/nnDiagram.png}
\caption{\centering EEF1-NN\ Architecture}
\label{fig:EEF1-NNArchitecture}
\Description{For 10 agents, valuation drawn randomly from a uniform distribution, plotting-TBD }
\end{figure}
\subsection{Training Details}
\label{ssec:training}
There are certain challenges with network training, especially in the setting of indivisible resource allocation.
\noindent\textbf{Integral Allocations. } The global optima of the optimization problem in Eq. \ref{eq:our_prb} might lie in a continuous allocation setting, i.e., similar to allocating divisible items. The training starts if a network learns to distribute an item equally among all agents, and the gradient vanishes. For a fair allocation, assigning an equal partition of each item is indeed an optimum. Converting these non-integral probabilities to integral allocation is non-trivial. Hence we set a \emph{temperature} parameter in the softmax layer of the network to prevent getting stuck at such optima.
Let $p_{j} = \{p_{j_1}, \ldots, p_{j_n} \}$ denote the output of our network before the final layer which represents the probability of assigning item $j$ to all the agents. The final allocation for agent $i$, i.e., $\mathcal{A}^w_i(\{j\})$is given by,
$$ \mathcal{A}^w_i(\{j\}) = \mbox{softmax}(p_{j_i}) = \frac{e^{p_{j_i}/T}}{\sum_{k=1}^n e^{p_{j_k}/T}}$$
It is common to start with a large temperature value for initial exploration and gradually reduce the temperature to reach the global optima.
While training, when we set the temperature value to 1, we get fractional allocations.
As we decrease the value of $T$, the network outputs allocation close to discrete. The approach we want is while training, allocation output is almost discrete, but not exactly discrete. When we keep the value of $T$ too low, the output is exact discrete allocations, and there is no learning because of the vanishing gradients \cite{elibendersky_2016}. So we appropriately choose $T$ based on our experiments. Once the network learns, we set the parameter low enough to ensure discrete allocations.
\noindent\textbf{Inefficient Local Optima. } Due to the low-temperature value, the training of EEF1-NN\ is highly unstable and often gets stuck at inefficient local optima. To overcome this, we use the technique of \emph{Bootstrap Aggregation} or Bagging \cite{bagging}. It combines the predictions from multiple classifiers to produce a single classifier. Hence we train multiple weak networks with varied hyper-parameters on the same data set, capturing different sets of local optima. While testing, the final allocation is aggregated from these networks. We pass a test sample through all networks and select the allocation that is EF1 with maximum USW.
Usually, in Bagging, we train neural networks with different training data sets, however in our case, varying $\lambda$ produces different models. In total, we bag seven networks with varied $\lambda \in [0.1,2]$ for increased performance. We further analyze how Bagging affects our results in the ablation study provided in the experimental section.
We implement EEF1-NN\ using Pytorch.
The network weights are initialized using Xavier Initialization \cite{xavierinit}. To train, we use Adam Optimizer \cite{Adamopt} with fixed learning rate $0.001$. We use a batch size of 256 samples. We sample valuations from $U[0,1]$ (goods), $U[-1, 0]$ (chores) and $U[-1,1]$ (combination). We sample $150k$ training data for both $10 \times 20$ and $13 \times 26$ for goods, chores, and combinations, so in total, we have $300k$ training samples, and we sample $10k$ testing samples for each setting. We transform these valuations into six-channeled input and feed to the network. We set the temperature parameter to 0.01. We train our network for 1000 epochs. We use our Lagrangian loss as the objective function to train our network. We train seven networks with varied $\lambda \in [0.1,2]$ and bag them for enhanced performance. The training process took 5-6 hours to train a single network using GPU.
We are training the network for $10 \times 20$ and $13 \times 26$. however, we show our test results for various $n \times m$. We test for network performance for $n \in [7,15]$. Further, we also train a individual network over different distributions such as Gaussian ($\mu$=0.5,$\sigma$=1), Log-normal ($\mu=0.5$,$\sigma=1$), and Exponential ($\lambda = 1$), with $150k$ training samples for $n=10$.
\section{Experiments and Results}
\label{sec:experiemtnsandresults}
For reporting the network performance on the test set, we define the following two metrics, one is the measure of fairness and the other of efficiency.
\smallskip
\noindent\textbf{Evaluation Metrics. }
\begin{enumerate}[noitemsep, leftmargin=*]
\item $\alpha_{EF1}^{ALG}$ - It measures the probability with which an algorithm $ALG$ outputs, EF1 allocation. $\alpha_{EF1}$ is the ratio of the number of samples that are EF1 to the total number of samples.
\item $\beta_{SW}^{ALG}$ - It measures the ratio of expected USW of an algorithm $ALG$ by expected USW of MUW allocation. $\beta_{SW}^{ALG} = \mathbb{E}(sw^{ALG}) / \mathbb{E}(sw^{MUW})$.
\end{enumerate}
Using the above metrics, we conduct the following ablation study to set appropriate hyper-parameters.
Our network is trained across various types of items (goods or/and chores) and types of distribution.
The test set consists of $10k$ samples.
Note that $\beta^{ALG}_{SW} \in [0,1]$ for goods, $\beta^{ALG}_{SW} \ge 1$ for chores, and will depend on the overall social welfare (positive/negative) for a combination of goods and chores.
We will use that notation $(\alpha_{EF1},\beta_{sw})$ to write network/algorithm's performance.
\subsection{Ablation Study}
\label{subsec:ablaS}
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{figures/AAMASablation_study-1.png}
\caption{Ablation Study over hyper-parameters}
\label{fig:ablationstudy}
\Description{For 10 agents, valuation drawn randomly from a uniform distribution, plotting-TBD }
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{figures/AAMAS-ablation_study2.png}
\caption{Ablation Study over Input channels and $\lambda$}
\label{fig:ablationstudy2}
\Description{For 10 agents, valuation drawn randomly from a uniform distribution, plotting-TBD }
\end{figure}
We illustrate the effect of specific hyper-parameters in the performance of EEF1-NN\ in Fig. [\ref{fig:ablationstudy},\ref{fig:ablationstudy2}].
We sample the valuations from the uniform distribution, set $n=10$, only goods, for all the ablation study experiments, and observe the $\alpha_{EF1}$ as $m$ increases.
In the plots, the red line with the label EEF1-NN\ denotes the $\alpha_{EF1}$ for optimal parameters.
Corresponding to EEF1-NN, a single network from this bagged network is labeled as \emph{Single Network} in the graph. This \emph{Single Network} trained with six-channeled input, $\lambda=1$, and temperature $T=0.01$ is the baseline to compare across this ablation study. Only one parameter is changed w.r.t. the \emph{Single Network} training for all the networks used in this study.
We only compare $\beta_{sw}$ for the network across varied $\lambda$ values, as $\alpha_{EF1}$ values are close to the \emph{Single Network}.
\smallskip
\noindent \emph{(i) Effect of Temperature $T$. }
Keeping other parameters fixed, we vary the $T=\{1,0.1,0.001\}$ in Fig. \ref{fig:ablationstudy}. When $T=1$, our network converges to global optima, i.e., fractional allocation, unable to learn EEF1 discrete allocation represented by the blue line at the bottom of the plot. Also, we empirically observed that when networks learn to allocate an equal fraction of an item among agents (0.1 in case of 10 agents), the gradient vanishes, thus stuck in a bad local optimum. When $T=0.001$ (violet line), it is too low, and performs sub-optimally compared to when $T = 0.01$ in single network (no bagging) (light blue line). We also noticed that the network's performance for $T=0.01$ and $T=0.1$ are close to each other. We set $T=0.01$ for all the bagged networks in EEF1-NN.
\smallskip
\noindent\emph{(ii) Effect of series of Conv-UpConv layers.} We select three series of Conv-UpConv For EEF1-NN\ as illustrated in Fig. \ref{fig:ablationstudy}. We plot the $\alpha_{EF1}$ for one Conv-UpConv series green dashed line. It is less than what we obtain for a 2-series red dotted line, which is less than the optimal 3-series (light blue line). As seen from Fig. \ref{fig:ablationstudy}, an increase between 1-series and 2-series is significant compared to 2-series and 3-series (single network without bagging). The complexity of the network having 4-series is far more than the performance improvement. We have limitations over the number of layers in Conv-UpConv, as we are working with a low dimensional matrix, such as $10\times 20$, and having such a series increases performance. Briefly stating, while training a 4-layered feed-forward fully functional network for $10 \times 20$, $\alpha_{EF1}$ was roughly 0.008.
\smallskip
\noindent \emph{(iii) Effect of loss function }
We analyze how different envy definitions in our loss function represented in Eq. \ref{eq:our_prb} affects the training of EEF1-NN. As shown in Fig. \ref{fig:ablationstudy}, when we train our network using EF, i.e., Eq. \ref{eq:EFenvy} (\emph{Single Network}, light blue line), the network performs significantly better than when trained using EF1, i.e., Eq.\ref{eq:E1envy} (orange dashed line).
For example, for $10 \times 20$, the performance of \emph{Single Network} is (0.3358,17.9611), whereas the performance of the EF1 trained network is (0.1530,17.8708). Given a distribution, one way of interpreting this behavior can be that when we train the neural network to maximize social welfare w.r.t. to Envy-free, the best fairness it can have is EF1 while maintaining efficiency close to MUW.
\smallskip
\noindent \emph{iv) Number of Input Channels. }
When training with just one channel input, i.e., valuations, the results we obtained were quite poor; for $10\times20$, we get $(0.2113,17.8976)$ as shown in Fig~\ref{fig:ablationstudy2}. Thus we changed our input representation into more channels. We tested for 2,6, and $(n+1)$ channels for $n=10$. For two channeled input, we set the first channel of input tensor as the valuation matrix and the second channel to matrix $X$, described in Section ~\ref{ssec:networkdetails}. Like six-channeled input, we expand the matrix $X$ to 11 channels, i.e., the number of channels here is equal to $n+1$. The learning of a six-channeled network, i.e., \emph{Single Network} is better than the two-channeled network. The performance of a two-channeled network is $(0.2365,17.8991)$, \emph{Single Network} is $(0.3358,17.9611)$, and 11-channeled network is $(0.3925,17.9395)$. We didn't plot $\alpha_{EF1}$ of the 11-channeled network because Even though $\alpha_{EF1}$ is high of the 11-channeled network, the network is dependent on $n$; cannot be generalized for $n$, along with an increase in input representation complexity.
\smallskip
\noindent \emph{v) Effect of Bagging. }
We try different combinations of networks, each trained for varied $\lambda$ values. The Lagrangian Loss, as described in Eq. ~\ref{eq:loss_per_sample}, $\lambda$ corresponds to the fairness constraint. More the value of $\lambda$, more penalty is given to envy in the loss. When $\lambda$ is too small, the penalty for allocating an unfair allocation is less, so the network learns a more efficient but less fair allocation. As we increase $\lambda$ up to a certain value, the network learns less efficient but more fair allocations. Beyond a certain value, if we increase $\lambda$, we get a degraded performance overall. In Fig. ~\ref{fig:ablationstudy2}, we compare $\alpha_{EF1}$ and $\beta_{sw}$ of \emph{Singe Network} ($\lambda$=1), lambda=0.5, and lambda=0.1.
We observed that varying $\lambda$ value results in converging to the different optimum.
We bagged seven networks trained on with $\lambda$ values of $\lambda \in [0.1,2]$. We choose a mix of (low efficiency, high fairness) and (high efficiency, low fairness). We feed six-channeled input to EEF1-NN, and it outputs the fairest and efficient, i.e., if more than one network gives EF1 allocation, then it will select the one with maximum social welfare.
In Fig. \ref{fig:ablationstudy}, we find combining the networks (red line) outperforms the performance of a single network (light blue line).
\subsection{Experiment Details and Observations}
We select the best training parameters for EEF1-NN\ based on the above ablation study for the following experiments. We conduct three types of experiments to compare existing approaches across, \textsc{Exp1}: Different kinds of resources, \textsc{Exp2:} Different input distributions, and \textsc{Exp3:} Scalability to samples with large $n$. In all three experiments, we compare EEF1-NN\ with the following existing methods,
\begin{itemize}[noitemsep, leftmargin=*]
\item \noindent \textsc{MUW}
Since we don't have Optimal EEF1 allocations to compare our results, we compare our results with MUW allocations. We also analyze after which value of $m$, an MUW allocation is likely to be EF1.
\item \noindent
\textsc{Round Robin (RR)} \cite{Caragiannis2016} finds EF1 allocations under additive valuations for pure goods and pure chores. \emph{Double Round Robin }\textsc{(D-RR)} \cite{Caragiannis2018} extends RR to find EF1 allocation for the case with a combination of goods and chores under additive valuations.
\item \noindent \textsc{CRR}
Based on paper \cite{aziz2019constrained}, we implement CRR to find RB sequence such that it allocate items to the agent who values it the most for goods. As mentioned in \cite{aziz2019constrained}, an RB sequence is EF1 when all items have positive valuations.
\end{itemize}
Note that approaches like using parametric ILP solver \cite{Highmultiplicitypaper2019} and the algorithm provided by \cite{ef1inumHaris} are exponential. Therefore, it is infeasible to use these for the configurations we report our results on, so we do not compare them.
Further, in Table ~\ref{tab:mvalueanalysis}, we study the convergence of different approaches towards EEF1 for uniform distribution, i.e., after which value of $m$, a method converges to EEF1.
\smallskip
\noindent\textbf{\textsc{EXP1:} Performance across differed resources for Uniform Distribution.} For $n=10$, we compare the $\alpha_{EF1}$ in Fig. \ref{fig:n10} (a1, b1, c1) and $\beta_{SW}$ in Fig. \ref{fig:n10} (a2, b2, c2) as $m$ increases across the existing approach.
Irrespective of the resource type, as the number of items increases, all the approaches will move closer to EEF1. We observe that MUW allocation (blue dotted line) converges towards EEF1 allocations much faster for chores or combinations than goods.
While Round Robin converges to EEF1 allocations much faster in goods compared to chores or combinations. We discuss this convergence in detail in Table ~\ref{tab:mvalueanalysis}.
We observe that EEF1-NN\ consistently has better $\alpha_{EF1}$ than MUW allocation and better $\beta_{sw}$ than RR/CRR.
The baseline \textsc{RR} is designed to find EF1 allocations and hence, by construct has $\alpha_{EF1}^{RR} = 1$. We observe that $\alpha_{EF1}^{EEF1-NN}$ is close to $\alpha_{EF1}^{RR}$ after a certain $m$. At the same time, the allocation returned by EEF1-NN\ is far more efficient than $RR$ as represented by the $\beta_{SW}$ values. (Fig \ref{fig:n10} (a2,b2,c2)). Though $\alpha_{EF}^{CRR} = 1$ (Fig. \ref{fig:n10} a1), note that the baseline \textsc{CRR} only works for goods. Even for goods, we observe that compared to \textsc{CRR}, EEF1-NN\ obtains marginally better $\beta_{SW}$, in Fig. \ref{fig:n10}(a2).
\begin{figure*}[!tbh]
\centering
\includegraphics[width=0.9\linewidth]{figures/AAMAs-10_agents.png}
\caption{\textsc{Exp1} ($n=10$, Uniform Distribution)}
\label{fig:n10}
\Description{For 10 agents, valuation drawn randomly from a uniform distribution, plotting }
\end{figure*}
\smallskip
\noindent\textbf{\textsc{EXP2:} Performance across different distributions.}
We provide the performance of EEF1-NN\ when the valuations are sampled from different distributions such as Gaussian ($\mu$=0.5,$\sigma$=1) in Fig \ref{fig:dist}(a1, a2), Log-normal ($\mu=0.5$,$\sigma=1$) in Fig \ref{fig:dist}(b1, b2), and Exponential ($\lambda = 1$) in Fig \ref{fig:dist}(c1, c2). Note that when we sample valuations from Gaussian distribution, it corresponds to allocating a combination of goods and chores. From Fig. \ref{fig:dist}, we observe that in all three distributions, $\alpha_{EF1}^{EEF1-NN}$ is more than 0.99 and $\beta_{SW}^{EEF1-NN}$ is more than 0.99 for $m \ge 40$.
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.9\linewidth]{figures/AAMAS-distributions.png}
\caption{\textsc{Exp2} ($n=10$, different distributions)}
\label{fig:dist}
\Description{For 10 agents, valuation drawn randomly from a uniform distribution, plotting-TBD }
\end{figure*}
\smallskip
\noindent\textbf{\textsc{EXP3:} Scalability to larger number of agents.}
EEF1-NN\ is trained only for $10\times20$ and $13\times26$.
As we have seen in the previous results and in Fig. \ref{fig:n13}, the performance scales across varying $m$ seamlessly. In this section, we provide the performance of EEF1-NN\ when $n=7$, $n=12$, and $n=14$ in Fig. \ref{fig:n13}.
Our network will not run when $n < 10$, given the four $3\times3$ Convolution-UpConvolution. Hence to report performance for $n \in [7,9]$, we reduce a Convolution-UpConvolution layer from our network and train accordingly with $7\times14$ and $10\times20$ valuation profiles.
EEF1-NN\ scales appropriately across $n$; however, if we test the network performance of $n=14$ and $n=20$, the network performs better for $n=14$, solely because we trained just using $10\times20$ and $13\times26$. For a much higher value of $n$, we need to expand our training samples.
\smallskip
\noindent\textbf{Analysis of Convergence to EEF1 Allocations (Uniform Distribution)}
\begin{definition}[$m^{\star}(n)$]
For a given $n$, we say an algorithm converges to EEF1 allocation at $m^{\star}(n)$ if $\forall m >m^{\star}(n)$, \\(i) \emph{For goods}: $\alpha_{EF1}^{ALG} \ge 0.99$ and $\beta_{sw}^{ALG} \ge 0.99$. \\(ii) \emph{For chores}: $\alpha_{EF1}^{ALG} \ge 0.99$, and $\beta_{sw}^{ALG} \le 1.02$.
\end{definition}
We empirically study the value of $m^{\star}(n)$ after which EEF1-NN, RR, and MUW start converging towards EEF1 for goods/chores for uniform distribution in Table ~\ref{tab:mvalueanalysis}. We don't report CRR in this Table; as we see fluctuations in $\beta_{sw}$, it doesn't increase smoothly in Fig. [\ref{fig:n10}\ref{fig:n13}]; However, note that CRR results better than RR for goods, and EEF1-NN\ performs marginally better than CRR. We observe that in the case of goods, EEF1-NN\ reaches close to EEF1 allocations faster than MUW and RR, and RR reaches close to EEF1 faster than MUW.
As seen in Table ~\ref{tab:mvalueanalysis}, EEF1-NN converges first, then MUW, and finally RR in the case of chores.
We report the value of $m^{\star}$ for RR we use $\beta_{sw} \le 1.064$ since these $m$ values are already significantly high than MUW and RR, concluding that RR will converge after a considerably larger $m$.
We also observed as $m$ increases, $\alpha_{EF1},\alpha_{EFX}$, and $\alpha_{EF}$ of MUW keeps getting closer. For example, for $9\times530$ goods uniform distribution, $\alpha_{EF1}=0.989$,$\alpha_{EFX}=0.9834$, and $\alpha_{EF}=0.9834$; while for $9\times200$ goods uniform distribution, $\alpha_{EF1}=0.6436$,$\alpha_{EFX}=0.5086$, and $\alpha_{EF}=0.5032$.
Note that the actual value of $m^{\star}(n)$ may be slightly different from the exact point of convergence mentioned in Table as we do not perform experiments for all possible values of $m$. Our goal here is to observe a pattern among approaches to compre the different approaches to achieve EEF1.
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.9\linewidth]{figures/AAMAS-differentn.png}
\caption{\textsc{Exp3} ($n=7,12,14$ goods, Uniform Distribution)}
\Description{For 10 agents, valuation drawn randomly from a uniform distribution, plotting-TBD }
\label{fig:n13}
\end{figure*}
\begin{table}[H]
\caption{Value of $m^{\star}(n)$ as different approaches converge to EEF1 allocations }
\centering{
\begin{tabular}{l|lll|lll}
\toprule
\multicolumn{1}{c}{\multirow{2}{*}{$n$}} & \multicolumn{3}{c}{ ($m$) Goods} & \multicolumn{3}{c}{($m$) Chores} \\
\multicolumn{1}{c}{} & EEF1-NN\ & RR & MUW & EEF1-NN\ & RR & MUW \\
\midrule
7 & 38 & 159 & 380 & 44 & 195 &112 \\
8 & 46 & 172 & 450 & 44 & 240& 120 \\
9 & 57 & 186 & 530 & 53 &295 &130 \\
10 & 70 & 196 & 610 & 60 & 340 &148 \\
11 & 82 & 206 & 660 & 68 &400 & 160 \\
12 & 94 & 214 & 740 & 75 & 455 &167 \\
13 & 110 & 220 & 840 & 83 &505 &180 \\
14 & 134 & 228 & 940 & 87 & 565 & 190 \\
\bottomrule
\end{tabular}}
\label{tab:mvalueanalysis}
\end{table}
\smallskip
\noindent\textbf{Discussion}
$\alpha_{EF1}^{EEF1-NN}$ reaches 1 much faster than $\alpha_{EF1}^{MUW}$, and $\beta_{sw}^{EEF1-NN}$ reaches close to $\beta_{sw}^{MUW}$ much faster than RR, D-RR, CRR. So the results from EEF1-NN\ show a better trade-off between EF1 and efficiency than the existing approaches for different input distributions.
We trained our network with fixed $n \times m$ and goods or/and chores, still the performance scales for any $m$ and a large $n$.
For smaller $n$ and $m$, one can use integer programming or any pseudo-polynomial approach, and when $m>>n$. We observed that MUW converges towards EEF1 faster than RR in goods, while in chores, it's the other way around.
Hence we conclude that our approach effectively learns and provides a better trade-off when $m$ is not too large or very small compared to the $n$ but is in a specific range.
\section{Conclusion}
In this paper, we addressed finding fair and efficient allocations for goods, chores, or combinations. In general, the problem is NP-hard. We proposed a neural network EEF1-NN\ to find EEF1 allocations. To overcome the issue of finding optimal discrete allocations, we designed appropriate architecture and input representation combined with other training heuristics. We studied the effect each proposed constituent has on performance. Our experiments demonstrated the efficacy of EEF1-NN\ for different input distributions over existing approaches. It finds reasonably fair and close to optimal solutions in real-time. Can we improve it further?
\bibliographystyle{ACM-Reference-Format}
| {
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Q: which is the best layout to represent matrices in Java swing? I'm trying to show the interface for
Matrix Multiplication - A*B*C - using Java concurrency and Using java.swing for the interface.
Meaning there will be 4 windows, 3 input one output. The output matrix cells will change its values dynamically as you enter the inputs in the 3 GUI input windows. I have solved until the logical part of the program but I'm stuck at showing the interface by using swing and binding these two things together?
Which is the best layout to represent matrices in Java swing? And where to start learning Swing for this?
A: I think GridLayout is perfect for your solution:
I recommend you to learn swing using Oracle's tutorials: Creating a GUI With JFC/Swing
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,377 |
Gazoduq preliminary route unveiled
Posted on April 24, 2019 Author Alan Carter Comment(0)
Gazoduq unveiled a preliminary outline of its proposed underground natural gas pipeline of more than 750 kilometers between northeastern Ontario and the Saguenay. In Saguenay-Lac-Saint-Jean, the Preferred Development Zone (ZAP) will extend over 235 km, affecting three RCMs and the city of Saguenay.
The company unveiled the work of the public consultations held in recent weeks, in a statement issued Tuesday night, unveiling first the ZAP, "identified with a desire to minimize social and environmental impacts," said Gazoduq.
The interior of the zone is an average width of 400 meters and at the end of the process the permanent right of way should be 30 meters.
In its press release, Gazoduq states that 78 percent of the PAZ is on public land and occupies 32 kilometers of permanent agricultural zone. No protected area is found in the projected area and it is learned that a compression station is planned on the territory of Lac-Ashuapmushuan, in the RCM Domaine-du-Roy.
According to data from the Ministry of the Environment of the Fight against climate change, there are 6613 km2 of protected areas in Saguenay-Lac-Saint-Jean, which represents a little more than six percent of the total area.
The PAZ also covers areas belonging to Sainte-Hedwidge, Roberval, Chambord, Saint-Francois-de-Sales, Saint-André-du-Lac-Saint-Jean, Métabetchouan-Lac-à-la-Croix, Hébertville, Saguenay and at Lac-Ministuk (Fjord-du-Saguenay RCM).
Gazoduq mentions that the PAZ largely avoids the populated areas that were within the corridor under study. Lakes, catchment and groundwater protection areas and protected areas were taken into consideration during the various meetings with citizens.
During the public consultations held in Saguenay-Lac-Saint-Jean in February, more than 200 people traveled to the Saguenay (Chicoutimi), Alma and Roberval meetings. Aboriginal communities had also been approached early in the process.
The next steps will be to inventory the fauna and flora along the PDA. The various mitigation measures to minimize the impacts of the project will be documented and the file will be submitted to the National Energy Board before the end of 2019, according to forecasts by Gazoduq. The environmental impact study of the company will follow during the same period.
"We are determined that our project will make a significant contribution to the fight against climate change on a global scale, thus enabling Québec to play a leading role in this regard. We are committed to continuing the dialogue with all communities throughout the development of our project to minimize its impacts and maximize local benefits, "said Gazoduq president Louis Bergeron in the statement.
Alan Carter
Alan Carter has been a reporter on the news desk since 2015. Before that she wrote about young adolescents and family dynamics for Styles and was the legal affairs correspondent for the Metro desk. Before joining The Koz Post, Alan Carter worked as a staff writer at the Village Voice and a freelancer for Newsday, The Wall Street Journal, GQ and Mirabella.
A dozen participating restaurants in the SkipTheDishes delivery service
Posted on April 27, 2019 Author Alan Carter
The SkipTheDishes delivery service has been operating in Saguenay since April 11th. A dozen restaurants are currently active in the area, including McDonald's, Thai Express, KFC, Salvatore, Sushi Shop and Passion Café. A great success, says the Canadian company. S kipTheDishes is a web application that offers a delivery service to participating restaurateurs. These are […]
Investments and job creation record for Sherbrooke Innopole
Investments of $ 223 million were made in 2018 in businesses in key sectors of Sherbrooke, a record that beats that of $ 210 million recorded in 2017. Also, despite the shortage of manpower, a "phenomenal" Employment was created at 1128, almost double the number of jobs created the previous year. This is according to […]
Gilbert Products invests $ 5 million
Products Gilbert, of Roberval, specializes in the production of equipment for sawmills, all-terrain surfacers, equipment for the forest industry and for construction, invests $ 5 million to acquire new robotic machining units to increase productivity. The investment was announced Friday afternoon by the president of the company, Sylvain Gilbert, in the presence of Richard Hébert, […]
Desjardins remits $ 5.5 million in rebates
Jess Moskaluke at the new country brome festival this summer | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"http:\/\/commons.apache.org\/proper\/commons-math\/apidocs\/org\/apache\/commons\/math3\/complex\/RootsOfUnity.html","text":"org.apache.commons.math3.complex\n\n## Class RootsOfUnity\n\n\u2022 All Implemented Interfaces:\nSerializable\n\npublic class RootsOfUnity\nextends Object\nimplements Serializable\nA helper class for the computation and caching of the n-th roots of unity.\nSince:\n3.0\nVersion:\n$Id: RootsOfUnity.java 1416643 2012-12-03 19:37:14Z tn$\nSerialized Form\n\u2022 ### Constructor Summary\n\nConstructors\nConstructor and Description\nRootsOfUnity()\nBuild an engine for computing the n-th roots of unity.\n\u2022 ### Method Summary\n\nMethods\nModifier and Type Method and Description\nvoid computeRoots(int\u00a0n)\nComputes the n-th roots of unity.\ndouble getImaginary(int\u00a0k)\nGet the imaginary part of the k-th n-th root of unity.\nint getNumberOfRoots()\nReturns the number of roots of unity currently stored.\ndouble getReal(int\u00a0k)\nGet the real part of the k-th n-th root of unity.\nboolean isCounterClockWise()\nReturns true if computeRoots(int) was called with a positive value of its argument n.\n\u2022 ### Methods inherited from class\u00a0java.lang.Object\n\nclone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait\n\u2022 ### Constructor Detail\n\n\u2022 #### RootsOfUnity\n\npublic\u00a0RootsOfUnity()\nBuild an engine for computing the n-th roots of unity.\n\u2022 ### Method Detail\n\n\u2022 #### isCounterClockWise\n\npublic\u00a0boolean\u00a0isCounterClockWise()\nthrows MathIllegalStateException\nReturns true if computeRoots(int) was called with a positive value of its argument n. If true, then counter-clockwise ordering of the roots of unity should be used.\nReturns:\ntrue if the roots of unity are stored in counter-clockwise order\nThrows:\nMathIllegalStateException - if no roots of unity have been computed yet\n\u2022 #### computeRoots\n\npublic\u00a0void\u00a0computeRoots(int\u00a0n)\nthrows ZeroException\n\nComputes the n-th roots of unity. The roots are stored in omega[], such that omega[k] = w ^ k, where k = 0, ..., n - 1, w = exp(2 * pi * i \/ n) and i = sqrt(-1).\n\nNote that n can be positive of negative\n\n\u2022 abs(n) is always the number of roots of unity.\n\u2022 If n > 0, then the roots are stored in counter-clockwise order.\n\u2022 If n < 0, then the roots are stored in clockwise order.\n\nParameters:\nn - the (signed) number of roots of unity to be computed\nThrows:\nZeroException - if n = 0\n\u2022 #### getReal\n\npublic\u00a0double\u00a0getReal(int\u00a0k)\nthrows MathIllegalStateException,\nMathIllegalArgumentException\nGet the real part of the k-th n-th root of unity.\nParameters:\nk - index of the n-th root of unity\nReturns:\nreal part of the k-th n-th root of unity\nThrows:\nMathIllegalStateException - if no roots of unity have been computed yet\nMathIllegalArgumentException - if k is out of range\n\u2022 #### getImaginary\n\npublic\u00a0double\u00a0getImaginary(int\u00a0k)\nthrows MathIllegalStateException,\nOutOfRangeException\nGet the imaginary part of the k-th n-th root of unity.\nParameters:\nk - index of the n-th root of unity\nReturns:\nimaginary part of the k-th n-th root of unity\nThrows:\nMathIllegalStateException - if no roots of unity have been computed yet\nOutOfRangeException - if k is out of range\n\u2022 #### getNumberOfRoots\n\npublic\u00a0int\u00a0getNumberOfRoots()\nReturns the number of roots of unity currently stored. If computeRoots(int) was called with n, then this method returns abs(n). If no roots of unity have been computed yet, this method returns 0.\nReturns:\nthe number of roots of unity currently stored","date":"2014-04-21 02:46:03","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\": 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\": 0.3094245493412018, \"perplexity\": 1898.8239151521718}, \"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-2014-15\/segments\/1397609539447.23\/warc\/CC-MAIN-20140416005219-00637-ip-10-147-4-33.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/quant.stackexchange.com\/questions\/34459\/approximations-for-quanto-options-pricing\/34462","text":"# Approximations for Quanto Options pricing\n\nOn page 4 of this paper, the auhor provides two good approximations for quanto options pricing: $V^d_{black}$ and $V^d_{blackATM}$. These approximations consist of using the ATM and\/or stike volatilities (of the underlying asset and FX rate) for the pricing procedure. Is there a mathematical reason for this? What I got is that when we have no better options, we run toward the ATM volatilities. But mathematically, I see no reason for this (maybe because it is an average volatility...). Could you please provide mathematical justification for these approximations?\n\nThank you.\n\nIf you compute the quanto adjustment $\\exp(-\\rho \\sigma_X \\sigma_S)$ from the vol $\\sigma_S(K)$ at the option strike $K$ then the quanto forward obtained by call\/put parity becomes strike dependent and that does not make sense.\nIn addition the quanto adjustment depends on the correlation parameter $\\rho$ which is difficult to estimate, and if you imply it from quoted quanto options then you might as well use the ATM vol for $\\sigma_S$ since the thing you're really interested in is the term $\\rho \\sigma_X \\sigma_S$.","date":"2019-07-16 16:16:16","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\": 0.8217524290084839, \"perplexity\": 938.7834116233727}, \"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-2019-30\/segments\/1563195524679.39\/warc\/CC-MAIN-20190716160315-20190716182315-00277.warc.gz\"}"} | null | null |
Mugena ist Teil der Tessiner Gemeinde Alto Malcantone im Kreis Breno, im Bezirk Lugano im oberen Malcantone.
Geographie
Das Dorf liegt auf 818 m ü. M. im obern Val Magliasina und sechs Kilometer südwestlich der Station Taverne der Linie Bellinzona-Lugano-Chiasso der Schweizerischen Bundesbahnen.
Gemeindefusion
Mugena fusionierte am 13. März 2005 mit den früheren Gemeinden Breno, Fescoggia, Arosio und Vezio zur neuen Gemeinde Alto Malcantone.
Geschichte
Eine erste Erwähnung findet das Dorf im Jahre 1214 unter dem damaligen Namen Megiadina. Überliefert ist die Römerstrasse von Ponte Tresa zum Monte Ceneri durch das Gemeindegebiet.
Bevölkerung
Sehenswürdigkeiten
Pfarrkirche Sant'Agata
Wohnhaus mit Malereien von Giovanni Pellegrinelli
Bildung
Fondazione Portugalli
Persönlichkeiten
Literatur
Virgilio Chiesa: Mugena. In: Lineamenti storici del Malcantone. Tipografia Gaggini-Bizzozero, Lugano 1961.
Laura Facchin: I Portugalli di Mugena a Livorno. In: Arte&Storia. Edizioni Ticino Management, 14. Jahrgang, Nummer 62, Lugano August 2014, S. 302–315.
Simona Martinoli u. a.: Guida d'arte della Svizzera italiana. Hrsg. Gesellschaft für Schweizerische Kunstgeschichte GSK, Edizioni Casagrande, Bellinzona 2007, ISBN 978-88-7713-482-0, S. 386.
Giovanni Maria Staffieri: Mugena. In: Malcantone. Testimonianze culturali nei comuni malcantonesi. Lugano-Agno 1985, S. 73, 74–75.
Celestino Trezzini: Mugena. In: Historisch-Biographisches Lexikon der Schweiz, Band 5, Monopole – Neuenkirch., Attinger, Neuenburg 1929, S. 205 (Digitalisat).
Weblinks
Offizielle Website der Gemeinde Alto Malcantone
Amt für Statistik des Kantons Tessin: Alto Malcantone (italienisch)
Alto Malcantone-Mugena: Kulturgüterinventar des Kantons Tessin
Mugena auf elexikon.ch
Pfarrkirche Sant'Agata auf www.flickr.com
26. Novembre 1214 Verkauf, Carta vendicionis in Lugano (italienisch) auf ti.ch/archivio-pergamene (abgerufen am 22. Januar 2017)
12. Februar 1331 Verkauf, Carta vendicionis (italienisch) auf ti.ch/archivio-pergamene (abgerufen am 26. Januar 2018).
Einzelnachweise
Alto Malcantone
Ort im Kanton Tessin
Ehemalige politische Gemeinde in der Schweiz
Alto Malcantone
Ersterwähnung 1214
Gemeindeauflösung 2005 | {
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\section{Introduction}
In theory, commons-based peer production projects like Wikipedia and GNU/Linux allow for diverse contributions on a global scale. In practice, would-be contributors face widely varying barriers when they seek to participate. Although potential contributors may seek privacy online in order to mitigate perceived threats, such as government oppression or personal harassment \citep{forte_privacy_2017, kang_why_2013}, contributing to peer production projects while maintaining strong anonymity is frequently disallowed \citep{mcdonald_privacy_2019}.
Are barriers to anonymity seekers' contributions warranted? What kinds of contributions can be expected from anonymity seekers? What happens when anonymous contributors interact with and work beside others in a community where anonymous contributions are often distrusted?
To address these questions, we examined contributions to English Wikipedia made through Tor, a secure privacy network that conceals IP addresses and geographic location. Although Wikipedia attempts to block editing through Tor (see Figure \ref{fig:editBlock}), the blocks have sometimes missed Tor addresses or failed to recognize new addresses quickly. As a result, Tor users have managed to edit articles thousands of times \citep{tran_tor_2019}.
We use these digital trace data as forensic evidence to construct narratives that provide a thick description of contributions to Wikipedia made by Tor users. In turn, we use these narratives as material for a ``contextualist'' thematic analysis \citep{braun_using_2006} that attempts to reflect the limitations of material, question our own assumptions, and maintain awareness of how social context may shape the meaning of what we read and see.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{editBlocked.png}
\caption{When users attempt to edit Wikipedia while using the Tor network, they are presented with a message like the one shown. Users of Tor are told that their IP address ``has been automatically identified as a Tor exit node'' and that ``Editing through Tor is blocked to prevent abuse.'' The concept of exit nodes is described in §\ref{sec:tor}.}
\label{fig:editBlock}
\end{figure}
This paper makes several contributions. First, we describe \textit{forensic qualitative analysis}, an extension of existing qualitative methodologies that we argue can help provide thick descriptions about participants---like anonymity seeking users of Tor---who cannot be interviewed or observed directly but who leave behind rich, if superficially decontextualized digital traces. As our primary contribution, we present the results of a thematic analysis of narratives constructed using this new method on a dataset of contributions made by Tor users to English Wikipedia. We use our knowledge of technology, history, culture, and the Wikipedia community itself to assist us in our interpretation and identify seven themes that suggest editors' intention. Third, we reflect on the challenges of developing trust online, and consider how contribution types carry different risks to contributors and the Wikipedia community. We position these themes with respect to previous work to characterize both motivations to seek anonymity and reasons that may lead service providers and communities to block them.
\section{Background}
In the following sections, we situate our study in the broader literature on why people seek out anonymity online, the means they employ to do so, and the challenges they face from the communities they seek to join. In particular, we consider how anonymity is operationalized and defined in our overlapping empirical settings: the Tor network and Wikipedia.
Our study joins a growing body of work that seeks to understand online anonymity. Although Wikipedia itself characterizes all individuals who contribute without an account as ``anonymous,'' we follow the lead of
previous authors who treat anonymity as a multidimensional concept. For example, \citet{marx_whats_1999} theorizes that anonymity entails obscuring seven types of identifying information: ``(1) legal name, (2) locatability, (3) pseudonyms that can be linked to legal name and/or locatability...(4) pseudonyms that cannot be linked to other forms of identity knowledge...(5) pattern knowledge, (6) social categorization, and (7) symbols of eligibility/noneligibility'' (p. 100). Online communities often allow people to obscure some types of identity information, while requiring disclosure of others. For example, some communities enforce a ``real name'' policy or require a home address or phone number for verification. Some dimensions of anonymity can be moderated by what someone chooses to share or do within the platform. One can carefully choose a pseudonym to avoid name recognition, decline to disclose personal details, and try to avoid making oneself identifiable through behavior patterns.\footnote{One important caveat here: individuals may be unaware of their `tells'---and given the active work in machine learning and fields such as stylometry, e.g., \citet{brennan_adversarial_2012}, maintaining privacy in this dimension can be extremely difficult.} Obscuring one's location online typically requires additional effort \citep{forte_privacy_2017}, and some service providers may either deny access or limit the capabilities of individuals who choose not to be locatable \citep{mcdonald_privacy_2019}.
The locatability dimension of privacy reflects an important challenge for Internet users because location information is systematically revealed through IP addresses. An IP address is a unique number used to identify every computer on the Internet. Like a ``to'' and ``from'' address on a piece of mail, IP addresses for both senders and receivers are associated with every piece of traffic sent over the Internet. IP addresses reveal location because they are assigned as part of a larger block to some identifiable and registered unit such as a university, company, or service provider. Using freely available databases, IP addresses can be mapped to approximate address or location by anybody. Network providers can associate individual subscriber homes or even individual computers with the IP address being used, and hence, directly identify the household if not the individual. The EU privacy regulation known as GDPR recognizes the relevance of IP address as a personal identifier
and requires that it be treated as such \citep{information_commissioners_office_what_2019}.
Reducing locatability is important to many Internet users because what people post and do online can lead not only to harassment \citep{menking_heart_2015, kang_why_2013}, but also threats to reputation, employment, and harm to self or loved ones \citep{forte_privacy_2017, kang_why_2013}.
Threats associated with privacy loss may originate from individuals, institutions, or governments, and can have a chilling effect on online expression. Harassed and doxed individuals may be forced off the Internet and into hiding, and journalists and activists may find themselves in jeopardy.
To the extent that anonymous contributors represent minority viewpoints, they may be sources of valuable contributions.
This account of the benefits of privacy online should not be interpreted as suggesting that those who seek privacy online are only those escaping censorship and oppression. Internet users may seek privacy in order to violate laws and norms about free expression or to violate copyright laws \citep{kang_why_2013}. The potentially disinhibiting effects of anonymity and pseudonymity have been linked to increased negative behavior \citep{suler_bad_1998, suler_online_2004, kiesler_social_2009}.
Despite the variety of reasons that Internet users may seek locatability privacy, it is clear that the use of high-quality privacy services is critical to some individuals' ability to participate in public life online and to contribute to peer production projects. We describe one such tool in the next section.
\subsection{IP-Privacy Through Tor}
\label{sec:tor}
Our study concerns users of the Tor network, which protects the privacy of its users' IP addresses. Despite the many reasons that a person might seek anonymity, media accounts of Tor have often described Tor in association with criminal activity \citep{bilton_digital_2013}, or emphasize this type of activity in sensationalist headlines of more nuanced articles \citep{kobie_what_2019, mcgoogan_dark_2016}. Other coverage of Tor reflects a more nuanced view and calls this reputation a matter of ``image'' for a ``useful privacy tool'' \citep{perlroth_tor_2016}. Others describe Tor as an ``internet boogeyman'' that is ``misunderstood'' \citep{menegus_dark_2017}.
Certainly privacy can also be used to conceal criminal activity \citep{cullum_is_2018}, and multiple studies have sought to measure the extent of illegal material and activity in the Tor network \citep{faizan_exploring_2019,moore_cryptopolitik_2016,owenson_tor_2015}.
Using Tor represents an explicit decision by a user to employ a privacy tool.
In conventional network routing, the route---including the IP address reflecting the point of origin---is visible to the recipient of the traffic. Combined with logs from a service provider, access to information about the IP address of the point of origin can allow the unique identification of the location, and often the specific computer, where the traffic originated. By contrast, Tor uses a multi-layered ``onion-routing'' structure that obscures the route to and from, and therefore, the location of the sender. To do this, Tor relies on people worldwide to volunteer machines to act as nodes in the network through which traffic bounces. Each node only knows the step just before and just after it, and no node can see the entire route \citep{huang_onion_2016}.
When someone uses Tor, the places they visit on the Internet can know only the final step in the sequence, known as the Tor exit node.
Tor dynamically reassigns users to a new exit node IP address as often as every 10 minutes to further obscure the trail back to the user. The list of exit nodes is published and refreshed regularly by Tor.
\subsection{Anonymity and Identifiability in Wikipedia}
Wikipedia allows what is described within the community as ``anonymous'' editing by permitting individuals to contribute without creating an account. These users' contributions are publicly associated with their IP address rather than a username. Although posting publicly as a traceable IP address is not a very effective means of achieving anonymity, this policy lowers barriers to participation for ``newbies'' on Wikipedia \citep{mcdonald_privacy_2019}. Scholars
have touted the value of such ``anonymous'' contributors, finding that their work may be more likely to persist \citep{anthony_reputation_2009}. Other work has shown that a significant number of IP-based contributors provide work of high quality \citep{javanmardi_user_2009, anthony_reputation_2009}
and that naive contributions may serve as an indicator of public attention that draws in the efforts of experienced editors \citep{gorbatai_paradox_2014}.
Anonymity seeking users face a number of challenges on Wikipedia. Contributors without an account are hampered in their ability to accumulate social capital and may be perceived by the community to be less trustworthy.
\citet{oxley_what_2010} observed that contributors to Wikipedia may make six types of authority claims: they may assert their (1) expertise, (2) life experience, or (3) institutional affiliation; (4) use their policy familiarity; (5) cite outside authorities; or (6) leverage social expectations from the community. Of these, we observe that any claim to expertise, life experience, or institutional affiliation would tend to diminish anonymity.
As a result, they may struggle to successfully assert their position in discussions about what belongs in an article.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{torBanner.png}
\caption{This banner is sometimes placed on user profile pages within Wikipedia to indicate that an IP address has been the source of contributions by Tor users.}
\label{fig:torBanner}
\end{figure}
Although Wikipedia allows edits from users without accounts, it systematically blocks edits from users of any ``open proxy'' system that seeks to obscure locatability by hiding contributors IP addresses---including Tor. The practical impact of this is that if Wikipedia identifies an address as coming from Tor, which it now does with high speed and accuracy, the individual using that address cannot edit or create an account. Tor users also cannot edit using an existing account unless they first generate a strong positive record of editing without the privacy protection of Tor---a very high bar for anyone to clear if they need privacy protection in order to safely contribute at all \citep{tran_tor_2019}.
Wikipedia blocks systems like Tor because it relies heavily on IP-level blocking to fight spam and vandalism. Because allowing edits from Tor would provide an easy way for users to evade bans, Wikipedia attempts to block all contributions from systems like Tor \citep{tran_tor_2019}.
Once an IP address is identified as part of the Tor network, edits from that address are blocked and profile pages associated with it are marked, as in Figure \ref{fig:torBanner}, so that it is clear that any contributions made from that IP address may have been made by a Tor user.
Given the barriers to Tor editing and the limited anonymity afforded by editing with an IP address, it is difficult to empirically assess what kinds of value anonymity seekers might bring to Wikipedia and similar projects.
We attempt to do so by taking advantage of a unique dataset of edits to Wikipedia made by Tor users that was produced by \citet{tran_tor_2019}.
Tran et al.~take advantage of the fact that although Wikipedia has sought to block Tor since at least 2005, the technology blocking Tor has been imperfect. As a result, over 11,000 edits have been made to Wikipedia using Tor. Tran et al.'s data includes contributions from as far back as 2007 when Tor data became available through 2018. The rate of edits is irregular, and as Tran et al.~speculate, the flaws in Wikipedia's blocking technology may have resulted from multiple factors: delays in the Tor network publishing new nodes, delays in Wikipedia ingesting updated lists of current nodes, irregularities in timing as new Tor nodes joined and left the network, and other timing or stability issues generated by Wikipedia's blocking mechanisms.
In their analysis of Tor users who circumvent the ban, \citet{tran_tor_2019} describe a series of quantitative analyses that suggest that those who contribute to Wikipedia using Tor are similar to other kinds of users, especially those contributing without accounts and new contributors.
Although a useful first comparison, Tran et al.'s quantitative comparison ignores the content and context of contributions in a way that makes evaluating their value extraordinarily difficult.
For example, if we examine the details and context of Tor editors' contributions as a Wikipedia community member might, we might recognize ways in which the contributions create unusual risk for, or offering unique benefits to, Wikipedia.
To help better understand the potential value and risk associated with anonymity seekers' contributions to social computing systems, we conduct a thematic analysis of Tor edits to English Wikipedia using a novel qualitative analysis approach that we introduce below.
\section{Methodology}
Our methods of reconstructing and interpreting Tor-based Wikipedia contributions comprised several steps. First, we constructed a random sample of 500 edit sessions to English Wikipedia made by users of Tor. We then used this sample to conduct what we call a \textit{forensic qualitative analysis}. This analysis involved examining digital traces within a broader context of other traces as well as making use of our understanding of the community in which they occurred. Next, we conducted thematic coding of the re-contextualized edit sessions by generating, discussing, revising, and applying a set of thematic codes. Finally, we produced a set of detailed narrative memos that described the digital traces, their antecedents, and their effects. This multi-step semantic analysis was conducted iteratively and interactively among a team of four analysts. Although initially session analysis was randomly assigned among the analysts, as narrative threads emerged from our discussion, some sessions were reassigned and an analyst took the lead in exploring the topic in more depth. For example, one edit war spanned multiple pages related to the same conflict about the naming conventions and relative virtues of South Asian schools (see §\ref{sec:thm-editwars}).
\subsection{Sample Construction}
\label{sec:data}
We developed our sample in two stages. First, we grouped edits from the original Tran et al.~dataset of English Wikipedia edits made using Tor into 7,786 edit ``sessions'' \citep{geiger_using_2013}. Second, we drew a random sample of 500 sessions.
This random sampling approach can only yield a sample which is representative of the Tor-based edits that were made despite the block. It is possible that the sample is not representative of the kinds of edits that were attempted or that might have been attempted if Tor were not blocked. For example, the most determined or most savvy Tor-based Wikipedia editors might be overrepresented in the population from which we sampled. Similarly, users who knew that Tor was blocked on Wikipedia might have stopped trying. We discuss this further in §\ref{sec:limitations}.
\citet{geiger_using_2013} define an edit session as a collection of all edits made
from \textit{the same account}
to \textit{any page or article}
as long as there is no more than an hour between edits. We altered this definition
in two ways and included edits made from \textit{any active Tor node} to the \textit{same page or article} as long as there is no more than an hour between edits. The allowance for multiple Tor IP addresses was made because Tor rotates the exit node in use after 10 minutes.\footnote{\url{https://www.torproject.org/docs/faq.html.en} \textit{archived at} \url{https://perma.cc/75V3-KD4N}} We limit our session to edits made to a single page because considering all edits to all pages would imply that only a single individual using Tor was actively editing on Wikipedia at a given time.
We chose edit sessions as our sampling unit because, while some contributors to Wikipedia may work for extended periods of time without saving their work, others make multiple subsequent edits, saving repeatedly as they go. The use of sessions, rather than isolated edits, supports the forensic qualitative methodology described in §\ref{sec:forensic}.
Our random sample of 500 edit sessions included 738 individual edits to 438 articles. Some sessions were composed of multiple edits, but most were not. Some articles in the sample were edited in multiple sessions, but most were not. The earliest session was from 2007, and the most recent was from 2017.
Our ethical commitment to the population of study and the broader community, a challenge noted by \citet{rotman_extreme_2012}, includes the use of only public data that is readily visible to anyone who uses or contributes to Wikipedia. As a result, our work was conducted entirely with logs, article histories, and public IP registrations, and involved no interaction or intervention with research subjects who remained unidentified throughout the process. The research was determined to not be human subjects research by the IRB of the lead author's institution. Despite this, we also pseudonymize names and articles, and in some cases paraphrase quotes, to make reidentification more difficult.
\subsection{Qualitative Forensic Analysis}
\label{sec:forensic}
The distributed and vast nature of online communities can make it difficult to provide the kind of thick descriptions of social phenomena common in traditional ethnographic research \citep{blomberg_reflections_2013, coleman_ethnographic_2010}.
\citet{geiger_work_2010} have advocated combining the analysis of digital traces with participant interviews as part of digital trace ethnography. In empirical contexts like ours, the inability to identify users yields fragmented traces and makes interviewing and the collection of other forms of rich ethnographic data impossible. How can researchers make meaning from traces of people's online activity in settings like ours?
To do so, we conduct what we call \textit{forensic qualitative analysis}. This approach involves attempting to imbue digital trace material with meaning through a detailed study of the trace materials themselves, along with their context and connections with other events and materials. The approach is centered on the experience of the individuals who left the traces and is informed by our own knowledge of the material affordances, behavior patterns, and community values and norms of the empirical setting.
Our approach draws upon innovative methodology used in previous studies in social computing. We are inspired by \citet{twyman_black_2017}, who combined quantitative analysis with careful qualitative reading of discussion pages and historical events to understand interactions of the Wikipedia community with the Black Lives Matter movement. We are also inspired by \citet{nagar_what_2012}, whose careful reading of edit histories was used to observe interactions on Wikipedia as editors participated in collective sensemaking through policy development and interpretation. Finally, we draw from forensic ethnography, which has been used to study corporate crime \citep{van_rooij_toxic_2018}, as well as network forensics, which describes a collection of techniques for reconstructing online events through network traffic analysis \citep{corey_network_2002}.
\subsubsection{Forensic qualitative analysis in contrast}
Our forensic qualitative analysis methodology builds on others' interpretive and technical strategies but has two unique qualities. First, it directly tackles the problem of the absence of physical participants by inviting the researcher to reconstruct the participant's experience. Second, it is informed by the measures, tools, and strategies employed by community participants who are themselves seeking to interpret the behaviors of others. Wikipedia editors routinely delve into the type of data using techniques similar to our methodology to investigate and interpret the actions of editors when deciding whether to give awards \citep{kriplean_articulations_2008},
when evaluating whether an individual should be made an administrator \citep{burke_taking_2008}, and when investigating complaints, rule violations, or content disputes.\footnote{\url{https://en.wikipedia.org/wiki/Wikipedia:Dispute\_resolution\_requests} \textit{archived at} \url{https://perma.cc/X7V5-UA48}}
Wikipedia is organized to support this type of work. Its public archives include a wealth of data about what changes were made and by whom. User-designed tools for querying this data are shared and hosted on Wikimedia Foundation servers.
Forensic qualitative analysis is particularly useful for studying anonymous actions. As \citet{scott_reveal_1998} describes in his theoretical model of anonymous communication: faced with an anonymous author, recipients may seek to reconstruct the identity and intentions of the author based on clues or details contained within the text itself.
Our method may be useful in other circumstances where research questions concern factors internal to the participant (e.g. motivations and intentions) and where subjects are not identifiable or reachable but where observational digital traces are rich. Examples include discussion boards and chat applications where anonymity is allowed or may even be normative \citep{bernstein_4chan_2011,schoenebeck_secret_2013}. Forensic qualitative analysis may also be useful in contexts where the platform allows participation from individuals both with and without identifiers such as an account.
\subsubsection{Forensic qualitative analysis in Wikipedia}
We began building context for each edit in our sample by looking roughly ten contributions forward and backward within each page's edit history.
For every edit, we attempted to explore Wikipedia's extensive backstage area for additional context. This includes the discussion (i.e., Talk) pages associated with each article. It also included the User and User Talk pages of any editors involved. User pages are often used as personal home pages for Wikipedia contributors and User Talk pages are pages used for interpersonal communication between editors. An individual need not have an account to have a User page or User Talk page.
We went further forward and backward in these edit histories if necessary.
We examined edits from all editors (registered, IP-based, or Tor-based) in these histories as well as edits made to other articles from the same Tor IP address at about the same time.
We aimed to remain curious for as long as possible, to chase down leads that presented themselves, and to consider other sources within Wikipedia such as block logs and noticeboards. We also used general Internet resources, such as the Internet Archive, the WHOIS IP registration database, and external sites which contributors added links. We sought to reconstruct the timeline and the effects of editing activity.
We tried to gain insight into the state of mind of all the participants. We used the web-based Wikipedia interface that Wikipedia contributors use including the article history page in Wikipedia that allows anyone to navigate all edits to an article in chronological order.
We kept an open mind and considered that vandalism on Wikipedia may reflect mistakes by inexperienced newbies \citep{halfaker_dont_2011, mcdonald_privacy_2019}. To the extent that innocent or unwitting vandalism is part of joining a community of practice, anti-normative behavior may simply be a feature of the learning environment \citep{bryant_becoming_2005, jackson_did_2018}. Other activities that might be labeled as vandalism include the activities from overzealous contributors passionate about a topic.
Our process involved creating detailed notes on each edit session that reflected our attempts to gather context.
Four of the authors conducted an analysis as described above across overlapping sub-samples drawn from our 500 edit sessions. All the authors have experience using Wikipedia. The first author analyzed over 200 sessions, two authors analyzed just under 150 sessions, and a fourth author analyzed 50 sessions. Some sessions were examined by multiple authors as part of our iterative process. In addition to revision IDs and hyperlinks to the archival location of each edited page in each session, we authored metadata including the time and date of edits, indicators of whether edits were reverts and whether the edit was later reverted, and an assessment from the ORES machine learning algorithm that assesses whether an edit might be considered good faith or damaging \citep{halfaker_ores:_2016}.
Using the sessions themselves and the notes we created while exploring context, we coded our data using an iterative, open thematic coded process as described by \citet{braun_using_2006}.
While coding our data, we met repeatedly to collaboratively generate inductive codes and discuss emerging themes. We eventually arrived at a consensus set of emergent codes and applied these codes to our selection of the data. We discussed definitional issues and questions of interpretation as they arose. Through this iterative process, we assigned one or more of our emergent codes to each edit in our dataset. On some occasions, a single session was composed of different kinds of edits. Once the edits were fully coded, the first and second authors began a process of revisiting our data to write a series of memos that described important themes in our open coding. We returned to our data to create a series of narratives that provided comprehensive descriptions of examples corresponding themes described in our memos. Finally, we grouped these narratives into the seven themes presented below.
\section{Findings}
In this section, we present the most important themes that emerged from our analysis. Each theme reflects a type of contribution made repeatedly by Tor editors to Wikipedia in terms of the intention of the person editing through Tor as well as the reaction of other editors. We observed seven major themes described in sections below:
quotidian contributions (§\ref{sec:thm-quotidian}),
bad faith contributions (§\ref{sec:thm-badfaith}),
activism (§\ref{sec:thm-activism}),
quality maintenance (§\ref{sec:thm-quality}),
edit wars (§\ref{sec:thm-editwars}),
non-article discussion (§\ref{sec:thm-nonarticle}),
and
protests against mistrust (§\ref{sec:thm-mistrust}).
For each theme, we consider potential ways in which the use of a privacy technology may or may not play a role in shaping the different risks faced by communities and individuals.
\subsection{Quotidian Contributions}
\label{sec:thm-quotidian}
The most conspicuous theme that emerged from our analysis reflected the fact that many edits were quotidian in nature. Tor-using contributors frequently engaged in the basic everyday tasks of English Wikipedia and did the same work as other kinds of contributors. These Tor users appear motivated to contribute in the same way many others are: they see a problem they are able to fix or a typo they know how to correct and they hit the edit button in order to do so \citep{bryant_becoming_2005}. Our sense is that these editors may be unaware that their use of Tor to edit Wikipedia is forbidden.
Edit sessions that reflect this theme included Tor-based contributors adding new details to plot summaries of television shows, fixing capitalization errors, and updating the details of bus and train schedules. On the surface, these edits do not appear to be controversial, damaging to the community, or ``high risk'' to an individual. An example of quotidian contributions that we observed is captured in a one-sentence narrative:
\begin{quotation}
On January 31, 2008 at 07:52 UTC, a Tor-based contributor updated the exchange rate for an Asian currency.
\end{quotation}
\noindent Changing numbers in an article with no explanation and no references might constitute vandalism if the information is incorrect or misleading. A numerical change might go unnoticed, and many articles have only a few citations against which specific facts might be verified. In this case, we conducted a search of historical exchange rates and found records that the exchange rate the Tor editor described was correct to all four decimal places they included. We found no evidence in the contributions made prior to this one that suggested that this edit was anything beyond what it appeared to be. We found no subsequent history to indicate objections or concerns from the community of Wikipedia editors.
It was easy for us to forget that these contributors chose keep their IP address private by using Tor. As we observed these seemingly mundane activities, we remained aware that we could not be sure if these edits were only ``quotidian'' to observers with one set of life experiences while controversial to others. Although we knew that Tor-based contributors were quantitatively similar to other kinds of editors along several dimensions \citep{tran_tor_2019}, we expected that qualitative work would reveal more dramatic differences. Given that people pursue privacy because their perspective places them at risk \citep{forte_privacy_2017}, we imagined that Tor-based contributors would reflect minority viewpoints or making unusual kinds of edits.
Instead, the most important theme to emerge from our analysis is consistent with \citet{anthony_reputation_2009} and \citepos{javanmardi_user_2009} description of persistent, high-quality contributions from people participating without accounts.
The plurality of the edits we analyzed were quotidian (see §\ref{sec:quant}). In almost all cases, other Wikipedia users gave no indication that they were aware that the Tor edits were made by an anonymity seeking user. In a few cases, an administrator would comment on the fact that the individual was using Tor while removing the edit or banning the user.
We describe the trivial example of the Tor-based edit to a currency exchange rate both to demonstrate the theme and also to offer an example of the level of detail at which our method operates. In an attempt to reconstruct the narrative around an action, we examine the timing, other edits from the same address, other edits before and after, and the content of the edit. We consider our own experience in Wikipedia to interpret what we observe.
\subsection{Bad Faith Contributions}
\label{sec:thm-badfaith}
Although we found many instances of helpful Tor-based contributors, we also found cases in which Tor users attempted to do harm to the community. Although we identified a range of actions taken in bad faith, the most common forms took the form of vandalism and harassment. As a warning to readers, several of the examples that we describe later in this section include descriptions of violence and self-harm. Examples of harmful contributions from our sample include:
\begin{quotation}
On March 31, 2011 at 22:02 UTC, a Tor-based contributor replaced the opening sentences of a section of an article about a type of vehicle with the text, ``IM THE BAET DANCER INT UHE WORLD.'' A registered contributor reverted this change 15 minutes later.
\end{quotation}
\noindent For reasons that should be obvious to anybody that has read an encyclopedia, this text---goofy, all capitals, off-topic, and misspelled---is not helpful or appropriate. In Wikipedia, contributions like this would be called ``vandalism'' and would be quickly and uncontroversially reverted. Other examples of vandalism were those that seemed designed to taunt administrators, set off alarms within the monitoring structures and systems of the Wikipedia community, or to attract attention. For example:
\begin{quotation}
On November 18, 2011 at 06:00 UTC, a Tor-based contributor updated the User Talk page associated with their Tor IP address seven times over the course of two minutes, with the message ``[name]\footnote{Here, the Tor-based editor typed in the syntax for linking to a user name.} feel like he may commit suicide he needs assistance''.
The Tor IP address, which had been identified as a Tor node and been blocked on 12 separate occasions since 2006, was blocked again at that point by an administrator who also reverted the Tor-based contributor's edits.
\end{quotation}
\noindent We cannot judge for certain whether the Tor-based editor was making a sincere cry for help or engaging in vandalism designed merely to provoke others. However, the response of the registered editor suggests that the contribution was understood by other Wikipedians as unproductive. The edit was quickly undone by a registered contributor, who we observe engaged in high-volume vandal-fighting in a process akin to the one described by \citet{geiger_work_2010}. The reverting editor was at the time making many edits per hour and leaving edit summaries that described their work as reverting vandalism.
We also observed Tor-based contributors engaging in harassment. This sometimes involved adding insults and attacks to other users' User pages.
While insults posted in articles might constitute vandalism if they do not reference a specific Wikipedia contributor, attacks placed on a User page or the associated User Talk page may be interpreted as an attack on the User page's owner.
Vandalism and harassment in our sample typically resulted in IP addresses being banned from contributing. In many cases, banning corresponded to a discovery by an administrator that the IP address in question was associated with Tor.
We observed examples of harassment as a response to administrators who banned Tor IPs. For example:
\begin{quotation}
On June 4, 2008 at 02:23 UTC, a Tor-based contributor edited the User Talk page associated with their IP address by inserting an appeal to their having been blocked, with the following text listed in the ``request reason'' portion of the appeal: ``I am [Wikipedia administrator name]. Unblock this IP address, or I will cut off your balls, eat them in front of you so that you an[sic] see it, then chop off your head.''
\end{quotation}
\noindent By placing this threatening comment in an appeal template as a reason for their address to be unblocked, the Tor-based contributor's text was included in a page in Wikipedia dedicated to discussing requests to be unblocked---where it would be read by administrators.
Attacks against hard-working community-selected leaders
suggest a risk posed to communities by contributors using privacy-protecting technology. Although unwelcome, spam, vandalism, and harassment are not unusual in online communities like Wikipedia. Many groups, policies, and technologies exist to counter these types of unwelcome contributions, including artificial intelligence and human moderation of comments and the rapid review of complaints. In Wikipedia, the creation of anti-vandalism automation tools that seek to lessen the burden on vandal fighters is an area of active study and engineering \citep{asthana_few_2018}.
\subsection{Activism}
\label{sec:thm-activism}
A third theme we observed reflects community tension of a different kind. Wikipedia's community standards include a requirement that all article text evinces a neutral point of view (NPOV) \citep{reagle_good_2010}. Establishing neutrality is a difficult and fraught enterprise that was on full display among edits involved in what we call ``activism.''
Maintaining an NPOV involves a constant collective effort and that can draw editors into direct conflict with one another. One person's definition of neutrality may seem a gross distortion of facts to another. Dominant narratives may be widely documented while subaltern points of view struggle to be heard. Some contributors may engage in what is termed ``POV Pushing''---contributions that unfairly or unreasonably bias content in the direction of one's own opinion \citep{reagle_good_2010}.
A contributor acting as a reviewer of others' content may likewise reflect their own point of view in choosing to support one side or another.
Definitions of NPOV and POV pushing are inherently subjective and a common subject of debate among Wikipedia contributors. For example, people might edit Wikipedia in order to systematically undermine a field of science such as physics or medicine or to change language usage. Other editors may seek to counter these agendas. To a supporter of modern physics or medicine, attacks on those fields may seem like POV pushing, and defending them may be understood as activism. To the skeptic of physics or medicine, casting doubt on these fields likewise may be experienced as activism.
Examples in our sample included a Tor-based contributor who removed the term ``allopathic'' from multiple articles. ``Allopathic'' is described in its Wikipedia article as a pejorative term that supporters of alternative medicine use to describe evidence-based medicine. In this case, the Tor-based editor acted in defense of evidence-based medicine. They were responding to the action of some individual or group which had systematically added the word ``allopathic'' to medicine related articles in order to qualify that the medical knowledge reflected in these articles only reflected one of several legitimate types of medicine.
In another example, a Tor-based contributor updated a politician/lawyer/liberal activist's biographical article to describe him as a ``Democrat politician'' rather than a ``Democratic politician.'' This subtle linguistic shift is part of a larger trend among some supporters of the American Republican Party to describe the Democratic Party as the ``Democrat Party'' in order to distance the party from the adjectival form of the word ``democracy'' \citep{abadi_trump_2017}.
In another example, we see what could be described as POV pushing morph into activism. Given the tone and timing of the edits, we believe that in this circumstance, a single individual engaged in activism via an IP address then perceived themselves to be at risk and migrated from using an IP address to Tor. Because this is an inductive leap and we may be observing the actions of two individuals, we make note of the evidence in the narrative.
\begin{quotation}
On November 14, 2013 at 17:49 UTC, an IP-based contributor edited an article about a mining company. A search of the WHOIS registration of this IP address states that it is registered to a home Internet service provider in a US state. Links in the company article at that time reveal that the company was experiencing financial difficulties and engaged in a conflict with environmentalists who sought to block the company's next project in Europe. The IP-based contributor's addition to the mining company encyclopedia entry accused the company of being complicit with genocide and torture in an Asian country by adding the following text under a new section header with the title ``Support of Genocide and Torture in [Asian Country A]'':
\begin{quote}
``As one of the profitable companies working with [Asian Country B] in the colonization of [Asian Country A], [mining company] is successfully funding and helping to promote genocide in [Asian Country A]. This includes but is not limited to torture (beatings with weapons and steel boots) of [religious figures], rape, imprisonment (with continued torture), electric shock to genitals, relocation and murder. Psychological warfare also plays an important role as [nationals of Asian Country A] are to be kept far away from mining machinery so profits can continue.''
\end{quote}
Less than two hours later, someone made an account on Wikipedia with an account name that matched the name of the mining company. This new contributor added updated personnel information and removed both the content added by the activist IP-based contributor and the section of the article describing the company's financial troubles and environmental controversy. The company-named contributor included an edit summary---metadata created by text entered in a box adjacent to the `Publish changes' button labeled ``(Briefly describe your changes).'' Frequently omitted by new users, the use of an edit summary suggests prior experience with Wikipedia. In the edit summary, the company-named contributor claimed that they were ``updating key personnel, subsidiary webpages and deleted section on Genocide in [Asian Country A] which is incorrect information that is not referenced.''
Three days later, at 00:33 UTC, 00:35 UTC, and 00:36 UTC, the same activist IP address again updated the mining company page to reference genocide. They inserted text stating that the mining company ``needs love and prayers so that they may overcome this immense greed which does not make them happy.'' They added statements that the company financials included ``funding for an ongoing genocide in [Asian Country A]'' and suggested that environmental approvals were only complete because ``officials and inspectors are paid off.''
Then the behavior of the activist IP editor shifts. At 00:45, 00:50, and 00:51 UTC that same day, they remove their own changes. We see edits from a Tor-based editor who adds text back to the same location in the article at 01:02 UTC and removes it again at 01:05 UTC. Although the specific phrasing of the new text is different (``where it funds rape, murder, and torture of innocent [citizens of Asian Country A]'' versus ``[in Asian Country A] helping continue torture, rape and murder''), the substance of the edits is extremely similar.
Then, between 01:06 and 01:27 UTC, the same Tor-based IP made seven edits continuing the same line of protest. In these later edits, the contributions adopt a more encyclopedic tone. They add encyclopedic statements describing percentage ownership of mines by Western companies and state that the mining company had not responded to a questionnaire from an ethical mining advocacy group. Each of their additions was accompanied by a citation to an external reference. Three days later on November 21, 2013 at 00:23 UTC, the original home IP added text with a reference to an article from Reuters which described the leader of Asian Country B as subject to arrest over genocide allegations if they were to travel abroad.
A little over two weeks later, on December 10, 2013, at 19:04 UTC, an IP-based contributor deleted the ``Controversies Involving [Asian Country A] Genocide'' section. A search of the public WHOIS registry of this IP address states that these edits came from an IP address registered to the offices of the mining company.
The company IP-based contributor deleted the ``Controversy'' section and the References section containing evidence for the controversy.
Although no further edits were made from the Tor network at this point, the article received several additional updates. Company-associated IPs and other IPs with no apparent relation to the company removed controversy. Registered contributors and bots applied formatting fixes and added new content that was unrelated to human rights activism. The original home IP used by the activist re-emerged on December 19, reverted changes from the mining company, and manually reinstated some of the same text that the Tor-based editor had added. The activist's changes were again removed by an IP-based contributor, and no further materials about controversies were added to the article over the next year.
This controversy seems to have played out quietly. We were not able to find any arbitration records or Talk page discussion related to the dispute. We do not know why it ended. We do know that by early 2014, Wikipedia's effort to block edits from Tor-based contributors became substantially more successful and the number of edits from Tor-based contributors to the encyclopedia dropped to near zero\citep{tran_tor_2019}.
\end{quotation}
\noindent All sides of this conflict were involved in what Wikipedia might call ``POV pushing'' and ``POV fighting'' and what we call ``activism.'' Both sides violated Wikipedia's rules repeatedly. The initial IP-based contribution that set off the conflict violated Wikipedia norms: its tone was not encyclopedic and it did not include references for its controversial claims.
The edits by people who appear to be associated with the mining company violated many rules as well including Wikipedia's policy on conflict of interest,\footnote{\url{https://en.wikipedia.org/wiki/Wikipedia:Conflict\_of\_interest} \textit{archived at} \url{https://perma.cc/5J92-NX4U}} Wikipedia's policy against accounts associated with organizations,\footnote{\url{https://en.wikipedia.org/wiki/Wikipedia:Username\_policy\#Shared\_accounts} \textit{archived at} \url{https://perma.cc/L4BM-FH9Q}} and rules against paid editing of Wikipedia.\footnote{\url{https://en.wikipedia.org/wiki/Wikipedia:Paid-contribution\_disclosure} \textit{archived at} \url{https://perma.cc/Q8LV-SVTM}}
In this narrative, we infer that the same person was responsible for all the activist-oriented changes critical of the company. It is also possible that two people were working in close collaboration with one another, one using Tor, the other using an IP address. Given textual similarities in their edits and proximities in time, we think the former is more likely. The transition from an IP address to Tor, coupled with removing their own work, is evocative of someone negotiating their approach to a topic, a platform, and their own identifiability.
This example serves to suggest the kinds of discourse that can be protected or limited by lack of access to privacy enhancing technology. Making accusations and raising awareness of controversy regarding powerful entities like corporations can lead to loss of employment or worse. It may be that the editor who used Tor feared some form of retaliation and was reluctant to continue their activism without protection from a privacy service.
\subsection{Quality Maintenance}
\label{sec:thm-quality}
Another type of contribution frequently made by Tor users involved the application of English Wikipedia's policies and conventions designed to ensure that articles maintain a standard of quality. This includes both removing materials that violate Wikipedia's policies as well as engaging in collaborative efforts with other contributors.
As Wikipedia has grown, the work of maintaining quality has likewise increased, with a growing proportion of effort going to coordination and upkeep. This a trend was observed as early as 2007 \citep{kittur_he_2007}.
Edits in this theme were typically catalyzed by low quality edits made by others.
When a low quality edit is made to Wikipedia, other contributors can ``revert'' the contribution by undoing it completely, they can try to improve it on their own, and or they can invite collaboration. We found evidence of Tor editors engaging in all three practices. As with quotidian contributions, we saw no evidence that other Wikipedia editors were aware that these contributors were using Tor.
One example of quality maintenance done by Tor-based contributors is the removal of links that violate Wikipedia's external links policy (referred to with the shorthand ``WP:EL''). WP:EL governs what can be placed in the list of links at the end of each article and states that contributors should avoid including links to sites which are, for example, misleading, repetitive, promotional, blogs, social network pages, composed of search results, and, in English Wikipedia, sites which primarily contain non-English content.
We found evidence of Tor-based contributors removing external links that violated this policy that invoked the policy explicitly by including the WP:EL shorthand in their edit summary.
We found WP:EL invoked by Tor users in edit summaries implicitly and explicitly in a series of edits made to 12 different articles related to religious conspiracy theories and minority religious practices between January 31, 2008 and February 5, 2008. For each article related to the conspiracy theory, at least one of the edits removed a link to the same external website. These edit summaries included:
\begin{quotation}
On January 31, 2008, at 02:35 UTC, a Tor user edited a biography article about a person allegedly involved in a conspiracy theory with an edit summary that read, ``remove links to self-published personal website; author's qualifications not provided, site may not be reliable''
On January 31, 2008, at 02:38 UTC, a Tor user edited an article referring to a secret cabal with an edit summary reading, ``need a better reference than a spam link to a self-published personal website.''
\end{quotation}
\noindent We inspected the website to which links were removed and confirmed that it was a personal page. This is consistent with the statements made by the Tor-based contributor in their edit summaries. We also found examples of Tor-based edits to the same group of articles adding and fixing reference links and adding the ``\texttt{\{\{fact\}\}}'' tag to an article which causes the phrase ``\texttt{[[Citation Needed]]}'' to appear in a specific place in an article's text.
We interpret these actions as examples of Tor-based contributors engaged in deliberate efforts to improve the overall sourcing of a group of pages. We observed the use of authority claims from the ontology suggested in \citet{oxley_what_2010} consistent with maintaining dimensions of anonymity including the use of community social expectations (edit summaries), policies (quoting WP:EL), and external authorities (use of links to sources agreed to be reliable). Subsequently, one of the two Tor IP addresses used by this Tor-based contributor was banned indefinitely (according to Wikipedia policy) by a Wikipedia administrator with the three-word explanation: ``No open proxies.''
In other cases, we found Tor-based editors and registered users collaborating to make a stronger contribution than either was able to make alone:
\begin{quotation}
At 23:13 UTC, on June 20, 2013, a Tor user updated a telecommunications privacy policy article, adding ``However, this has been postponed'' in a subsection discussing the implementation of the policy in a Scandinavian country. The contributor did not include any references or sources for this information.
Less than an hour later, a registered contributor updated the article to say that the law ``was implemented in 2011...after being postponed'' and added an out-of-date reference. The registered contributor also included an edit summary stating, ``The last information I could find on this is from 2010 (three years ago). The article already cited is from 2011 (two years ago). Please find a better citation if you disagree. A [Scandinavian]-language one is okay.''
The Tor user made another edit 19 minutes later, stating in the same subsection of the article, ``But this will not be in effect before 1. jan. 2015.'' and adding a more recent, non-English reference.
23 minutes later, the same registered contributor responded, incorporating what the Tor contributor had added, with an edit summary of: ``Cheers. grammar, date formatting, making sure that language is noted in citation.'' Their edit corrected the grammar of the Tor-based contribution and expanded the metadata for the reference that the Tor-based contributor had provided.
\end{quotation}
\noindent In this example, both parties made attempts to signal their collaborative intent.
The registered contributor used edit summaries to offer explanations for their actions and to give suggestions to their collaborator.
Although the Tor-based contributor did not use edit summaries, the text of their edits revealed that they were responsive to feedback.
Tor-based contributors' familiarity with norms like adding the \texttt{\{\{fact\}\}} tag suggests familiarity with Wikipedia rules and procedures. An experienced editor doing quality maintenance work might choose to protect their privacy for many reasons. For example, they may do so specifically to avoid harassment or stalking. They might also simply be someone who seeks location privacy routinely. Research has documented that those who uphold community policies can face harassment and threats of rape and violence \cite{forte_privacy_2017}.
The editor contributing to a series of articles about a conspiracy theory in our example may have topic-specific reasons to seek to conceal their location and to contribute without an account. In the telecommunications privacy policy article, it might simply reflect the fact that users of privacy technologies like Tor may have expertise in, and a desire to tell others about, privacy-related topics.
These examples suggest that the privacy-seeking community may have value to offer a peer production community by shouldering policy enforcement in circumstances where harassment is a concern, by drawing attention to dubious claims, and through nuanced topical contributions.
An irony of policy-enforcing contributions from Tor-based editors is the fact that these contributors are themselves policy violators simply by using Tor. According to Wikipedia policy, contributing despite being banned is itself considered grounds for having one's contributions reverted, regardless of the merit or intent of one's contributions.
Although the Tor editors in our sample may not realize this until they find themselves blocked and their contributions removed, the evidence that at least some of them are familiar with Wikipedia conventions suggests that they may be aware of their contingent access.
\subsection{Edit Wars}
\label{sec:thm-editwars}
A number of the sessions in our sample that contained multiple edits were examples of what Wikipedians call ``edit wars.'' Edit warring is described by Wikipedia policy as unconstructive back-and-forth editing where two or more editors repeatedly undo each others' contributions. Edit wars are strongly discouraged by Wikipedia policy.\footnote{\url{https://en.wikipedia.org/wiki/Wikipedia:Edit\_warring} \textit{archived at} \url{https://perma.cc/CW3A-45EG}}
Edit wars may result in all editing to an article being limited or locked. In some cases, participants in an edit war are enjoined to discuss the dispute first and arrive at some agreement on the article's Talk page before they begin editing again. Some edit wars in which Tor-based contributors participate are resolved in this fashion while others remain contentious.
We imagine that participants in edit wars would re-tell their participation in a different manner than an outside observer. Although they might come across as being frustrated, unreasonable, or angry, they may describe themselves as brave, righteous, or trying to uphold fairness. Despite the seeming futility of two people repeatedly undoing each others' work, the circumstance of an edit war are that the first person to step out of the war in essence loses in that the article's text will reflect the update made in the final round. We observed edit wars regarding whether an individual should be listed as an economist, the relative rankings of sports teams in an international contest, the classification of a widely-illegal behavior as a crime, the expansion of an article about a field in physics, and what kind of information should be highlighted in the infobox about characters in a television show. Edit warring often co-occurs with other themes, including activism and harassment, but is characterized by the presence of at least two factions repeatedly undoing each other's work.
One example of an edit war in our dataset is a dispute that reflects an offline conflict spilling into Wikipedia. This edit war broke out over a set of articles for several rival South Asian schools. The conflict centered around a registered contributor we will call Cassidy, who engaged in edit warring with one or several people using Tor.
\begin{quotation}
We reviewed dozens of edits related to this edit war which first appear in our sample at October 27, 2010 at 02:58 UTC when a Tor user expressed frustration about the contribution patterns of a registered editor. The Tor-based editor placed their complaint on a WikiProject\footnote{Wiki projects are groups of contributors who coordinate their efforts through dedicated group pages.} page corresponding to a South Asian country.
We observed that back-and-forth editing between one or more Tor-based users and Cassidy continued until at least May 5, 2013. The conflict often involved naming of schools, whether the word ``Royal'' should appear in the school name, whether and how to translate school names from a South Asian language to English, and the correct order of words when trying to disambiguate the schools from one another in name. Conflict also arises over which page will host the definition of a sports match between two schools.
We learn, through Tor-based contributors posting news article links, that this tension is part of some larger conflict that at one point includes violence in the streets between students from two of the schools that are the subject of the edit war.
Cassidy suffers significant abuse throughout this edit war. On July 3, 2012 at 01:37 UTC, he is accused of being a ``[unemployed] sick wiki f*ker'' in a comment made by a Tor-based contributor on Cassidy's user page.
\end{quotation}
\noindent This multi-article edit war largely comprised a Tor-based editor or editors warring with a single registered editor. The conflict was frequently adjudicated by other registered editors who did not always conclude that Cassidy was acting in good faith or contributing with a neutral point of view. The subject of dispute was difficult for other community members, and for us, to fully comprehend.
We were able to gather evidence of numerous policy violations on both sides. Although attacks and harassment of Cassidy was clearly an unacceptable norm violation, Cassidy's edits consistently favor a single contested point of view about naming conventions in defiance of guidance from multiple uninvolved contributors.
The fact that Cassidy's attackers had recourse to Tor to conceal their identity may have allowed them to do more damage with their harassment than if they had been identifiable. On the other hand, it may be that the underlying complaint about naming conventions might not have caught the attention of uninvolved Wikipedia contributors without the objections raised by Tor users.
As a registered editor, Cassidy was able to receive direct replies from administrators regarding conflicts and was able to consistently defend his point of view. Tor editors without consistent IP addresses would have likely found it difficult to locate conversations in order to participate in conflict resolution processes.
\subsection{Non-Article Discussion}
\label{sec:thm-nonarticle}
Another theme described a group of sessions that focused on non-article discussions in Wikipedia. In sessions reflecting this theme, Tor-based contributors participated in activities that range from social chatter, to requests for help with technical problems posed at the Wikipedia Reference Desk, to discussion of policies and misconduct. An example of a narrative that falls into this category is:
\begin{quotation}
On November 13, 2007 at 07:57 UTC, a Tor user posted an accusation to an administrative board, stating that two Wikipedia administrators were engaged in a form of sockpuppeting called ``strawpuppeting.''
The two administrators accused of involvement in strawpuppeting were engaged in a discussion as to whether or not their work to advise marketers on contributing to Wikipedia might represent a conflict of interest.
The Tor-based contributor includes several links to provide circumstantial support for their claim.
\end{quotation}
\noindent The term ``strawpuppet'' is a portmanteau of the ``strawman'' bad faith form of argumentation and the term ``sockpuppeting''. ``Sockpuppeting'' is the practice of creating multiple accounts to be used in bad faith. Sockpuppets might be used to post messages in agreement with oneself or to create the appearance of strong support for a proposal. A sockpuppet might also be used to evade rules that restrict individual contributions, such as the rule that limits the number of times any individual can revert changes to an article in a given day. A ``strawpuppet'' is a sockpuppet which is used not to post support of one's own arguments, but rather to post a strawman attack on an argument which can then be dismantled by the puppeteer. The editor accused of being a strawpuppet was indeed making sweeping claims which disagreed with the two administrators about whether their conduct was appropriate.
While the narrative we developed appears to be a case of a community member calling for accountability, we do not have a view into what follow-up if any occurred. Although this example may be a case of a Tor-based contributor unfairly attacking Wikipedia administrators, legitimate accusations of administrative misconduct may carry risk for identifiable participants. Anonymous individuals may feel more able to criticize community leaders in these ways.
\subsection{Protests Against Mistrust}
\label{sec:thm-mistrust}
One of our examples in §\ref{sec:thm-badfaith} involved a Tor-based contributor responding to being blocked with vandalism and harassment. We also observed instances in which Tor-based contributors protested their treatment by the Wikipedia community while remaining civil. In a series of edit sessions, we saw Tor-based contributors grappling with the mistrust that their contributing without an account generated as well as protesting the policies that prohibit them from contributing. In some cases these protests seem naive and suggest a lack of awareness. In other cases, Tor users appear to be very informed about Wikipedia's policies and processes.
The following provides an example of a narrative reflecting this theme:
\begin{quotation}
On June 26, 2011, at 20:28 UTC, a Tor user posted a question in a discussion about whether or not it is forbidden to use Wikipedia's Reference Desk to ask about administrative rulings about individuals. The Tor-based contributor asking this question expanded the inquiry about whether or not this was forbidden in all cases by asking, ``what if someone wanted to ask a question about a notable editor, like for example [name].''[Here, the Tor user invoked the name of a registered editor.]
A registered user responded, ``If you can't ask the question except as a brand-new IP, I suggest the answer is yes.'' [That is, yes, it would be forbidden.]
In the discussion that followed, the Tor-based contributor defended the fact that they asked their question from an IP without an edit history, denied the charge that they were pretending to be a newbie, and described their use of Wikipedia conventions as evidence that they were not pretending to be new.
Other registered editors respond to the Tor-based contributor's question, while yet another asks the Tor-based contributor to give their IP or account name. A registered contributor defends the Tor-based contributor, asking why this information is ``any of [their] business?''
\end{quotation}
\noindent This exchange provides insight into the difficulties faced by contributors participating without an account when they interact with others. The absence of a stable identifying account name, the lack of an edit history to which their respondents can refer, and limited (or non-existent) external cues or relationships to corroborate an initial impression all make productive engagement less likely.
We also observe occasions when Tor-based contributors object to being blocked or banned and try to negotiate access. Two examples include:
\begin{quotation}
On August 21, 2010 at 23:47 UTC, a Tor user wrote the following using a template that placed their message on a noticeboard for administrators, ``This isn't fair. I am trying to post from the library and I am being blocked.''
On January 28, 2012, at 10:14 UTC, a Tor user posted an unblock request using a template
with the unblock reason: ``Please unblock this address I would like to create an account.''
\end{quotation}
\noindent Other protests to being banned express confusion:
\begin{quotation}
On January 20, 2008, at 08:07 UTC, a Tor user writes in their unblock request on their User Talk page, in Chinese, ``What did I do wrong?''[author's translation] The blocked IP in this instance had been blocked since November 12, 2007.
\end{quotation}
\noindent We found no evidence that anybody on English Wikipedia understood the user's request. Other protests make use of Wikipedia-specific understanding:
\begin{quotation}
On August 30, 2007 at 01:31 UTC, a Tor user writes in their unblock request on their User Talk page, ``tor should be soft blocked.''\footnote{A ``soft block'' of an IP address in Wikipedia is one in which Wikipedia bans account creation and blocks any edits from a given IP address by individuals who are not logged in, but which does not block editing from that IP for individuals with an account. By contrast, a ``hard block'' does not allow individuals with accounts to contribute from the IP; Tor is hardblocked.}
15 minutes later, a Wikipedia administrator responds to this request stating, ``There has been no community consensus that the enforcement of \texttt{Wikipedia:No open proxies} is to be overturned yet. While I do sympathize with the situation in mainland China and the Middle East regarding Internet censorship, I have dealt with too many sockpuppets who have abused open proxies in the past and until a technical solution is found, I am not willing to expose this website and its community to such an unacceptable security risk.''
The Tor-based contributor twice posted a response to this denial, including links to comments from a founder of Wikipedia to support their position. These responses were reverted.
\end{quotation}
\noindent In this exchange, we see a lack of trust of the Tor-based editor and a closing-off of further discussion. The administrator's reference to ``sockpuppets'' was a common feature we observed in responses to Tor editors.
\subsection{Theme Prevalence}
\label{sec:quant}
\begin{table}[t]
\centering
\caption{Prevalence of themes in Tor edit sessions, observed in a sample of 500 sessions. Because multiple codes can be applied to a single observation, percentages do not sum to 100.
}
\begin{tabular}{lcc}
Theme & Number of Sessions & Prevalence\\
\hline
Quotidian & 184 & 37\% \\
Bad faith & 152 & 31\% \\
Activism & 56 & 11\%\\
Quality maintenance & 50 & 10\%\\
Edit wars & 39 & 8\%\\
Non-article discussion & 20 & 4\%\\
Protesting mistrust & 12 & 2\% \\
\end{tabular}
\label{tab:quantThemes}
\end{table}
The goal of this study is to reconstruct activity through discussion among multiple coders and to interpret the meaning of these activities. It is not to report their frequency with respect to an established standard. As a result, no measure of inter-rater reliability is calculated. We report on the prevalence of themes in Table \ref{tab:quantThemes} to provide requested context for readers but urge readers to interpret these numbers carefully and with some skepticism. Readers should keep in mind that the ordering of theme prevalence varied somewhat between coders, and session assignment to coders was not fully randomized; although initially assignment was random, when narratives spanned multiple sessions and articles (e.g. the South Asian school edit war), a single coder took the lead in assessing those sessions.
Although each session was coded in terms of at least one of our themes, sometimes multiple codes applied equally well so we applied multiple codes. The application of multiple codes to a single session means the prevalence of all themes do not sum to 100\%.
Table \ref{tab:quantThemes} shows that the most common theme we observed was quotidian editing which we observed in 184 of the 497 sessions we inspected (37\%). The second most common theme we observed was bad faith, which we observed in 152 sessions (31\%). The third most common theme we observed was activism, in 56 sessions (11\%).
The remaining themes, in rank order, were: quality maintenance (50 sessions, 10\%), edit wars (39 sessions, 8\%), non-article discussion (20 sessions, 4\%), and protesting mistrust (12 sessions, 2\%). Three sessions in our sample set had been deleted by Wikipedia administrators and removed from the public logs such that they could not be characterized.
\section{Discussion}
The themes that emerged from our forensic qualitative analysis suggest that Tor-based contributors make everyday contributions, they work to uphold quality, they collaborate and argue, they violate norms and misbehave, they grapple with questions of trust and credibility online, and they are drawn into edit wars. In all these ways, Tor editors are like other contributors to Wikipedia. We also found examples where Tor-based contributors introduce specific perspectives that are marginalized or at risk in ways that are consistent with self-protecting anonymity seekers.
To the extent that policies which block privacy tools discourage contributions from people who might make unique contributions, these policies cause a loss of value to the community.
\subsection{Risks to Anonymity Seekers and the Value to Communities}
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{bubble_plot-mako_20190812.pdf}
\caption{An exploratory mapping of our themes in terms of the value a type of contribution represents to the Wikipedia community and the importance of anonymity in facilitating it. Anonymity protecting tools play a critical role in facilitating contributions on the right side of the figure while edits on the left are more likely to occur even when anonymity is impossible. Contributions toward the top reflect valuable forms of participation in Wikipedia while edits on the bottom reflect damage.
}
\label{fig:bubbleChart}
\end{figure}
We began this paper by discussing the perceived threats that compel would-be contributors to seek anonymity. Throughout our paper, we have shown how Tor users' participation can reflect both value and damage to the Wikipedia community.
In Figure \ref{fig:bubbleChart}, we explore how the need for anonymity varies in relation to the value brought by anonymity seekers to members in the communities that must decide whether and how to allow anonymous participation. The x-axis in Figure \ref{fig:bubbleChart} attempts to represent each theme in terms of how important a role anonymity plays is facilitating those types of contributions. The y-axis attempts to reflect the value brought to the community if a type of contribution is allowed to occur. The specific locations of the themes are meant to be evocative and we acknowledge that there are valid arguments for many other arrangements.
Although reflecting only an exploratory synthesis, we are confident that edits in our sample occupy all four quadrants.
Contributions in the top left benefit the community and seem unlikely to require anonymity---though newbies might benefit from such a cloak for trial and error. For example, many of the quotidian edits we observed reflect unambiguous contributions of value to the Wikipedia community and occur frequently among fully identified contributors.
It seems possible that some of these contributions would occur in a context in which Tor was completely blocked. Because they are not placed at risk by these types of contributions, some of the anonymity seeking contributors we observed making mundane edits might have just edited using a pseudonymous account.
Banning contributions from anonymity seekers seems even less problematic for contributions that fall into the bottom right quadrant: they are damaging and seem likely to be at least partially deterred by barriers to anonymous participation. For example, edit wars cause frustration and wasted effort all around. Contributions in this quadrant would likely be welcome casualties of a policy to block anonymous participation.
Bad faith contributions span the bottom two quadrants. They detract value from the community in that they demand attention and resources to counteract while varying in the risk they pose to an identified bad faith actor. To the extent that using racial slurs or making specific threats online could threaten employment status or result in a police investigation, a strong requirement of identification might reduce some forms of harassment. On the other hand, many forms of bad faith acts, such as the ``BAET DANCER'' instance of goofy vandalism, would pose little risk for the vandal if connected to their identity. Blocking Tor might reduce actions like harassment more effectively than vandalism.
Engaging in non-article discussion is also a complex case. Social discussion and participation in shared governance activities is good for the community in so far as it is pursued in a genuine and transparent manner, but can be dangerous for identifiable individuals.
However, increased identifiability also places limits on harmful forms of participation, especially manipulations of perceptions of identity such as sockpuppetry and strawpuppetry.
The most important quadrant of the figure is the top right corner which includes contributions that add value to Wikipedia but will not occur if contributors cannot remain anonymous.
For example, many instances of activism we observed fall clearly into this quadrant. Our example of a user documenting the history of notable human rights abuses from a powerful mining company furthers Wikipedia's mission, but places any identified contributor doing so at risk.
Quality maintenance work is likewise supportive of community values and goals but may in some circumstances be triggers for harassment or threats from the individual who authored the removed material. Previous research has highlighted avoiding harassment as a motivation for using Tor ~\citep{forte_privacy_2017}.
These types of contributions will be less common if barriers to anonymity are kept high.
Although we offer Figure \ref{fig:bubbleChart} as one conceptual tool to help think through our findings, we put less stock in the specific locations that our themes occupy on the chart.
When assessing threats and value, we rely on our interpretation, which does not reflect the wide range of risk contexts that may be hard for us to discern from our vantage point as privileged US-based researchers. Some of the world's largest linguistic communities live under various forms of autocracy where edits that seem innocuous or mundane in a democracy with relatively little censorship may present much higher risk. For example, we wondered whether mundane edits we found documenting transportation routes in South Asian countries that are known to censor and surveil the Internet were acts of sedition. We recalled that editing articles about American films or video games might be punishable in oppressive religious regimes. Individuals living in a range of political environments might find themselves under threat from other individuals from whom the law cannot or will not defend them. We examine instead those cases where there is some variation in potential for risk based on the contribution itself.
\subsection{Unique Contributions}
We conducted this analysis with the expectation that we would see some evidence of unique contributions from anonymity seeking contributors, but we did not know what form they might take. We found that even quotidian edits have some potential to be unique to the extent that they represent niche topics. Given that English Wikipedia now contains well over five million pages, there are many specialist topics from which to choose.
We found unique contributions not only in specialist topics, but also in how privacy seekers may engage with epistemological issues and norm violations.
We see the strongest potential for unique contributions from anonymity seeking contributors in our activism theme. Activists may place themselves at risk to raise awareness of important issues and may seek to keep their activist work separate from their public persona as a result. Likewise, any community relies on robust debate to maintain accountability and we saw evidence of how anonymity can support productive debate. Journalists cite unnamed sources, newspapers publish anonymous editorials, and ballots are often private. Privacy tools like Tor allow individuals to create privacy for themselves when and where they decide they need and want it. We found that despite rich digital trace data these circumstances are difficult for others---including ourselves---to gauge.
The decision to block Tor leaves out contributors who require more stringent anonymity including those who contribute activism, quality control, and quotidian edits from locations, or with identities, that put them at elevated risk. We found evidence that these contributions add value that, in some cases, may not be possible to elicit otherwise in that the reflect the perspective of a person at risk. Even edit wars, like those we uncovered in South Asia, may contribute value because they expose an ethnic and (anti-) colonial divide that may not reach western Wikipedia editors otherwise.
\section{Limitations and Future Work}
\label{sec:limitations}
Our work is limited in several important ways.
For example, the way we selected our field site generates certain limitations \citep{blomberg_reflections_2013}. For one, conducting research on individuals who have taken explicit steps to conceal their identity introduces challenges and the potential for error. While we can determine that a given IP address was acting as a Tor node at a given time, we cannot be certain if the edits coming from that node were made using the Tor network or if the Tor node operator was also using the same computer to browse the web directly. We likewise cannot determine definitively whether multiple edits from the same IP are the same person or different Tor users doing similar types of work. We hope that this is mitigated by the fact that we have conducted a careful manual analysis of each edit. We believe that our narrative description would not be substantially altered if a group rather than an individual were responsible for a tightly-spaced series of similar edits.
Our work is also limited in its ability to reflect on what might happen if Wikipedia were to unblock Tor in that the population from which our sample is drawn is unlikely to be representative of people who might want to edit Wikipedia from Tor. For example, regular Tor users who attempted to edit Wikipedia once and received a message explaining that Tor was blocked may never have tried again. In this way, new Tor users might be overrepresented in our sample. In that switching exit nodes repeatedly or requesting specific nodes could make it more likely for Tor users to work around Wikipedia's block, our sample might disproportionately reflect the efforts of savvy Tor users who know how to customize Tor's behavior to take advantage of flaws in Wikipedia's blocking approach.
If these groups have systematically different interests or goals than the types of people who might edit Wikipedia using Tor in the absence of the block, our analysis might provide limited insight into the question of what might happen in this setting.
Our work is limited in that we have not corroborated our narratives with the community we were observing. We share our interpretation as a laboriously constructed version of what anyone might do when faced with an anonymous message by assembling clues to explain what has occurred. We believe that our narrative-building approach is a practical way to shed light on the behaviors of a hard-to-observe population.
Additionally, it is limited by our decision to sample randomly and then to recontextualize edits as best we could. This means that some elements of long-running narratives, like the South Asian schools edit war, might not be fully captured in our account. As is always the case in qualitative and interpretive research, our account is necessarily partial and incomplete. There are many fascinating narratives that that have unavoidably been omitted.
\section{Conclusion}
Building on a number of existing methodologically approaches,
we propose a new qualitative methodology---\textit{forensic qualitative analysis}---that extends existing methods to provide us with an understanding of the behavior and intent of people who cannot or will not participate in more direct research techniques.
Using this technique, we contribute to our understanding of a very difficult-to-observe population. We construct our field site around the contributions of individuals protecting their privacy through Tor and the responses of the community to these contributions. Building on prior work that examined the degree to which Tor-based contributors act in good faith and make contributions that are non-damaging, we use our new methodology to establish a series of themes.
We found that Tor-based contributors make quotidian, good-faith, non-damaging contributions to building the encyclopedia. They use policies to uphold quality and use platform affordances to collaborate with others with varying degrees of success. We also found that they violate policies in ways that damage the quality of the resource: they vandalize, harass, and participate in edit wars.
Although anonymity seemed incidental or important in many of the types of contributions we examined, it appeared to play a critical in making the contributions possible in others. We saw examples of activists challenging power structures who may have had good reasons for seeking to protect their privacy. Resources which seek to reflect all the world's knowledge may have good reason to embrace those who would champion perspectives outside the status quo that might not be heard in the absence of strong anonymity protection.
With the deeper look afforded by forensic qualitative analysis, our results suggest that anonymity seekers may add additional value to peer production projects through their work on controversial topics as well as through their ability to challenge prevailing power structures. Our work also suggests that the risks to communities of allowing anonymous contributions may vary enormously. Indeed, different contributors and communities may value particular contributions differently when weighing them against these risks.
We saw many examples of both productive and unproductive engagement through Tor and believe, from the context that were able to gather, that at least some of both types of edits would never had occurred if Tor were blocked completely.
\citet{myagmar_threat_2005} suggest four alternatives for managing risk: accept, transfer, remove, and mitigate. Wikipedia initially accepted participation from Tor before following a strategy of removal with progressively effective techniques. New technology such as automated filtering systems and damage detection capabilities may allow communities to pursue a mitigation strategy in the future. We believe that our findings provide evidence in support of mitigation-based approaches that attempt to maximize value while minimizing damage. We also believe our work can provide some insight into the nuanced ways that value may flow into peer production communities from anonymity seeking users that we hope will inform these approaches in the future.
\begin{acks}
This work was supported by the National Science Foundation (awards CNS-1703736 and CNS-1703049) and included the work of two undergraduates supported through an NSF REU supplement. Feedback and support for this work came from members of the Community Data Science Collective, participants in the Critical and Creative Thinking Studio and in the Science in a Changing World Workshop (both at University of Massachusetts Boston), and from the University of Washington Department of Communication. The manuscript benefited from excellent feedback from several anonymous referees and associate chairs at CSCW.
The project was only possible because Chau Tran generously shared the dataset of Tor edits he had constructed over more than a year of work. The creation of that dataset was facilitated though the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system at the University of Washington.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,484 |
(11216) Billhubbard (1999 JG1) – planetoida z pasa głównego asteroid okrążająca Słońce w ciągu 3,39 lat w średniej odległości 2,26 j.a. Została odkryta 8 maja 1999 roku w ramach projektu Catalina Sky Survey.
Zobacz też
lista planetoid 11001–12000
lista planetoid
Linki zewnętrzne
Planetoidy pasa głównego
Nazwane planetoidy
Obiekty astronomiczne odkryte w 1999 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,071 |
by Chip Knappenberger
from MasterResource Website
Editor Note:
Using mainstream models and assumptions, Mr. Knappenberger finds that in the year 2050 with a 83% emissions reduction (the aspirational goal of Waxman-Markey, the beginning steps of which are under vigorous debate), the temperature reduction is nine hundredths of one degree Fahrenheit, or two years of avoided warming by 2050.
A more realistic climate bill would be a fraction of this amount.
The author will respond to technical questions on methodology and results and invites input on alternative scenarios and analyses.
The IPCC-Based Arithmetic of No Gain
The economics and the regulatory burdens of climate change bills are forever being analyzed, but the bills' primary function - mitigating future climate change - is generally ignored.
Perhaps that's because it is simply assumed.
After all, we are barraged daily with the horrors of what the climate will become if we don't stop emitting greenhouse gases into the atmosphere (the primary focus being on emissions from the combustion of fossil fuels).
So doing something as drastic as that proposed by Waxman-Markey - a more than 80% reduction of greenhouse gas emissions from the United States by the year 2050 - must surely lessen the chances of climate catastrophe. Mustn't it?
But if that were the case,
Why aren't the climate impacts being touted?
Why aren't Representatives Waxman and Markey waving around the projected climate success of their bill?
Why aren't they saying:
"Economics and regulations be damned. Look how our bill is going to save the earth from human-caused climate apocalypse"?
That reason is that it won't. And they know it...
That is why they, and everyone else who supports such measures, are mum about the outcome.
The one thing, above all others, that they don't want you to know is this:
No matter how the economic and regulatory issues shake out, the bill will have virtually no impact on the future course of the earth's climate.
And this is even in its current "pure" form, without the inevitable watering down to come.
So discussion of the bill, instead of focusing on climate impacts, is shrouded in economics and climate alarm.
Getting a good handle on the future climate impact of the proposed Waxman-Markey legislation is not that difficult. In fact, there are several ways to get at it.
But perhaps the most versatile is the aptly named MAGICC: Model for the Assessment of Greenhouse-gas Induced Climate Change. MAGICC is sort of a climate model simulator that you can run from your desktop (available here).
It was developed by scientists at the National Center for Atmospheric Research (primarily by Dr. Tom Wigley) under funding by the U.S. Environmental Protection Agency (EPA) and other organizations.
MAGICC is itself a collection of simple gas-cycle, climate, and ice-melt models that is designed to produce an output that emulates the output one gets from much more complex climate models.
MAGICC can produce in seconds, on your own computer, results that complex climate models take weeks to produce running on the world's fastest supercomputers. Of course, MAGICC doesn't provide the same level of detail, but it does produce projections for the things that we most often hear about and care about - for instance, the global average temperature change.
Moreover, MAGICC was developed to be used for exactly the purpose that we use it here - the purpose for which Representatives Waxman and Markey and everybody else who wants a say in this issue should be using it.
That purpose is, according to MAGICC's website,
"to compare the global-mean temperature and sea level implications of two different emissions scenarios",
...for example, scenarios both with and without the proposed legislative emissions reductions.
So that is what we'll do. We'll first use MAGICC to produce a projection of global average temperature change through the 21st century under two of the Intergovernmental Panel on Climate Change's future emissions scenarios (which assume no explicit policy implementation).
The two are:
a mid-range emissions scenario (SRES A1B for those interested in the details)
a high-end emissions scenario (SRES A1FI)
Then, we'll modify these IPCC scenarios by entering in the emissions reductions that will occur if the provisions outlined in the Waxman-Markey Climate Change Bill are fully met (leaving aside whether or not that could be done).
Basically, Waxman-Markey calls for U.S. emissions to be reduced to 20% below the 2005 emissions level by 2020, 42% below 2005 levels by 2030, and 83% below 2005 levels by 2050. We'll assume that U.S. emissions remain constant at that reduced value for the rest of the century.
We'll then use MAGICC to produce temperature projections using these modified scenarios and compare them with the original projections.*
And here is what we get all rolled into one simple figure.
The solid lines are the projections of the change in global average temperature across the 21st century from the original IPCC A1FI (red) and A1B (blue) high and mid-range emissions scenarios, respectively (assuming a climate sensitivity of 3ºC).
The dotted lines (of the same color) indicate the projected change in global average surface temperature when the emissions reductions prescribed by Waxman-Markey are factored in.
By the year 2050, the Waxman-Markey Climate Bill would result in a global temperature "savings" of about 0.05ºC regardless of the IPCC scenario used - this is equivalent to about 2 years' worth of warming. By the year 2100, the emissions pathways become clearly distinguishable, and so to do the impacts of Waxman-Markey.
Assuming the IPCC mid-range scenario (A1B) Waxman-Markey would result in a projected temperature rise of 2.847ºC, instead of 2.959ºC rise - a mere 0.112ºC temperature "savings."
Under the IPCC's high-emissions scenario, instead of a projected rise of 4.414ºC, Waxman-Markey limits the rise to 4.219ºC - a "savings" of 0.195ºC. In either case, this works out to about 5 years' worth of warming.
In other words, a full implementation and adherence to the emissions restrictions provisions described by the Waxman-Markey Climate Bill would result only in setting back the projected rise in global temperatures by a few years - a scientifically meaningless prospect.
(Note: I present the results to three significant digits, not that they are that precise when it comes to the real world, but just so that you can tell the results apart).
Now, various aspects of the MAGICC model parameters can be tweaked, different climate models can be emulated, and different scenarios can by chosen. And different answers will be obtained.
That is the whole purpose of MAGICC - to be able to examine the sensitivity of the output to these types of changes.
But if you take the time to download MAGICC yourself and run your own experiments, one thing that you will soon find out is: No matter what you try, altering only U.S. emissions will produce unsatisfying results if you seek to save the world by altering its climate.
We have calculated only the climate impact of the United States acting alone. There is no successor treaty to the Kyoto Protocol to bind other countries to greenhouse gas emissions reductions. But, truth be told, the only countries of any real concern are China and India.
The total increase in China's emissions since the year 2000 is 50 percent greater than the total increase from rest of the world combined and is growing by leaps and bounds. And consider that India carbon dioxide emissions haven't started to dramatically increase yet.
But it is poised to do so, and an Indian official recently stated that,
"It is morally wrong for us to agree to reduce [carbon dioxide emissions] when 40 percent of Indians do not have access to electricity."
Without a large reduction in the carbon dioxide emissions from both China and India - not just a commitment but an actual reduction - there will be nothing climatologically gained from any restrictions on U.S. emissions, regardless whether they come about from the Waxman-Markey bill (or other cap-and-trade proposals), from a direct carbon tax, or through some EPA regulations.
This is something that should be common knowledge. But it is kept carefully guarded.
The bottom line is that a reduction of U.S. greenhouse gas emissions of greater than 80%, as envisioned in the Waxman-Markey climate bill will only produce a global temperature "savings" during the next 50 years of about 0.05ºC.
Calculating this isn't all that difficult or costly. All it takes is a little MAGICC.
There are many parameters that can be altered when running MAGICC, including the climate sensitivity (how much warming the model produces from a doubling of CO2 concentration) and the size of the effect produced by aerosols. In all cases, we've chosen to use the MAGICC default settings, which represent the middle-of-the-road estimates for these parameter values.
Also, we've had to make some assumptions about the U.S. emissions pathways as prescribed by the original IPCC scenarios in order to obtain the baseline U.S. emissions (unique to each scenario) to which we could apply the Waxman-Markey emissions reduction schedule.
The most common IPCC definition of its scenarios describes the future emissions, not from individual countries, but from country groupings.
Therefore, we needed to back out the U.S. emissions. To do so, we identified which country group the U.S. belonged to (the OECD90 group) and then determined the current percentage of the total group emissions that are being contributed by the United States - which turned out to by ~50%. We then assumed that this percentage was constant over time.
In other words, that the U.S. contributed 50% of the OECD90 emissions in 2000 as well as in every year between 2000 and 2100. Thus, we were able to develop the future emissions pathway of the U.S. from the group pathway defined by the IPCC for each scenario (in this case, the A1B and the A1FI scenarios).
The Waxman-Markey reductions were then applied to the projected U.S. emissions pathways, and the new U.S. emissions were then recombined into the OECD90 pathway and into the global emissions total over time.
It is the total global emissions that are entered into MAGICC in order to produce global temperature projections - both the original emissions, as well as the emissions modified to account for the U.S. emissions under Waxman-Markey.
Global Sign-Up
Yesterday's MasterResource post (Part I above) looked at the potential climate impacts of the proposed Waxman-Markey Climate Bill. But I limited my analysis to only U.S. actions - after all, Waxman-Markey can't mandate international man-made greenhouse gas reduction timetables.
But, what would happen if the rest of the world wanted to join in?
The ability of the industrialized world, through emissions reductions alone, to impact the future course of global climate is minimal.
the former Soviet countries,
...all limited their emissions of greenhouse gases according to the schedule laid out under Waxman-Markey - a monumental, unexpected development - it would, at most, avoid only a bit more than one-half of a °C of projected global warming (out of 4.5°C - or only about 10%).
And this is under worst-case emissions assumptions; middle-of-the-road scenarios and less sensitive climate models produce even less overall impact.
To make any significant in-roads to lowering the rate (and thus final magnitude) of projected global temperature rise, the bulk of the emissions reduction needs to come from other parts of the world, primarily,
The problem is, is that these governments are not inclined to restrict the energy usage of its citizens - in fact, they either are in the process of, or are soon hoping to, significantly expand the amount of energy available to their (growing) populations - and in the process, subsuming all potential emissions savings from the (current) industrialized world.
If supporters of large greenhouse gas emissions restrictions were really interested in "saving the world," they would be putting all of their effort into getting China and India to buy into their plan - and then turning to the U.S. up in mop up duty.
As it stands now, they are talking to the wrong end of the horse.
Over the first decade of the 21st century, global carbon dioxide emissions have been growing a pretty good clip - in fact, they've been growing at a rate which exceeds the projected rate from the most extreme scenario envisioned by the Intergovernmental Panel on Climate Change (IPCC).
It is also the scenario which, when fed into the world's climate models, produces the greatest warming by the end of the century - about 4.5ºC (although the world abounds with observations that suggests that this temperature rise is overblown, but that is the subject of a different analysis).
The question I want to explore here, is,
"if we wanted to do something to ameliorate this projected temperature rise, what could we do?"
And more specifically, who are "we"?
The proposed Waxman-Markey Climate Bill is aimed to reduce the projected rise in global temperature. This bill calls for a reduction in greenhouse gases from the United States according to the following schedule - a 20% reduction (below the 2005 emissions level) by the year 2020, a 58% reduction by 2030 and a 83% reduction by 2050.
So, let's take "we" to be Americans bound by the emissions reduction schedule laid out under Waxman-Markey and see what effect that "we" would have on the projected global temperature increase if "we" followed the Waxman-Markey plan.
Then, we'll look at what would happen if "we" were able to get other parts of the world to go along with the plan.
The extreme IPCC scenario is the A1FI scenario and is described as a fossil-fuels intensive scenario of a,
"future world of very rapid economic growth, global population that peaks in mid-century and declines thereafter, and the rapid introduction of new and more efficient technologies" and that the "[m]ajor underlying themes are convergence among regions, capacity building and increased cultural and social interactions, with a substantial reduction in regional differences in per capita income."
What this all means in terms of the IPCC's vision of future CO2 emissions is shown in Figure 1.
Projected carbon dioxide emissions
from four country groupings as defined by the IPCC's A1FI scenario.
For a description of the country groupings, see the text.
(source: IPCC SRES)
The IPCC breaks the world down into four general classifications:
OECD90 (industrialized countries including the U.S., Western Europe, Australia and Japan)
REF (countries undergoing economic reform including Eastern Europe, former Soviet Union and Sub-Saharan Africa)
ALM (North Africa, Latin America and the Middle East)
ASIA (Asian countries including China and India)
As can be seen in Figure 1, the emissions from each of the groups increase, with most of the increase in the first half of the century coming from the ASIA.
In the last few decades of the second half of the 21st century, the IPCC projects the emissions from the OECD90 countries to quickly ramp upwards, despite slowed growth or even declines among other groups and despite little population growth. This seems like an odd expectation, but I digress…
Now, what I am going to do, through the help of MAGICC (a simple climate model which was developed to emulate the large-scale output of more complex climate models and which was designed to explore the impacts of different emissions scenarios on projected global temperatures), is show you what happens to future global temperature projections if the Waxman-Markey emissions limitation provisions were adopted (and adhered to) by the U.S. And while I'm at it, I'll take you through the impacts of the adoption by the other regions as well.
Figure 2 is the same as Figure 1, except that I have adjusted the future OECD90 emissions to account for a reduced contribution from the U.S. assuming we stick to the Waxman-Markey emissions schedule.
Same as Figure 1, except the original OECD90 pathway
(dotted pink line) has been modified to account for
the U.S. adherence to the Waxman-Markey emissions schedule (solid pink line).
Figure 3 shows what happens to global temperature projections when the MAGICC model is run with the original A1FI emissions pathways (shown in Figure 1) as well as when it is run under the modified A1FI scenario to include U.S. reductions (shown in Figure 2).
The net result on the projected future global temperatures of a full adherence to the stipulations of the Waxman-Markey Climate Bill is a temperature "savings" of 0.06ºC by the year 2050, increasing to about 0.20ºC by the end of the century.
Projected global temperatures under the A1FI scenario (blue)
and the A1FI scenario modified for a U.S. adherence to
the Waxman-Markey emissions reductions schedule (red).
So, there you have it - going it alone, the U.S. succeeds at only managing to knock off two-tenths of a global temperature rise projected to be nearly 4.5ºC by 2100. Not a whole lot of bang for the buck.
So, clearly we (Americans) need a little, er, a lot of help.
In Figure 4, I depict what happens to the A1FI emissions pathways if every country of the world decided that the plan drawn up by Representatives Waxman and Markey was something that it could not live without and joined in the effort.
Most notably, instead of the rapid rises in ASIA emissions that are projected to occur through the half of the 21st century, the emissions there top out by 2010 and decline sharply thereafter - despite a growing population and rapid industrialization - that'll be a neat trick to pull off!
Same as Figure 1, except that all groups adhere to
the Waxman-Markey emissions reduction schedule.
Dotted lines are the original A1FI pathways, solid lines are the modified pathways.
Figure 5 shows the projected global temperatures with the different country groups signing on (i.e. MAGICC run with the modified emissions scenario depicted in Figure 4).
and the A1FI scenario modified for an adherence to the Waxman-Markey
emissions reductions schedule by all countries in the world in succession.
The top curve in Figure 5 (the greatest temperature rise) is projected to occur under the unfettered A1FI scenario.
The bottom curve (the least temperature rise) occurs with everyone on-board. The curves in the middle show who contributes what.
The U.S. acting alone under Waxman-Markey (as we have seen) reduces the projected global temperature rise by the year 2100 by 0.195ºC, if the rest of the OECD90 countries come along, the reduction increases to 0.402ºC - still less than 10% of the total projected rise.
Even with the help of the REF countries, we only get a reduction of 0.602ºC. When the temperature rise really starts to show a decent slowdown is with the cooperation of the ALM countries (a reduction 1.241ºC). And, of course, the biggest impact, nearly as large as everyone else combined, comes from the ASIA countries.
If they alone reduce emissions in line with Waxman-Markey suggestions, they will produce a 1.129ºC decline, and when acting along with everyone else they bring the total temperature reduction to 2.37ºC - a rise that is more than 50% smaller than projected under the original A1FI scenario.
Nothing to sneeze at.
(Again, let me stress that I am describing the impacts on projected global temperatures. There is growing evidence that actual global temperatures are not evolving the way projections indicate that they should. So, the degree to which these temperature projections described above reflect what really will happen in the future, is far from certain.)
So, the key to producing a meaningful change in the course of projected global temperatures is to make sure that those countries of the world which are projected to have the greatest contributions to future emissions growth - primarily the countries in the ALM and ASIA group - take the actions to insure that those growth projections are not met.
The United States has an extremely limited direct role to play in projected future global climate - internal emissions reductions do virtually nothing.
So, plans like the Waxman-Markey Climate Bill really don't serve to change the climate in and of themselves. Instead, their purpose is to attempt to spur technological innovation and set an example as to what can be done to reduce emissions - with Americans serving both as the experimenters and the guinea pigs.
It is not the climate impact of our experiment that is of any significance, but instead it is the tools that we may develop in attempting to achieve major emissions reductions, for the only truly effective course of action we have available to us in attempting to control the future course of global climate is to tell the rest of the world what to do and how to do it.
Let's hope they are agreeable - for "we" (Americans) are setting ourselves up to take a great risk for which the outcome, both internally and externally, is far from certain. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,226 |
{"url":"https:\/\/askthetask.com\/2410\/give-the-exact-area-of-the-following-triangle","text":"0 like 0 dislike\nGive the exact area of the following triangle\n\n0 like 0 dislike\n9 sqrt 3 in^2\n\nStep-by-step explanation:\n\nAssuming this is a equilateral triangle\n\nfind height\n\nsin 60 = height\/6\n\n6 sqrt (3)\/2 = height\n\n3 sqrt 3 = height\n\narea = 1\/2 base * height\n\n= 1\/2 * 6 * 3 sqrt 3\n\n= 9 sqrt 3 in^2\nby","date":"2022-09-29 18:27:19","metadata":"{\"extraction_info\": {\"found_math\": false, \"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\": 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\": 0.9119795560836792, \"perplexity\": 13478.62384161564}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"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-2022-40\/segments\/1664030335362.18\/warc\/CC-MAIN-20220929163117-20220929193117-00381.warc.gz\"}"} | null | null |
Theranos Founder Elizabeth Holmes, Formerly Worth $4.5 Billion, Now Worth 'Nothing'
Joanna Rothkopf
Last year, Elizabeth Holmes, the founder of troubled blood-testing company Theranos, was at the top of Forbes' Richest Self-Made Women list with an estimated net worth of $4.5 billion, making her the first youngest self-made female billionaire. On Wednesday, the magazine revised its estimate to "nothing."
Forbes wrote that its estimate was based solely on her 50 percent ownership in Theranos, which was valued at $9 billion in 2014. Without it, Holmes's net worth looks quite different:
Forbes spoke to a dozen venture capitalists, analysts and industry experts and concluded that a more realistic value for Theranos is $800 million, rather than $9 billion. That gives the company credit for its intellectual property and the $724 million that it has raised, according to VC Experts, a venture capital research firm. It also represents a generous multiple of the company's sales, which Forbes has learned about from a person familiar with Theranos' finances.
The company has recently become the subject of a criminal investigation and potential forced moratorium in services after its blood testing methods were found to be suspect and the company had recklessly put tens of individuals (edit: 81) in danger by failing to disclose a specific hematology test's deficiencies. Last week, Theranos announced it had voided two years of blood test results because of potential inaccuracies.
Elizabeth Holmes, Theranos May Be Facing a 2-Year Ban From the Blood Testing Business
Elizabeth Holmes, Theranos CEO and founder, our favorite blood-fearing, turtleneck-loving,…
Because Theranos investors own preferred shares, Forbes explains, they would get reimbursed before Holmes, who owns common stock. Were the company to be liquidated at its current value, Holmes wouldn't get anything—as Forbes puts it: "Holmes' stake is essentially worth nothing."
Senior Editor, Jezebel
fighting polish, white sox rememberer
honestly becoming a billionaire on the back of absolutely nothing and stealing money from Silicon Valley dumb dumbs is sort of my dream so YOU LIVE YOUR LIFE LIZZIE HOLMES | {
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} | 647 |
LSA-code
Ergine, ook lyserginezuuramide of LSA genoemd, is een in de natuur voorkomend hallucinogeen
LSA-verbindingen, een systeem voor het maken van draadverbindingen in elektronische netwerken | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,535 |
cask 'xamarin-android' do
version '6.1.1-1'
sha256 'c89e6a390ef9d25e932b0cb4043831912cb67a1c335f19b7ddd654c8f2846c12'
url "https://download.xamarin.com/MonoforAndroid/Mac/xamarin.android-#{version}.pkg"
appcast 'https://static.xamarin.com/installer_assets/v3/Mac/Universal/InstallationManifest.xml',
checkpoint: '2cb6260bf1f5348e125fe9cab2effe9862fe76f0706f146dd91c827827cdda6b'
name 'Xamarin.Android'
homepage 'https://xamarin.com/android'
license :unknown # TODO: change license and remove this comment; ':unknown' is a machine-generated placeholder
pkg "xamarin.android-#{version}.pkg"
uninstall pkgutil: 'com.xamarin.android.pkg'
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,666 |
Path to the Spiders' Nests
I saw someone reading this on the subway the other day. He was at one end of the car and I was at the other. Can it be said that a cover is successful if it leaps out at you from across a crowded subway? Anyone know who designed it?
From Amazon:
Italo Calvino's debut novel, updated to include changes that the author made for the definitive Italian edition, previously censored passages, and his newly translated, unabridged preface. "The Path to the Spiders' Nests," written when Calvino was twenty-three and first published in 1947--tells the story of Pin, a cobbler's apprentice in a town on the Ligurian Coast during World War II. He lives with his sister, a prostitute, and spends as much time as he can at the lowlife bar where he amuses the grownups. After a mishap with a Nazi soldier, Pin becomes involved with a band of partisans. Calvino's portrayal of this band, seen through the eyes of the child, is not only a revealing commentary on the Italian resistance, but also an insightful coming-of-age story. A bold, adventurous novel, The Path to the Spiders' Nests is animated by the formidable imagination that made Italo Calvino one of the most respected writers of our time.
Posted by Blair at 3:23 PM
Ian Koviak said...
ooh. very fun. kind of russian/italian constructivist/socialist appeal to it all. They had to throw that spider in there huh? It works though.
like an old propaganda poster...
Gould said...
Michael Crichton just died of cancer.
His novels were guilty pleasure, and their design were great (Rising Sun is among my favorite).
Tal : that would deserve some quite of tribute don't you think? His book design for Knopf were really groundbreaking thanks to Kidd.
Tal said...
Hmmm....I already did a post for Rising Sun and Next. We shall see, Gould, we shall see.
All right Commander.
You should at least post something on the 2666 book from FSG. It's a marvel. Already pre-ordered my copy on amazon
OK, Gould, I'll post it. But as for this cover, is this an homage to Dada? It's growing on me, like a spider web. Interesting that the leading doesn't seem to change in the title, while the font size increase. This makes me hope that someone will write a novel about Spider-Man, like Tom de Haven's awesome novel It's Superman!
Christine Van Bree: Lincoln | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,576 |
Olive Garden Potato Sausage Kale Soup Recipe - This is the latest information about Olive Garden Potato Sausage Kale Soup Recipe, this information can be your reference when you are confused to choose the right design for your home.
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"redpajama_set_name": "RedPajamaC4"
} | 5,635 |
{"url":"http:\/\/www.helpteaching.com\/questions\/Polynomials_and_Rational_Expressions?pageNum=2","text":"Want to see correct answers?\n\nLooking for Algebra worksheets?\nCheck out our pre-made Algebra worksheets!\n Tweet\nBrowse Questions\n\u2022 Arts (4594)\n\u2022 English Language Arts (59085)\n\u2022 English as a Second Language ESL (42458)\n\u2022 Health and Medicine (9413)\n\u2022 Life Skills (2966)\n\u2022 Math (27569)\n\n\u2022 Vectors\n\n\u2022 Trigonometry\n\n\u2022 Physical Education (4006)\n\u2022 Science (50452)\n\u2022 Social Studies (22666)\n\u2022 Study Skills and Strategies (353)\n\u2022 Technology (2526)\n\u2022 Vocational Education (7199)\n\nPolynomials and Rational Expressions Questions - All Grades\n\nYou can create printable tests and worksheets from these Polynomials and Rational Expressions questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.\n\nPrevious Next\nGrade 9 Polynomials and Rational Expressions\nSimplify: 5x + 12x\n1. 7x\n2. $7x^2$\n3. 17x\n4. $17x^2$\nGrade 11 Polynomials and Rational Expressions CCSS: HSA-APR.A.1\nFind the greatest common factor of the terms in the polynomial.\n\n$9x^4 + 12x^3 +6x$\n1. $2x^2$\n2. $3x$\n3. $6x^3$\n4. $9x^4$\nGrade 11 Polynomials and Rational Expressions\nDivide and simplify. $(3x)\/(x+9)-:(x)\/(3x+27) \\ \\ \\ (x!=-9,0)$\n1. $x^2\/(x+9)^2$\n2. $9\/x$\n3. $x\/9$\n4. $9$\nGrade 11 Polynomials and Rational Expressions CCSS: HSA-APR.A.1\nWhat is the difference $(x^4 - 3x^2 + 4x + 7) - (x^4 - 6x^3 + 4x)?$\n1. $3x^3 + 7$\n2. $6x^3 - 3x^2 + 7$\n3. $6x^3 - 3x^2 + 8x + 7$\n4. $3x^3 + 8x + 7$\nGrade 11 Polynomials and Rational Expressions\nUse the GCF of the terms to factor the polynomial.\n\nGiven: $23x^4 + 46x^3$\n1. $x^3(23x+46)$\n2. $23x^3(x+2)$\n3. $23x^4(x+2)$\n4. $23x(x^3+2x^2)$\nGrade 9 Polynomials and Rational Expressions\nSimplify\n$2x + 3x^2 - 5x +4 - 7x^2$\n1. $7x^2 +4$\n2. $7x +10x^2 +4$\n3. $-4x^2 -3x +4$\n4. $-3x^6$\nGrade 8 Polynomials and Rational Expressions\nGrade 11 Polynomials and Rational Expressions\nDivide and simplify the expression $(x-2)\/(5x+10)-:(3)\/(3x+6)$.\n1. $5\/(x-2), \\ \\ x!=-2$\n2. $(3(x-2))\/(15(x+2)^2)$\n3. $(x-2)\/5, \\ \\ x!=-2$\n4. $(3(x-2))\/(15(x+2))$\nGrade 9 Polynomials and Rational Expressions\nSimplify the expression.\n$(5m^2 - 4m - 8) - ( 3m^2-6m+2)$\n1. $8m^2 - 10m +6$\n2. $8m^2 +2m -10$\n3. $2m^2 +2m -10$\n4. $-2m^2 - 10m +6$\nGrade 10 Polynomials and Rational Expressions\nSelect the polynomial that is written in standard form.\n1. $3x^2+4x^5-7x$\n2. $4x^5+3x^2-7x$\n3. $-7x+4x^5+3x^2$\n4. $7+4x^5-3x^4+2x^3$\nGrade 11 Polynomials and Rational Expressions\nWrite the given polynomial in standard form.\n\nGiven: $4g\u2013g^3 +3g^2 \u20132$\n1. $\u20132+4g+3g^2\u2013g^3$\n2. $3g^3 \u2013g^2 +4g\u20132$\n3. $g^3\u20133g^2+4g\u20132$\n4. $\u2013g^3 +3g^2 +4g\u20132$\nGrade 11 Polynomials and Rational Expressions CCSS: HSA-APR.A.1\nWhich polynomial has the factorization $(x - 10)(x^2 + 10x + 100)?$\n1. $x^3 - 1000$\n2. $x^3 - 100$\n3. $x^3 + 100$\n4. $x^3 + 1000$\nGrade 11 Polynomials and Rational Expressions\nSimplify the expression $(7x^2-63)\/(x+3)$.\n1. $(x+3)\/7, \\ x!=-3$\n2. $(7x^2-9)\/(x), \\ x!=-3$\n3. $7(x+3), \\ x!=-3$\n4. $7(x-3), \\ x!=-3$\nGrade 11 Polynomials and Rational Expressions\nFind the values that make the rational expression undefined:\n\n$(x^2+3)\/(x(x+5)$\n1. $x=0$ or $-5$\n2. $x=0$ or $7$\n3. $x=0$ or $5$\n4. $x=2$ or $4$\nGrade 11 Polynomials and Rational Expressions\nSimplify the rational expression:\n\n$(2y)\/(8y^2)$\n1. $y\/6$\n2. $(2y)\/3$\n3. $1\/(4y)$\n4. $2\/(5y)$\nGrade 9 Polynomials and Rational Expressions\nGrade 9 Polynomials and Rational Expressions\nGrade 9 Polynomials and Rational Expressions\nGrade 11 Polynomials and Rational Expressions\nFind the value that makes the rational expression undefined:\n\n$8\/(y+6)$\n1. $y= -4$\n2. $y=0$\n3. $y=3$\n4. $y=-6$\nGrade 9 Polynomials and Rational Expressions CCSS: HSA-APR.A.1\nSimplify the expression\n$9y ^2 - 2y - 3y^2 + 5y$\n1. $12 y^2 + 3y$\n2. $6y^2 + 3y^2$\n3. $6 y^2 + 3y$\n4. $6y^2 + 7y$\nPrevious Next\nYou need to have at least 5 reputation to vote a question down. Learn How To Earn Badges.","date":"2017-12-13 17:04:31","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\": 0.34115663170814514, \"perplexity\": 6068.731829484781}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2017-51\/segments\/1512948529738.38\/warc\/CC-MAIN-20171213162804-20171213182804-00199.warc.gz\"}"} | null | null |
\section{Quelques conjectures du programme de Langlands}
\subsection{Repr{\'e}sentations automorphes}
La r{\'e}f{\'e}rence standard pour la d{\'e}finition des formes et repr{\'e}sentations
automorphes est le texte de Borel et Jacquet dans Corvallis (\cite{BJ}), voir
aussi le livre de Moeglin et Waldspurger (\cite{MW}).
Soit $n$ un entier strictement positif. On note $\mathbb{A}=\mathbb{R}\times{\mathbb{A}_f}$ l'anneau des
ad{\`e}les de $\mathbb{Q}$, o{\`u}
\[{\mathbb{A}_f}=\mathbb{Q}\otimes_\mathbb{Z}\widehat{\mathbb{Z}}=\{(x_p)\in\prod_{p\ {\rm premier}}\mathbb{Q}_p|x_p\mbox{ pour
presque tout }p\}\]
est l'anneau des ad\`eles finies
(o{\`u} {\og pour presque tout $p$\fg} signifie {\og pour tout $p$ sauf un nombre fini\fg}).
On fixe une mesure de Haar sur le groupe topologique
\[{\bf{GL}}_n(\mathbb{A})=\{(g_\infty,(g_p))\in{\bf{GL}}_n(\mathbb{R})\times\prod_p{\bf{GL}}_n(\mathbb{Q}_p)|g_p\in{\bf{GL}}_n(
\mathbb{Z}_p)\mbox{ pour presque tout }p\},\]
et on note $L^2_{{\bf{GL}}_n}=L^2({\bf{GL}}_n(\mathbb{Q})\mathrm{A}\setminus{\bf{GL}}_n(\mathbb{A}),\mathbb{C})$,
o{\`u} ${\bf{GL}}_n(\mathbb{Q})$ est plong{\'e}
diagonalement dans ${\bf{GL}}_n(\mathbb{A})$ et $\mathrm{A}=\mathbb{R}_{>0}$ est la composante connexe
de $1$ dans le centre de ${\bf{GL}}_n(\mathbb{R})$.\footnote{On pourrait fixer un caract{\`e}re
unitaire quelconque $\xi$ de $\mathrm{A}$ et consid{\'e}rer l'espace des fonctions
$f:{\bf{GL}}_n(\mathbb{Q})\setminus{\bf{GL}}_n(\mathbb{A})\longrightarrow\mathbb{C}$ telles que $f(zg)=\xi(z)f(g)$ pour tous
$z\in\mathrm{A}$ et $g\in{\bf{GL}}_n(\mathbb{A})$ et qui sont $L^2$ modulo $\mathrm{A}$. Dans cet expos{\'e},
on prendra $\xi=1$, mais c'est uniquement pour all{\'e}ger les notations.}
Le groupe ${\bf{GL}}_n(\mathbb{A})$ agit sur $L^2_{{\bf{GL}}_n}$ par translation {\`a} droite sur
l'argument de la fonction. Une \emph{repr{\'e}sentation automorphe discr{\`e}te de
${\bf{GL}}_n(\mathbb{A})$} est une repr{\'e}sentation irr{\'e}ductible de ${\bf{GL}}_n(\mathbb{A})$ qui
appara{\^i}t comme facteur direct de la repr{\'e}sentation $L^2_{{\bf{GL}}_n}$. En fait, on a
\[L^2_{{\bf{GL}}_n}=L^2_{{\bf{GL}}_n,{\rm disc}}\oplus L^2_{{\bf{GL}}_n,{\rm cont}}\]
en tant que repr{\'e}sentation de ${\bf{GL}}_n(\mathbb{A})$,
o{\`u} $L^2_{{\bf{GL}}_n,{\rm cont}}$ n'a pas de facteur direct irr{\'e}ductible et
\[L^2_{{\bf{GL}}_n,disc}=\bigoplus \pi^{m(\pi)}\]
(somme directe compl{\'e}t{\'e}e),
o{\`u} la somme est sur les repr{\'e}sentations automorphes discr{\`e}tes et les $m(\pi)$
sont des entiers strictement positifs.
Soit $f\in L^2_{{\bf{GL}}_n}$ une fonction born{\'e}e. On dit que $f$ est \emph{cuspidale}
si, pour tout sous-groupe parabolique propre ${\bf P}$ de ${\bf{GL}}_n$, si on note
${\bf N}_P$ le radical unipotent de ${\bf P}$
et $dn$ une mesure de Haar sur ${\bf N}_P(\mathbb{A})$,
alors, pour tout $g\in{\bf{GL}}_n(\mathbb{A})$,
\[\int_{{\bf N}_P(\mathbb{Q})\setminus{\bf N}_P(\mathbb{A})} f(ng)dn=0.\]
L'espace $L^2_{{\bf{GL}}_n,{\rm cusp}}$ des fonctions born{\'e}es cuspidales est un sous-espace
de $L^2_{{\bf{GL}}_n}$ ferm{\'e} et stable par l'action de ${\bf{GL}}_n(\mathbb{A})$,
qui est contenu dans $L^2_{{\bf{GL}}_n,{\rm disc}}$, c'est-{\`a}-dire
somme directe compl{\'e}t{\'e}e de repr{\'e}sentations irr{\'e}ductibles de ${\bf{GL}}_n(\mathbb{A})$.
Les repr{\'e}sentations irr{\'e}ductibles de ${\bf{GL}}_n(\mathbb{A})$
qui apparaissent dans $L^2_{{\bf{GL}}_n,{\rm cusp}}$
sont dites \emph{automorphes cuspidales}.
De plus, si $\pi$ est une repr{\'e}sentation automorphe discr{\`e}te de ${\bf{GL}}_n(\mathbb{A})$,
on a
\[\pi=\pi_\infty\otimes\bigotimes'_{p\ premier}\pi_p,\]
o{\`u} $\pi_\infty$ (resp. $\pi_p$) est une repr{\'e}sentation irr{\'e}ductible de
${\bf{GL}}_n(\mathbb{R})$ (resp. ${\bf{GL}}_n(\mathbb{Q}_p)$) et, pour presque tout $p$, la repr{\'e}sentation
$\pi_p$ est \emph{non ramifi{\'e}e}, c'est-{\`a}-dire que $\pi_p^{{\bf{GL}}_n(\mathbb{Z}_p)}\not=0$
(cet espace est alors de dimension $1$). Voir l'article de Flath
\cite{F} pour la
d{\'e}finition du produit tensoriel restreint $\bigotimes'$ et pour des r{\'e}f{\'e}rences.
La classification de Langlands associe {\`a} la repr{\'e}sentation irr{\'e}ductible
admissible $\pi_\infty$ de ${\bf{GL}}_n(\mathbb{R})$ une repr{\'e}sentation du groupe de
Weil $W_\mathbb{R}$ de $\mathbb{R}$ dans ${\bf{GL}}_n(\mathbb{C})$. En restreignant cette repr{\'e}sentation
au sous-groupe $\mathbb{C}^\times$ de $W_\mathbb{R}$, on obtient un morphisme $r:\mathbb{C}^\times\longrightarrow
{\bf{GL}}_n(\mathbb{C})$. On dit que la repr{\'e}sentation automorphe $\pi$ est \emph{alg{\'e}brique}
si $r$ est un morphisme de groupes alg{\'e}briques sur $\mathbb{C}$.
Cette d{\'e}finition est due {\`a} Clozel (d{\'e}finition 1.8 de \cite{C1}), et peut aussi
se formuler comme une condition d'int{\'e}gralit{\'e} sur le caract{\`e}re infinit{\'e}simal de $\pi_\infty$, c'est-{\`a}-dire
le caract{\`e}re par lequel
le centre de l'alg{\`e}bre universellement enveloppante de $\Lie({\bf{GL}}_n(\mathbb{R}))\otimes_
\mathbb{R}\mathbb{C}$ agit sur $\pi_\infty$.
\footnote{En fait, la d{\'e}finition donn{\'e}e ci-dessus n'est pas tout {\`a} fait
celle de Clozel, car on a supprim{\'e} la torsion par $|.|^{(n-1)/2}$. La notion
que nous avons d{\'e}finie est celle de repr{\'e}sentation \emph{L-alg{\'e}brique}
au sens de Buzzard et Gee (cf. \cite{BG}), qui semble plus adapt{\'e}e au cas
d'un groupe r{\'e}ductif g{\'e}n{\'e}ral, et qui est celle qu'utilise Scholze.}
\subsection{Conjecture de r{\'e}ciprocit{\'e} de Langlands et Clozel}
\begin{conj}\label{conj:LC}
Soit $\pi=\pi_\infty\otimes\bigotimes'_p \pi_p$ une repr{\'e}sentation
automorphe alg{\'e}brique cuspidale de ${\bf{GL}}_n(\mathbb{A})$, et soit $\ell$ un nombre
premier.
Alors il existe une repr{\'e}sentation continue semi-simple\footnote{
En fait, la repr\'esentation $\rho_\pi$ devrait m\^eme \^etre irr\'eductible,
mais on ne sait le prouver que dans quelques cas particuliers, voir par
exemple \cite{BR} et le th\'eor\`eme D de \cite{BLGGT}.}
$\rho_\pi:\Gal(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow{\bf{GL}}_n(\oQ_\ell)$ telle que, pour tout
nombre premier $p\not=\ell$ tel que $\pi_p$ soit non ramifi{\'e}e, $\pi_p$
et $\rho_{\pi|\Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)}$ se correspondent par l'isomorphisme
de Satake.
\end{conj}
Donnons quelques explications sur l'{\'e}nonc{\'e}. Une r{\'e}f{\'e}rence pour l'isomorphisme
de Satake est le chapitre IV de l'article \cite{C} de Cartier, voir aussi
l'article introductif \cite{G} de Gross. Soit $p$ un nombre
premier. Rappelons que l'on dit que $\pi_p$ est non ramifi{\'e}e (ou que
$\pi$ est non ramifi{\'e}e en $p$) si $\pi_p^{{\bf{GL}}_n(\mathbb{Z}_p)}\not=0$. Cet espace
d'invariants est alors n{\'e}cessairement de dimension $1$, et d{\'e}finit donc
un caract{\`e}re de l'alg{\`e}bre de Hecke non ramifi{\'e}e $\mathcal{H}_p$ des fonctions
$f:{\bf{GL}}_n(\mathbb{Q}_p)\longrightarrow\mathbb{C}$ {\`a} support compact et invariantes {\`a} gauche et {\`a} droite
par ${\bf{GL}}_n(\mathbb{Z}_p)$ (le produit est le produit de convolution, d{\'e}fini en
utilisant la mesure de Haar sur ${\bf{GL}}_n(\mathbb{Q}_p)$ telle que ${\bf{GL}}_n(\mathbb{Z}_p)$ soit
de volume $1$). Il r{\'e}sulte de l'isomorphisme de Satake (sous la forme, par
exemple, du corollaire 4.2 de \cite{C}) que l'ensemble des caract{\`e}res
de $\mathcal{H}_p$ est en bijection avec l'ensemble des classes de conjugaison
semi-simples de ${\bf{GL}}_n(\mathbb{C})$.
Revenant {\`a} la conjecture, on note $a_{\pi_p}$ la classe de conjugaison
correspondant {\`a} $\pi_p$. Dire que $\rho_{\pi|\Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)}$ correspond
{\`a} $\pi_p$ par l'isomorphisme de Satake signifie d'abord que
$\rho_{\pi|\Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)}$ est non ramifi{\'e}e, c'est-{\`a}-dire se
factorise par le quotient $\Gal(\overline{\mathbb{F}}_p/\mathbb{F}_p)$ de
$\Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$, et ensuite que l'image par $\rho_\pi$ du morphisme
de Frobenius g{\'e}om{\'e}trique (i.e. le g{\'e}n{\'e}rateur
$x\longmapsto x^{1/p}$ de $\Gal(\overline{\mathbb{F}}_p/\mathbb{F}_p)$) est dans la classe de
conjugaison $a_{\pi_p}$.\footnote{Le plongement $\Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)
\longrightarrow\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ n'est pas canonique, puisqu'il d{\'e}pend du choix
d'un plongement $\overline{\mathbb{Q}}\longrightarrow\overline{\mathbb{Q}}_p$. Cependant, la d{\'e}finition
ci-dessus ne d{\'e}pend pas de ce choix.}
D'apr{\`e}s le th{\'e}or{\`e}me de densit{\'e} de {\v C}eboratev, $\rho_\pi$ est uniquement
d{\'e}termin{\'e}e par $\pi$. D'apr{\`e}s le th{\'e}or{\`e}me de multiplicit{\'e} un fort
de Piatetski-Shapiro et Jacquet-Shalika (cf. \cite{PS}), $\pi$
est uniquement d{\'e}termin{\'e}e par $\rho_\pi$.
\begin{rema} En combinant les conjectures de Langlands, Clozel et Fontaine-Mazur, on obtient en fait une bijection conjecturale entre les classes d'isomorphisme
de repr{\'e}sentations automorphes cuspidales alg{\'e}briques de ${\bf{GL}}_n(\mathbb{A})$ et les
classes d'isomorphisme de repr{\'e}sentations continues irr{\'e}ductibles
$\Gal(\oQ/\mathbb{Q})\longrightarrow{\bf{GL}}_n(\oQ_\ell)$ qui sont g{\'e}om{\'e}triques (c'est-{\`a}-dire
presque partout non ramifi{\'e}es et de Rham en $\ell$, voir \cite{FM}).
\end{rema}
\begin{rema} En fait, Langlands conjecture qu'il existe un groupe
pro-alg{\'e}brique $\Lf_\mathbb{Q}$ sur $\mathbb{C}$ (le groupe de Langlands de $\mathbb{Q})$ tel que,
pour tout entier $n$, les repr{\'e}sentations
alg\'ebriques irr\'eductibles de dimension $n$ de $\Lf_\mathbb{Q}$
classifient les repr{\'e}sentations
automorphes cuspidales de ${\bf{GL}}_n(\mathbb{A})$.
(En d'autres termes, les repr{\'e}sentations automorphes
cuspidales de ${\bf{GL}}_n(\mathbb{A})$ sont les objets simples de dimension $n$
d'une cat{\'e}gorie tannakienne, dont le groupe tannakien est $\Lf_\mathbb{Q}$.)
Le groupe de Galois motivique $\M_\mathbb{Q}$
de $\mathbb{Q}$ serait alors le quotient de $\Lf_\mathbb{Q}$
correspondant {\`a} la sous-cat{\'e}gorie des repr{\'e}sentations alg{\'e}briques. Si
$\pi$ est une repr{\'e}sentation automorphe cuspidale alg{\'e}brique de ${\bf{GL}}_n(\mathbb{A})$
et $\varphi:\Lf_\mathbb{Q}\longrightarrow\M_\mathbb{Q}
\longrightarrow{\bf{GL}}_{n,\mathbb{C}}$ est la repr{\'e}sentation alg{\'e}brique correspondante,
on devrait obtenir $\rho_\pi$ en {\'e}valuant $\varphi$ sur les $\oQ_\ell$-points
(on choisit un isomophisme $\oQ_\ell\simeq\mathbb{C}$ pour faire ceci), puis en
restreignant au sous-groupe $\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ de $\M_\mathbb{Q}(\oQ_\ell)$
(le plongement {\'e}tant donn{\'e} par la r{\'e}alisation $\ell$-adique).
\end{rema}
\subsection{Repr{\'e}sentations automorphes et cohomologie des espaces localement
sym{\'e}triques}
Toutes les preuves de cas particuliers de la conjecture \ref{conj:LC}\footnote{
Au moins celles connues de l'auteur.} passent pas l'{\'e}tude de la cohomologie
des espaces localement sym{\'e}triques. Si $K$ est un sous-groupe compact ouvert
de ${\bf{GL}}_n({\mathbb{A}_f})$ (par exemple un sous-groupe d'indice fini de ${\bf{GL}}_n(\widehat
{\mathbb{Z}})$), on pose
\[X_K={\bf{GL}}_n(\mathbb{Q})\setminus{\bf{GL}}_n(\mathbb{A})/K \mathrm{A} K_\infty,\]
o{\`u} $K_\infty=\bf{SO}(n)\subset{\bf{GL}}_n(\mathbb{R})$. C'est une vari{\'e}t{\'e} analytique r{\'e}elle si
$K$ est assez petit (sinon, c'est un orbifold). Dans la suite, on supposera
toujours $K$ assez petit.
Si $K$ et $K'$ sont deux sous-groupes ouverts compacts de ${\bf{GL}}_n({\mathbb{A}_f})$ et
$g\in{\bf{GL}}_n({\mathbb{A}_f})$ est tel que $g^{-1}K'g\subset K$, on a un morphisme
analytique fini $c_g:X_{K'}\longrightarrow X_K$ qui envoie la classe de $h\in{\bf{GL}}_n(\mathbb{A})$ sur
celle de $hg$. Donc, si on prend $K'=K\cap gK g^{-1}$, on obtient une
correspondance $(c_g,c_1):X_{K'}\longrightarrow X_K\times X_K$ (appel{\'e}e
\emph{correspondance de Hecke}), qui induit un morphisme
$u_g:H^*_c(X_K)\longrightarrow H^*_c(X_K)$, o{\`u} $H^*(X_K)$ est la cohomologie de Betti {\`a}
supports compacts et {\`a} coefficients dans $\mathbb{C}$ de $X_K$ (voir \cite{P} 1.2, 1.3).
Soit $\mathcal{H}$ l'alg{\`e}bre de Hecke globale, c'est-{\`a}-dire l'alg{\`e}bre des fonctions
localement constantes {\`a} support compact de ${\bf{GL}}_n({\mathbb{A}_f})$ dans $\mathbb{C}$, munie du
produit de convolution (pour une mesure de Haar fix{\'e}e sur ${\bf{GL}}_n({\mathbb{A}_f})$). C'est
une alg{\`e}bre associative non unitaire. Rappelons qu'une repr{\'e}sentation
$\pi:{\bf{GL}}_n({\mathbb{A}_f})\longrightarrow{\bf{GL}}(V)$ (o{\`u} $V$ est un $\mathbb{C}$-espace vectoriel) est appel{\'e}e
\emph{lisse} si $\pi$ est continue pour la topologie discr{\`e}te sur ${\bf{GL}}(V)$,
et \emph{admissible} si, pour tout sous-groupe compact ouvert $K$ de
${\bf{GL}}_n({\mathbb{A}_f})$, $V^K$ est de dimension finie. Par exemple, si $\pi=\pi_\infty\otimes\pi_f$ est une repr{\'e}sentation automorphe discr{\`e}te de ${\bf{GL}}_n(\mathbb{A})=
{\bf{GL}}_n(\mathbb{R})\times{\bf{GL}}_n({\mathbb{A}_f})$, alors $\pi_f$ est lisse admissible.
La cat{\'e}gorie des repr{\'e}sentations lisses de ${\bf{GL}}_n({\mathbb{A}_f})$ est naturellement
{\'e}quivalente {\`a} celle des repr{\'e}sentations $V$ de $\mathcal{H}$ telles que, pour tout
$v\in V$, on ait $1\!\!\mkern -1mu1_K.v=v$ pour $K\subset{\bf{GL}}_n({\mathbb{A}_f})$ assez petit. En
particulier, toute repr{\'e}sentation automorphe discr{\`e}te de ${\bf{GL}}_n(\mathbb{A})$
d\'efinit une repr{\'e}sentation de $\mathcal{H}$.
D'autre part, on d{\'e}finit une action de $\mathcal{H}$ sur $H^*_c:=\varinjlim_K H^*_c(X_K)$
en convenant que, pour tout sous-groupe compact ouvert $K$ de ${\bf{GL}}_n({\mathbb{A}_f})$
et tout $g\in{\bf{GL}}_n({\mathbb{A}_f})$, la fonction caract{\'e}ristique de $Kg^{-1}$ agit
sur $H^*_c(X_K)$ par l'op{\'e}rateur $u_g$.
Enfin, on dit qu'une repr{\'e}sentation automorphe discr{\`e}te $\pi=\pi_\infty\otimes
\pi_f$ est \emph{cohomologique} s'il existe une repr{\'e}sentation alg{\'e}brique
$W$ de ${\bf{GL}}_n(\mathbb{R})$ telle que la $(\ggoth,\mathrm{A}\mathrm{K}_\infty)$-cohomologie de
$\pi_\infty\otimes W$ soit non nulle (o{\`u} $\ggoth=\Lie({\bf{GL}}_n(\mathbb{R}))$).
Cela implique
que $\pi$~est alg{\'e}brique et que le caract{\`e}re infinitis{\'e}mal de $\pi_\infty$
v{\'e}rifie de plus une condition de r{\'e}gularit{\'e}.\footnote{Les
$\pi_\infty$ possibles ont
{\'e}t{\'e} classifi{\'e}es par Vogan et Zuckerman dans \cite{VZ}.}
Il r{\'e}sulte alors de la conjecture de Borel, prouv{\'e}e par Franke (th{\'e}or{\`e}me
18 de \cite{Fr}), que l'on a le th{\'e}or{\`e}me suivant :
\begin{theo}\label{th:Franke}
Les sous-quotients irr{\'e}ductibles de la repr{\'e}sentation de
$\mathcal{H}$ sur $H^*_c$ viennent tous de repr{\'e}sentations automorphes
cohomologiques de ${\bf{GL}}_n(\mathbb{A})$. De plus, si $\pi$ est une repr{\'e}sentation
automorphe cuspidale cohomologique de ${\bf{GL}}_n(\mathbb{A})$ sur laquelle $\mathrm{A}$
agit trivialement, et si on peut prendre $W=1\!\!\mkern -1mu1$ dans la d{\'e}finition
ci-dessus, alors la repr{\'e}sentation de $\mathcal{H}$ associ{\'e}e {\`a} $\pi$ appara{\^i}t comme
un sous-quotient de $\H^*_c$.\footnote{On obtiendrait les repr{\'e}sentations
cohomologiques pour les $W$ non triviales en prenant la cohomologie {\`a}
coefficients dans un syst{\`e}me local non trivial sur les $X_K$.}
\end{theo}
\subsection{Cohomologie des vari{\'e}t{\'e}s de Shimura et conjecture \ref{conj:LC}}\label{1.4}
Les espaces localement sym{\'e}triques associ{\'e}s au groupe ${\bf{GL}}_n$ sont seulement
des vari{\'e}t{\'e}s analyiques r{\'e}elles, mais, en utilisant d'autres groupes, on
peut obtenir des espaces avec plus de structure (c'est-{\`a}-dire des vari{\'e}t{\'e}s
alg{\'e}briques sur des corps de nombres, appel{\'e}es \emph{vari{\'e}t{\'e}s de Shimura}).
Soit ${\bf{G}}$ un groupe
alg{\'e}brique r{\'e}ductif connexe sur $\mathbb{Q}$. \`A l'exception du th\'eor\`eme
de multiplicit\'e un fort,
toutes les d{\'e}finitions et les r{\'e}sultats
ci-dessus restent valables pour ${\bf{G}}$
(il faut remplacer $K_\infty=\bf{SO}(n)$ par un sous-groupe
compact connexe maximal de ${\bf{G}}(\mathbb{R})$ et $\mathrm{A}$ par ${\bf S}(\mathbb{R})^\circ$, o{\`u}
${\bf S}$ est le sous-tore d{\'e}ploy{\'e} (sur $\mathbb{Q}$) maximal du centre de ${\bf{G}}$; pour
l'isomorphisme de Satake, il faut supposer ${\bf{G}}$ non ramifi{\'e} en $p$ et
remplacer ${\bf{GL}}_n(\mathbb{Z}_p)$ par un sous-groupe compact maximal hypersp{\'e}cial
de ${\bf{G}}(\mathbb{Q}_p)$, voir la section 1.10 de l'article \cite{T} de Tits).
Si par exemple ${\bf{G}}$ est le groupe symplectique ${\bf {Sp}}_{2n}\subset{\bf{GL}}_{2n}$
de la forme symplectique $x_1 y_{2n}+\dots x_n y_{n+1}-x_{n+1}y_n-\dots
-x_1 y_{2n}$ et $K=\Ker({\bf{G}}(\widehat{\mathbb{Z}})\longrightarrow{\bf{G}}(\mathbb{Z}/N\mathbb{Z}))$ pour $N$ un entier
$\geq 3$ (pour que $K$ soit assez petit), alors l'espace localement sym{\'e}trique
associ{\'e} $X_K^{{\bf{G}}}$ est l'espace de modules des vari{\'e}t{\'e}s ab{\'e}liennes de dimension
$n$ sur $\mathbb{C}$ principalement polaris{\'e}es et munies d'une structure de niveau
$N$ (voir la section 11 de l'article \cite{K1} de Kottwitz). On peut d{\'e}finir
le probl{\`e}me de modules sur $\mathbb{Q}$ (ou m{\^e}me $\mathbb{Z}$), et Mumford a montr{\'e} que
ce probl{\`e}me de modules est repr{\'e}sentable par un sch{\'e}ma quasi-projectif
(cf. le th{\'e}or{\`e}me 7.9 de \cite{GIT}). Les correspondances de Hecke ont
aussi une description modulaire, et sont donc d{\'e}finies sur $\mathbb{Z}$.
En utilisant le th{\'e}or{\`e}me de comparaison
entre cohomologie de Betti et cohomologie {\'e}tale et en utilisant
$\oQ_\ell$ au lieu de $\mathbb{C}$ comme corps de coefficients, on en d{\'e}duit que
$H^*_{c,{\bf{G}}}:=\varinjlim_{K}H^*_c(X_K^{{\bf{G}}})$
est muni d'une action de $\Gal(\oQ/\mathbb{Q})$ qui
commute {\`a} l'action de l'alg{\`e}bre de Hecke $\mathcal{H}$. On peut donc {\'e}crire la
semi-simplifi{\'e}e de cette repr{\'e}sentation de la mani{\`e}re suivante :
\[(H^i_{c,{\bf{G}}})^{ss}=\bigoplus_{\pi=\pi_\infty\otimes\pi_f}\pi_f\otimes\sigma^i(\pi_f),\]
o{\`u} $\pi$ parcourt l'ensemble des repr{\'e}sentations
automorphes cohomologiques de ${\bf{G}}(\mathbb{A})$ et les $\sigma^i(\pi_f)$ sont des
repr{\'e}sentations semi-simples de $\Gal(\oQ/\mathbb{Q})$. Il n'est pas {\'e}vident en
g{\'e}n{\'e}ral de d{\'e}terminer $\sigma^i(\pi_f)$ en fonction de $\pi_f$, mais, si
l'on utilise la cohomologie d'intersection (voir par exemple
\cite{BBD}) au lieu de la cohomologie {\`a} supports compacts, alors on une
formule conjecturale tr{\`e}s pr{\'e}cise pour $\sigma^i(\pi_f)$, due {\`a} Langlands,
Rapoport et Kottwitz (cf. la section 10 de \cite{K1}). Cette formule fait
intervenir la conjecture de r{\'e}ciprocit{\'e} de Langlands pour le groupe ${\bf{G}}$.
De plus, tout le paragraphe pr{\'e}c{\'e}dent
est en fait valable pour les groupes ${\bf{G}}$ dont les espaces
localement sym{\'e}triques sont des vari{\'e}t{\'e}s de Shimura de type PEL, par
exemple les groupes unitaires et certains groupes orthogonaux (voir la
section 5 de l'article \cite{K2} de Kottwitz), {\`a} la diff{\'e}rence que la structure
de vari{\'e}t{\'e} alg{\'e}brique des $X_K^{{\bf{G}}}$ est en g{\'e}n{\'e}ral d{\'e}finie non pas sur $\mathbb{Q}$,
mais sur une extension finie de $\mathbb{Q}$ appel{\'e}e corps reflex.
Si on choisit le groupe ${\bf{G}}$ (et le degr{\'e} $i$) correctement, la cohomologie
d'intersection en degr{\'e} $i$ (qui, dans le cas o{\`u} ${\bf{G}}^{der}$ est anisotrope
sur $\mathbb{Q}$, est simplement $\H^i_{c,{\bf{G}}}$, car les $X_K^{{\bf{G}}}$ sont alors des
sch{\'e}mas projectifs) r{\'e}alise conjecturalement une partie de
la correspondance de Langlands
pour le groupe ${\bf{G}}$. De plus, dans de nombreux cas, la conjecture est en fait
connue.
Si l'on veut obtenir des informations sur la conjecture \ref{conj:LC} pour
${\bf{GL}}_n$, on peut passer des repr{\'e}sentations automorphes de ${\bf{GL}}_n$ {\`a} celles
d'un autre groupe en utilisant le principe de fonctorialit{\'e} de Langlands
(conjectural lui aussi en g{\'e}n{\'e}ral, mais connu dans les cas que l'on veut
utiliser ici). Soit ${\bf{G}}$ un groupe r{\'e}ductif connexe sur $\mathbb{Q}$, d{\'e}ploy{\'e} pour
simplifier, et soit
$\widehat{{\bf{G}}}$ son dual de Langlands (le groupe r{\'e}ductif connexe sur $\mathbb{C}$
obtenu en {\'e}changeant le r{\^o}le des racines et des coracines dans
la donn{\'e}e radicielle de ${\bf{G}}$). Alors les repr{\'e}sentations automorphes cuspidales
de ${\bf{G}}(\mathbb{A})$ sont conjecturalement classifi{\'e}es par les param{\`e}tres de
Langlands, qui sont des morphismes alg{\'e}briques irr{\'e}ductibles\footnote{
C'est-{\`a}-dire dont l'image n'est contenue dans aucun sous-groupe parabolique.}
$\Lf_\mathbb{Q}\longrightarrow\widehat{{\bf{G}}}$. En g{\'e}n{\'e}ral, cette param{\'e}trisation n'est plus bijective,
et l'on s'attend {\`a} ce que chaque param{\`e}tre corresponde {\`a} un ensemble fini
de repr{\'e}sentations automorphes cuspidales, appel{\'e} un $L$-paquet.
En tout cas, si l'on a deux groupes r{\'e}ductifs connexes~${\bf{G}}$ et ${\bf{H}}$ et un
morphisme $\widehat{{\bf{G}}}\longrightarrow\widehat{{\bf{H}}}$, cette conjecture implique que l'on
a un {\og transfert\fg} qui envoie une repr{\'e}sentation automorphe cuspidale
de ${\bf{G}}(\mathbb{A})$ sur un $L$-paquet de repr{\'e}sentations automorphes
de ${\bf{H}}(\mathbb{A})$ (de
mani{\`e}re compatible {\`a} l'isomorphisme de Satake aux places o{\`u} les
repr{\'e}sentations sont non ramifi{\'e}es). On devrait aussi pouvoir caract{\'e}riser
l'image de ce transfert.\footnote{Le transfert
ne pr{\'e}serve pas la cuspidalit{\'e} en g{\'e}n{\'e}ral, il faut donc travailler avec
les repr{\'e}sentations automorphes discr{\`e}tes, qui sont (conjecturalement)
param{\'e}tr{\'e}es par des
param{\`e}tres d'Arthur $\psi:\Lf_\mathbb{Q}\times\bf{SL}_2(\mathbb{C})\longrightarrow\widehat{{\bf{G}}}$ et non des
param{\`e}tres de Langlands.}
Par exemple, si ${\bf{G}}={\bf {Sp}}_{2n}$ et ${\bf{H}}={\bf{GL}}_{2n+1}$, alors $\widehat{{\bf{G}}}=\bf{SO}_{2n+1}
(\mathbb{C})$ se plonge de mani{\`e}re {\'e}vidente dans $\widehat{{\bf{H}}}={\bf{GL}}_{2n+1}(\mathbb{C})$. Dans
ce cas, l'existence du transfert et ses propri{\'e}t{\'e}s ont {\'e}t{\'e} {\'e}tablies par
Arthur dans le livre \cite{A}; l'image du transfert est caract{\'e}ris{\'e}e par une
condition d'autodualit{\'e} et une condition sur les p{\^o}les d'une certaine
fonction~$L$, voir le th{\'e}or{\`e}me 1.5.3 de \cite{A}. Si ${\bf{G}}$ est un
groupe unitaire et ${\bf{H}}$ un groupe g\'en\'eral lin\'eaire, on a des r\'esultats
similaires, dus \`a Mok (\cite{M}) et Kaletha-Minguez-Shin-White (\cite{KMSW}).
En utilisant le transfert des groupes unitaires vers les groupes g{\'e}n{\'e}raux
lin{\'e}aires, le calcul de la cohomologie des vari{\'e}t{\'e}s de Shimura de certains
groupes unitaires, le lemme fondamental et des techniques d'interpolation
$p$-adique pour attraper certaines repr{\'e}sentations automorphes, on arrive
au r{\'e}sultat suivant (d\^u, au moins, {\`a} Kottwitz, Clozel, Labesse, Harris-Taylor,
Fargues, Mantovan, Shin, Laumon-Ng\^o, Waldspurger,
Bella{\"i}che-Chenevier\footnote{L'auteur
regrette de ne pouvoir garantir l'exhaustivit{\'e} de cette liste.}) :
\begin{theo}[\cite{PL}]\footnote{On a un r{\'e}sultat similaire pour les
repr{\'e}sentations du groupe ${\bf{GL}}_n(\mathbb{A}_F)$, o{\`u} $F$ est un corps de nombres
totalement r{\'e}el ou CM. Un {\'e}nonc{\'e} pr{\'e}cis est rappel{\'e} dans le th{\'e}ror{\`e}me V.1.4
de \cite{S}.}
La conjecture \ref{conj:LC} est vraie pour les repr{\'e}sentations
automorphes cuspidales cohomologiques autoduales (c'est-{\`a}-dire isomorphes
{\`a} leur contragr{\'e}diente).
\end{theo}
\subsection{Repr{\'e}sentations non autoduales}
\label{1.5}
Les m{\'e}thodes de la section pr{\'e}c{\'e}dentes ne peuvent s'appliquer aux repr{\'e}sentations
non autoduales de ${\bf{GL}}_n(\mathbb{A})$. En effet, toutes les repr{\'e}sentations venant
par transfert depuis un groupe ayant une vari{\'e}t{\'e} de Shimura v{\'e}rifient une
propri{\'e}t{\'e} d'autodualit{\'e}.
L'approche suivante a {\'e}t{\'e} sugg{\'e}r{\'e}e par Clozel.
Si ${\bf{G}}={\bf {Sp}}_{2n}\subset{\bf{GL}}_{2n}$ est
le groupe symplectique de la forme antidiagonale (comme plus haut), alors
le groupe \mbox{${\bf P}={\bf {Sp}}_{2n}\cap\begin{pmatrix}* & * \\ 0 & *\end{pmatrix}$} est un
sous-groupe parabolique maximal de ${\bf{G}}$, de quotient de Levi isomorphe
{\`a} ${\bf{GL}}_n$. Si $\pi$ est une repr{\'e}sentation automorphe cuspidale cohomologique
de ${\bf{GL}}_n(\mathbb{A})$, elle d{\'e}finit donc par induction parabolique une repr{\'e}sentation
automorphe\footnote{Ou plusieurs...} $\Pi$
de ${\bf {Sp}}_{2n}(\mathbb{A})$, cf. \cite{L}, qui n'est {\'e}videmment pas cuspidale, mais se
trouve {\^e}tre aussi cohomologique. Comme le groupe ${\bf {Sp}}_{2n}$ admet une
vari{\'e}t{\'e} de Shimura, on peut essayer d'appliquer les techniques de la section
pr{\'e}c{\'e}dente {\`a} $\Pi$. Malheureusement, les repr{\'e}sentations galoisiennes
qui apparaissent, dont la composante $\Pi_f$-isotypique (o{\`u} $\Pi=
\Pi_\infty\otimes\Pi_f$) de la cohomologie
des $X_K^{{\bf{G}}}$, ne sont pas tr\`es int\'eressantes
(on obtient quelque chose qui ressemble beaucoup {\`a} la puissance ext{\'e}rieure
$n$-i{\`e}me de la repr{\'e}sentation de dimension~$n$ que l'on essaie de construire).
Une autre id{\'e}e, au lieu d'utiliser directement la cohomologie de Betti
des $X_K^{{\bf{G}}}$, est d'utiliser l'autre r{\'e}alisation cohomologique des formes
automorphes (comme sections de certains fibr{\'e}s vectoriels, dits {\og automorphes\fg},
sur les espaces localement sym{\'e}triques) pour approcher $p$-adiquement la
repr{\'e}sentation
$\Pi$ par des repr{\'e}sentations automorphes cuspidales cohomologiques de
${\bf {Sp}}_{2n}(\mathbb{A})$ (o\`u $p$ est un nombre premier fix\'e arbitrairement).
Ces repr{\'e}sentations se transf{\`e}rent alors en des
repr{\'e}sentations automorphes autoduales de ${\bf{GL}}_{2n+1}(\mathbb{A})$ (voir la
discussion sur le transfert plus haut), qui ont des repr{\'e}sentations
galoisiennes associ{\'e}es ({\`a} coefficients dans $\oQ_p$). En prenant la limite
de ces repr{\'e}sentations galoisiennes (ou plut{\^o}t de leurs caract{\`e}res), on
obtient une repr{\'e}sentation galoisienne de dimension $2n+1$, dont il est
possible d'extraire la repr{\'e}sentation $\rho_\pi$ cherch{\'e}e. Cette strat{\'e}gie
a {\'e}t{\'e} men{\'e}e {\`a} bien de mani{\`e}re ind{\'e}pendante par Harris-Lan-Taylor-Thorne
(\cite{HLTT}),
Scholze (\cite{S}) et Boxer. En fait, Scholze et Boxer prouvent des
r{\'e}sultats plus forts, que nous allons expliquer ci-dessous.
D'abord, remarquons que le paragraphe pr{\'e}c{\'e}dent n'a pas de sens a priori,
car les repr{\'e}sentations automorphes sont {\`a} coefficients dans $\mathbb{C}$, et
non $\oQ_p$. Mais d'apr{\`e}s le th{\'e}or{\`e}me \ref{th:Franke} ci-dessus (qui est valable
pour un groupe r{\'e}ductif connexe quelconque), on peut
remplacer l'{\'e}tude des repr{\'e}sentations automorphes cohomologiques par celles
des sous-quotients de la repr{\'e}sentation $H^*_{c,{\bf{G}}}=\varinjlim_{K}\H^*_c(X_K^{{\bf{G}}})$
de l'alg{\`e}bre de Hecke globale $\mathcal{H}_{{\bf{G}}}$ de ${\bf{G}}$ (l'alg{\`e}bre des fonctions
localement constantes {\`a} support compact ${\bf{G}}({\mathbb{A}_f})\longrightarrow\mathbb{C}$, munie du produit
de convolution). Mais tant la cohomologe de Betti que l'alg{\`e}bre de Hecke
globale ont un sens si l'on remplace le corps de coefficients $\mathbb{C}$ par un
anneau commutatif de coefficients quelconque $A$;\footnote{Ce n'est pas
tout \`a fait vrai. A priori, il
faudrait avoir une mesure de Haar sur ${\bf{G}}({\mathbb{A}_f})$ \`a valeurs dans $A$, ce qui
est une condition non triviale. On verra dans le paragraphe suivant comment
faire si $A=\mathbb{Z}$.}
notons $\H^*_{c,{\bf{G}}}(A)$
et $\mathcal{H}_{{\bf{G}}}(A)$ les objets ainsi obtenus. Si l'on a deux repr\'esentations
de $\mathcal{H}_{{\bf{G}}}(\mathbb{Q}_p)$ apparaissant dans $\H^*_{c,{\bf{G}}}(\mathbb{Q}_p)$, cela a un sens
de demander qu'elles soient $p$-adiquement proches, mais ce n'est pas encore
exactement ce que l'on veut faire. En effet, on veut approximer la classe
d'isomorphisme d'une repr\'esentation et non la repr\'esentation elle-m\^eme.
Il est donc naturel de chercher \`a approximer le caract\`ere de la
repr\'esentation, et pour cela il est plus commode de fixer le niveau $K$
(afin d'avoir des repr\'esentations de dimension finie).
On fixe donc un sous-groupe compact ouvert $K$ de ${\bf{G}}({\mathbb{A}_f})$, et on suppose
que $K$ est de la forme $\prod_v K_v$, avec $v$ parcourant les nombres premiers
et $K_v$ un sous-groupe compact ouvert de ${\bf{G}}(\mathbb{Q}_v)$.
Il existe un
ensemble fini $S$ de nombres premiers tel que, pour tout $v\not\in S$,
le groupe ${\bf{G}}$ soit non ramifi\'e en $v$ et $K_v$ soit hypersp\'ecial (voir
le d\'ebut de la section \ref{1.4} pour la d\'efinition), et on fixe un tel $S$. Pour
tout $v\not\in S$, on note $\mathcal{H}_{v,{\bf{G}}}$ l'alg\`ebre de
Hecke locale non ramifi\'ee en $v$, c'est-\`a-dire l'alg\`ebre des
fonctions $f:{\bf{G}}(\mathbb{Q}_v)\longrightarrow\mathbb{Z}$ \`a support compact et bi-invariantes par
$K_v$, munie du produit de convolution (pour la mesure de Haar sur
${\bf{G}}(\mathbb{Q}_v)$ telle que $K_v$ soit de volume $1$; pour le fait que le
produit de convolution de deux fonctions \`a valeurs dans $\mathbb{Z}$ est bien une
fonction \`a valeurs dans $\mathbb{Z}$, voir par exemple la section 2 de \cite{G}).
On note aussi $\mathcal{H}^S_{{\bf{G}}}=\bigotimes_{v\not\in S}\mathcal{H}_{v,{\bf{G}}}$. Gr\^ace \`a
l'isomorphisme de Satake, les alg\`ebres $\mathcal{H}_{v,{\bf{G}}}$ et $\mathcal{H}^S_{{\bf{G}}}$ sont
commutatives, donc leurs repr\'esentations irr\'eductibles sont simplement
des caract\`eres.
De plus, l'alg\`ebre $\mathcal{H}^S_{{\bf{G}}}$ agit sur la cohomologie (de Betti) $\H^*_c(X_K^{{\bf{G}}},A)$, pour tout anneau commutatif $A$.
Notons que ces constructions sont possibles pour n'importe quel groupe
r\'eductif connexe ${\bf{G}}$.
On peut alors faire la chose suivante. On prend comme avant ${\bf{G}}={\bf {Sp}}_{2n}$,
et on voit ${\bf{GL}}_n$ comme le quotient de Levi d'un sous-groupe parabolique
maximal de ${\bf{G}}$.
Soient~$\pi$ et $\Pi$ comme plus haut.
On choisit le sous-groupe compact ouvert $K$ de ${\bf{G}}({\mathbb{A}_f})$ tel que $\Pi^\mathrm{K}\not=0$.
Alors $\Pi$ correspond \`a un caract\`ere $\varphi$ de $\mathcal{H}^S_{{\bf{G}}}$ qui appara\^it comme
un sous-quotient de $\H^*_c(X_K^{{\bf{G}}},\mathbb{C})$. Le choix d'un isomorphisme $\mathbb{C}\simeq
\oQ_p$ permet de voir $\varphi$ comme un caract\`ere \`a valeurs dans
$\oQ_p$, et on montre qu'il existe une extension finie $E$ de $\mathbb{Q}_p$ telle que
$\varphi$ soit \`a valeurs dans $\mathcal{O}_E$. La question devient alors de savoir
si l'on peut
trouver une suite $(\Pi_i)_{i\in\mathbb{N}}$
de repr\'esentations automorphes cohomologiques \emph{cuspidales} de
${\bf{G}}(\mathbb{A})$ telle que $(\Pi_i)^{K^S}\not=0$ pour tout $i$ (o\`u
$K^S=\prod_{v\not\in S}K_v$) et que, si
$\varphi_i:\mathcal{H}^S_{{\bf{G}}}\longrightarrow\oQ_p$ est le caract\`ere associ\'e \`a $\Pi_i$ comme
plus haut, on ait
\[\lim_{i\rightarrow\infty}\varphi_i(x)=\varphi(x)\]
pour tout $x\in\mathcal{H}^S_{{\bf{G}}}$, o\`u on prend la limite pour la topologie $p$-adique sur
$\oQ_p$.
Harris, Lan, Taylor et Thorne ont \'et\'e les premiers \`a donner une
r\'eponse affirmative \`a cette question, ce qui leur a permis de prouver
le th\'eor\`eme suivant :
\begin{theo}[\cite{HLTT}]\label{th:HLTT}
La conjecture \ref{conj:LC} est vraie si
$\pi$ est cohomologique.\footnote{Et on a un r\'esultat similaire pour les
repr\'esentations automorphes de ${\bf{GL}}_n(\mathbb{A}_F)$, si $F$ est un corps de
nombres totalement r\'eel ou CM. Notons que la preuve dans le cas CM utilise les
vari\'et\'es de Shimura des groupes unitaires quasi-d\'eploy\'es au lieu
de celles des groupes symplectiques.}
\end{theo}
Rappelons que la m\'ethode esquiss\'ee ci-dessus ne donne pas directement la
repr\'esentation $\rho_\pi$, mais plut\^ot quelque chose qui ressemble
\`a $\rho_\pi\oplus(\rho_\pi)^*$. Il y a une derni\`ere \'etape qui consiste
\`a extraire $\rho_\pi$ de cette repr\'esentation, et que nous ignorerons
totalement (voir la section V.3 \cite{S}).
Scholze et Boxer ont reprouv\'e ce r\'esultat (ind\'ependamment de Harris-Lan-Taylor-Thorne
et ind\'ependamment l'un de l'autre) et l'ont g\'en\'eralis\'e aux caract\`eres
de $\mathcal{H}^S_{{\bf{GL}}_n}$ apparaissant dans la torsion de $H^*_c(X_K^{{\bf{GL}}_n},\mathbb{Z})$. (Noter
qu'il s'agit maintenant de l'alg\`ebre de Hecke et de l'espace localement
sym\'etrique pour ${\bf{GL}}_n$, et non ${\bf {Sp}}_{2n}$.) Plus pr\'ecis\'ement, on a la
conjecture suivante, due \`a Ash :
\begin{conj}[\cite{As1},\cite{A2}] \label{conj:A}Soit $S$ un ensemble fini
de nombres premiers. Si $\varphi:\mathcal{H}^S_{{\bf{GL}}_n}\longrightarrow\mathbb{F}_p$ est un caract\`ere
qui appara\^it dans $\H^*_c(X_K^{{\bf{GL}}_n},\mathbb{F}_p)$, pour $K=\prod_v K_v$ un sous-groupe
compact ouvert de ${\bf{GL}}_n({\mathbb{A}_f})$ tel que $K_v={\bf{GL}}_n(\mathbb{Z}_v)$ si $v\not\in S$,
alors il existe une repr\'esentation semi-simple
$\rho:\Gal(\oQ/\mathbb{Q})\longrightarrow{\bf{GL}}_n(\mathbb{F}_p)$ telle que, pour tout $v\not\in S\cup\{p\}$,
$\varphi_{|\mathcal{H}_{v,{\bf{GL}}_n}}$ et $\rho_{|\Gal(\oQ_p/\mathbb{Q}_p)}$ se correspondent par
l'isomorphisme de Satake.\footnote{Voir les explications apr\`es la conjecture
\ref{conj:LC}.}
\end{conj}
\section{\'Enonc\'e du th\'eor\`eme principal et strat\'egie de la preuve}
\begin{theo}[Scholze \cite{S}, Boxer]\label{th}
La conjecture \ref{conj:A} est
vraie.\footnote{Comme dans le cas du th\'eor\`eme \ref{th:HLTT}, Scholze et
Boxer prouvent en fait un r\'esultat valable pour le groupe ${\bf{GL}}_n$ sur
un corps de nombres totalement r\'eel ou CM.}
\end{theo}
Notons que l'on a en fait une version du th\'eor\`eme ci-dessus pour la
cohomologie \`a coefficients dans $\mathbb{Z}/p^m\mathbb{Z}$, pour tout entier strictement
positif $m$ (mais l'\'enonc\'e est un peu plus compliqu\'e \`a formuler, voir le
th\'eor\`eme V.3.1 de \cite{S}). En passant \`a la limite sur $m$, on obtient
donc une nouvelle preuve du th\'eor\`eme \ref{th:HLTT}. Dans ce texte, nous
nous concentrerons pour simplifier sur le cas o\`u $m=1$.
On va pr\'esenter la preuve de Scholze (dans les grandes lignes),
qui utilise la th\'eorie des
espaces perfecto\"ides (voir l'article introductif \cite{S2} de Scholze ou
l'expos\'e \cite{Fo} de Fontaine au s\'eminaire Bourbaki). La preuve de
Boxer n'utilise pas cette th\'eorie, mais les d\'etails de cette preuve ne
sont pas connus de l'auteur.
On se ram\`ene tout d'abord \`a un \'enonc\'e sur la torsion dans la
cohomologie de certaines vari\'et\'es de Shimura, de la mani\`ere suivante.
Comme dans la section \ref{1.5}, on pose \mbox{${\bf{G}}={\bf {Sp}}_{2n}$,} et on voit
${\bf{GL}}_n$ comme le quotient de Levi d'un sous-groupe parabolique maximal ${\bf P}$
de ${\bf{G}}$. Pour tout nombre premier $v$, l'application {\og terme constant le
long de ${\bf P}$\fg}
(voir par exemple la formule (19) de \cite{C}) d\'efinit un morphisme
injectif d'alg\`ebres $\mathcal{H}_{v,{\bf{G}}}\longrightarrow\mathcal{H}_{v,{\bf{GL}}_n}$. D'o\`u, si $S$ est un
ensemble fini de nombres premiers, un morphisme $\mathcal{H}^S_{{\bf{G}}}\longrightarrow
\mathcal{H}^S_{{\bf{GL}}_n}$.
D'autre part, soit $K$ un sous-groupe ouvert compact de ${\bf{G}}({\mathbb{A}_f})$. L'espace
localement sym\'etrique $X_K^{{\bf{G}}}$ n'est pas compact, mais il admet
une compactification $\overline{X}_K^{{\bf{G}}}$ appel\'ee \emph{compactification
de Borel-Serre} et d\'efinie dans \cite{BS}, qui est une vari\'et\'e
analytique r\'eelle \`a coins ayant la m\^eme cohomologie
(sans supports) que $X_K^{{\bf{G}}}$
et telle que le bord $\overline{X}_K^{{\bf{G}}}-X_K^{{\bf{G}}}$ admette
une stratification par des sous-vari\'et\'es analytiques r\'eelles de la
forme $X_{K_Q}^{{\bf Q}}$, pour ${\bf Q}$ un sous-groupe parabolique de ${\bf{G}}$ et $K_Q$
un sous-groupe compact ouvert de ${\bf Q}({\mathbb{A}_f})$. En particulier, on a
des strates correspondant au sous-groupe parabolique maximal ${\bf P}$.
Comme on a un morphisme surjectif $\pi:{\bf P}\longrightarrow{\bf{GL}}_n$ (qui identifie ${\bf{GL}}_n$
au quotient de Levi de ${\bf P}$), on obtient, pour tout sous-groupe compact
ouvert $K_P$ de ${\bf P}({\mathbb{A}_f})$, un morphisme surjectif $X_{K_P}^{{\bf P}}\longrightarrow
X_{\pi(K_P)}^{{\bf{GL}}_n}$, qui se trouve \^etre un fibr\'e en $(S^1)^N$, o\`u $N$
est la dimension du radical unipotent de ${\bf P}$. En utilisant ce fait
et la suite exacte longue d'excision, on montre que tout caract\`ere
$\varphi$ de $\mathcal{H}^S_{{\bf{GL}}_n}$ qui appara\^it dans un $H^*_c(X_{K_{{\bf{GL}}_n}}^{{\bf{GL}}_n},
\mathbb{F}_p)$ appara\^it aussi dans un $H^*_c(X_K^{{\bf{G}}},\mathbb{F}_p)$ (c'est-\`a-dire
que le compos\'e $\varphi':\mathcal{H}^S_{{\bf{G}}}\longrightarrow\mathbb{F}_p$ de $\varphi$ et du
morphisme $\mathcal{H}^S_{{\bf{G}}}\longrightarrow\mathcal{H}^S_{{\bf{GL}}_n}$ ci-dessus
appara\^it dans
un $H^*_c(X_K^{{\bf{G}}},\mathbb{F}_p)$). Pour les d\'etails, voir le d\'ebut de la section
V.2 de \cite{S}.
On se ram\`ene donc \`a montrer le th\'eor\`eme suivant :
\begin{theo}[Th\'eor\`emes I.5 et IV.3.1 de \cite{S}]\label{th:princ}\footnote{Ce th\'eor\`eme
est en fait valable pour tous les groupes ${\bf{G}}$ d\'efinissant des vari\'et\'es
de Shimura de type Hodge.}
Soit $\varphi:\mathcal{H}^K_{{\bf{G}}}\longrightarrow\mathbb{F}_p$ un caract\`ere apparaissant dans un
$H^*_c(X_K^{{\bf{G}}},\mathbb{F}_p)$ (pour un $K$ hypersp\'ecial aux places hors de~$S$,
comme au-dessus du th\'eor\`eme \ref{th:HLTT}). Alors il existe
une repr\'esentation automorphe cuspidale cohomologique $\pi$ de
${\bf{G}}(\mathbb{A})$, non ramifi\'ee en les places hors de $S$ et telle que, si
$\psi$ est le
caract\`ere correspondant de $\mathcal{H}^S_{{\bf{G}}}$, vu comme un morphisme
$\mathcal{H}^S_{{\bf{G}}}\longrightarrow\overline{\mathbb{Z}}_p$, alors la r\'eduction modulo $p$ de $\psi$
est \'egale \`a $\varphi$.
\end{theo}
Pour prouver ce th\'eor\`eme, on veut
comparer un caract\`ere de $\mathcal{H}^S_{{\bf{G}}}$ qui appara\^it dans
un groupe de cohomologie de Betti (ou \'etale) $H^*_c(X_K^{{\bf{G}}},\mathbb{F}_p)$
avec un caract\`ere venant d'une repr\'esentation automorphe cuspidale.
On peut voir les formes automorphes cuspidales sur ${\bf{G}}(\mathbb{A})$ comme les
sections d'un certain faisceau coh\'erent sur $X_K^{{\bf{G}}}$, et Scholze
a justement un th\'eor\`eme de comparaison entre la cohomologie d'un
$\mathbb{F}_p$-syst\`eme local $\mathbb{L}$ sur un espace adique
propre et lisse $X$ sur $\mathbb{C}_p$
et celle de $\mathbb{L}\otimes\mathcal{O}^+_X/p$ (corollaire 5.11 de
\cite{S1} et th\'eor\`eme 3.3 de \cite{S2}). Voir la section \ref{comp}
pour des rappels sur ce th\'eor\`eme.
Dans notre cas, le syst\`eme local $\mathbb{L}$ sera le syst\`eme local
trivial, et on compare sa cohomologie \`a celle du {\og faisceau des formes
cuspidales\fg} sur $X_K^{{\bf{G}}}$, ou plut\^ot sur sa compactification de
Baily-Borel $X_{K}^{{\bf{G}},*}$. Voir la section \ref{cc}.
Il faut ensuite passer de la cohomologie du faisceau des formes cuspidales
\`a ses sections globales. Or, Scholze a prouv\'e que la cohomologie d'un
faisceau coh\'erent sur un espace perfecto\"ide affino\"ide est presque
nulle. Les vari\'et\'es de Shimura ne sont pas perfecto\"ides, mais
Scholze prouve le r\'esultat suivant : Soit $p$ un nombre
premier. Si on fixe
un sous-groupe compact ouvert $K^p$ de ${\bf{G}}({\mathbb{A}_f}^p)$, o\`u ${\mathbb{A}_f}^p=\prod'_{v\not=p}
\mathbb{Q}_v$, alors la limite projective
$X_{K^p}^{{\bf{G}}}$ des $X_{K_p K^p}^{{\bf{G}}}$ lorsque $K_p$ parcourt
les sous-groupes compacts ouverts de ${\bf{G}}(\mathbb{Q}_p)$ {\og est\fg} un espace
perfecto\"ide (dans un sens \`a pr\'eciser), qu'on appelle parfois {\og vari\'et\'e
de Shimura de niveau infini en $p$\fg}.
De plus,
on a un r\'esultat similaire
pour les compactifications de Baily-Borel de ces vari\'et\'es.
La preuve de ce r\'esultat utilise de mani\`ere essentielle le morphisme
des p\'eriodes de Hodge-Tate, qui n'est d\'efini que sur la vari\'et\'e
de Shimura de niveau infini. Voir la section \ref{HT}.
On se place donc sur la vari\'et\'e de Shimura de niveau infini en $p$.
En utilisant un recouvrement (explicite) par des ouverts affino\"ides
perfecto\"ides, on montre que l'on peut approximer la cohomologie du
faisceau des formes cuspidales par des sections de ce faisceau sur ces
ouverts, qui sont des vecteurs propres pour l'action de l'alg\`ebre de
Hecke $\mathcal{H}^S_{{\bf{G}}}$.
Il faut encore prolonger ces sections \`a toute la vari\'et\'e
de Shimura, sans changer les valeurs propres pour l'action de l'alg\`ebre de
Hecke. La m\'ethode classique utilise l'invariant de Hasse, qui ne suffit
pas ici. Cependant, en utilisant le morphisme des p\'eriodes de
Hodge-Tate (qui est \'equivariant sous l'action des correspondances de Hecke
en dehors de $p$), on construit de {\og faux\fg} invariants de Hasse qui
jouent le m\^eme r\^ole et permettent de finir la preuve. Voir la section
\ref{faux}.
Notons que la preuve du th\'eor\`eme \ref{th} n'est pas l'unique
application des m\'ethodes de \cite{S}. Voir la section \ref{appl} pour
quelques autres applications.
\section{Un th\'eor\`eme de comparaison}\label{comp}
Le th\'eor\`eme suivant est le d\'ebut de la preuve par Scholze du th\'eor\`eme
de comparaison entre cohomologie de de Rham et cohomologie \'etale $p$-adique.
\begin{theo}[Th\'eor\`eme 5.1 de \cite{S1} ou th\'eor\`eme 3.3 de \cite{S2}]
\label{th:comp1}
Soit $C$ une extension compl\`ete alg\'ebriquement close de $\mathbb{Q}_p$,
d'anneau des entiers $\mathcal{O}_C$, et soit $X$ un espace adique propre et lisse
\footnote{En fait, l'hypoth\`ese de lissit\'e est superflue, voir le th\'eor\`eme 3.17 de \cite{S2}.}
sur $(C,\mathcal{O}_C)$. Alors, pour tout $\mathbb{F}_p$-syst\`eme local $\mathbb{L}$
sur $X$, on a des presque isomorphismes
\[H^i(X_{\et},\mathbb{L})\otimes\mathcal{O}^a/p\simeq\H^i(X_{\et},\mathcal{O}_X^{+a}/p),\]
o\`u $\mathcal{O}_X^+\subset\mathcal{O}_X$ est le faisceau des fonctions born\'ees par
$1$ sur $X$.
\end{theo}
Pour des rappels et des r\'ef\'erences sur les espaces adiques, voir le
d\'ebut de la section~2 de \cite{Fo}. On peut par exemple prendre pour $X$
l'espace adique associ\'e \`a un sch\'ema propre et lisse sur $C$, mais une
des forces du r\'esultat de Scholze est qu'il s'applique aussi aux espaces
adiques ne venant pas d'un sch\'ema. Pour des rappels sur le langage des
presque math\'ematiques, voir la section 1.6 de \cite{Fo}; un {\og presque
isomorphisme\fg} (ou un isomorphisme de presque $\mathcal{O}_C$-modules)
est un morphisme de $\mathcal{O}_C$-modules dont le noyau et le
conoyau sont annul\'es par l'id\'eal maximal de $\mathcal{O}_C$, et les {\og $a$\fg} en
exposant sont l\`a pour rappeler que l'on consid\`ere les objets comme des
presque $\mathcal{O}_C$-modules.
Mentionnons aussi que le th\'eor\`eme ci-dessus est toujours valable si l'on
remplace $\mathcal{O}_C$ par un sous-anneau de valuation ouvert et born\'e $C^+$ de
$C$, et dans le cas relatif (c'est-\`a-dire pour les images directes par
un morphisme propre et lisse de vari\'et\'es analytiques rigides sur $C$,
cf. le corollaire 5.11 de \cite{S1}).
\noindent{\sc Preuve} (esquisse) --- Il y a trois \'etapes dans la preuve.
\begin{enumerate}
\item[(1)] (Cf. le th\'eor\`eme 4.9 de \cite{S1}.)
Si $X$ est une vari\'et\'e analytique rigide affino\"ide et
connexe sur $C$ (o\`u $C$ est comme dans l'\'enonc\'e), si $x\in X(C)$ et si
$\pi=\pi_1(X,x)$ (le groupe fondamental \'etale profini de $X$), alors, pour
tout $\mathbb{F}_p$-syst\`eme local $\mathbb{L}$ sur~$X$, les morphismes canoniques
\[H^i_{cont}(\pi,\mathbb{L}_x)\longrightarrow H^i(x_{\et},\mathbb{L})\]
sont des isomorphismes. (Autrement dit, $X$ est un $K(\pi,1)$ pour les coefficients
de $p$-torsion.)
Il faut montrer que toute classe de cohomologie dans un $H^i(X_{\et},\mathbb{L})$
pour $i>0$ est tu\'ee par un rev\^etement fini \'etale de $X$. On se ram\`ene
facilement au cas o\`u $\mathbb{L}$~est le syst\`eme local trivial $\mathbb{F}_p$.
Chaque rev\^etement fini \'etale de $X$ est affino\"ide, donc de la forme
$\Spa(A,A^+)$. En prenant une compl\'etion appropri\'ee de
la limite inductive de ces $C$-alg\`ebres~$A$, on obtient une
$C$-alg\`ebre perfecto\"ide $(A_\infty,A_\infty^+)$, dont le $\Spa$ est
un espace perfecto\"ide $X_\infty$, qui m\'erite le nom de rev\^etement
fondamental de $X$. (Voir \cite{Fo} pour une introduction aux espaces
perfecto\"ides.) Il s'agit maintenant de montrer que $\H^i(X_{\infty,\et},
\mathbb{F}_p)=0$ pour $i>0$. En utilisant le basculement (ou tilt), on se ram\`ene
\`a montrer le r\'esultat similaire pour l'espace perfecto\"ide $X_\infty^\flat$
sur le corps perfecto\"ide~$C^\flat$, qui est de caract\'eristique $p$.
Cela r\'esulte alors de la suite exacte d'Artin-Schreier $0\rightarrow\mathbb{F}_p
\rightarrow\mathcal{O}_{X_\infty^\flat}\rightarrow\mathcal{O}_{X_\infty^\flat}\rightarrow 0$ et
du fait que $X_\infty^\flat$ n'a pas de rev\^etement fini \'etale non trivial.
\item[(2)] On prouve ensuite que, si $X$ est un espace adique propre et
lisse sur $C$ et $\mathbb{L}$ est un $\mathbb{F}_p$-syst\`eme local sur $X$, alors
les groupes de cohomologie $H^i(X_{\et},\mathbb{L}\otimes\mathcal{O}_X^+/p)$
sont presque de type fini, et presque nuls pour $i>2\dim X$ (lemme 5.8 de
\cite{S1}). L'id\'ee de la preuve est classique (et d\'ej\`a utilis\'ee par
Cartan et Serre pour les vari\'et\'es analyiques complexes et par Kiehl pour
les vari\'et\'es analytiques rigides; bien s\^ur, il faut faire marcher cette
id\'ee) : on calcule la cohomologie de $X$ en utilisant le complexe de {\v C}ech
d'un nombre fini de recouvrements par des ouverts affino\"ides dont chacun
raffine assez le pr\'ec\'edent. (Les m\'ethodes de (1) sont utilis\'ees pour
prouver que tous les groupes de cohomologie qui apparaissent sont
presque de type fini.)
\item[(3)] Enfin, pour prouver le th\'eor\`eme, on utilise la topologie
pro-\'etale de $X$. C'est une topologie de Grothendieck plus fine que la
topologie \'etale; l'id\'ee de base est simplement
que l'on autorise des recouvrements par des
limites projectives d'espaces \'etales sur $X$, mais les d\'etails techniques
ne sont pas totalement \'evidents (voir la section 3 de \cite{S1} pour la
d\'efinition rigoureuse). Ce qui rend cette topologie si utile est le fait que
\emph{tout espace adique localement noeth\'erien est pro-\'etale localement
perfecto\"ide} (voir la proposition 4.8 de \cite{S1}). On peut donc introduire
le faisceau structurel compl\'et\'e bascul\'e $\widehat{\mathcal{O}}^+_{X^\flat}$, qui
est un faisceau (de $p$-torsion) sur le site pro-\'etale $X_{\proet}$, et
l'utiliser pour calculer les $H^i(X_{\proet},\mathbb{L})$ via la suite exacte
longue de cohomologie pro-\'etale de la suite exacte
d'Artin-Schreier $0\rightarrow\mathbb{L}\rightarrow\mathbb{L}\otimes
\widehat{\mathcal{O}}^+_{X^\flat}\rightarrow\mathbb{L}\otimes\widehat{\mathcal{O}}^+_{X^\flat}
\rightarrow 0$ (une suite exacte de faisceaux sur $X_{\proet}$). On utilise
le r\'esultat de finitude de (2) pour montrer que les morphismes
de connexion (c'est-\`a-dire ceux allant d'un $H^i$ dans un $H^{i+1}$) dans
la suite exacte longue ci-dessus sont nuls. Notons qu'on a aussi utilis\'e
sans le dire un th\'eor\`eme de comparaison entre cohomologies \'etale
et pro-\'etale, voir le corollaire 3.17 de \cite{S1}.
\end{enumerate}
Le th\'eor\`eme \ref{th:comp1} admet une version plus g\'en\'erale (au moins pour
les espaces adiques provenant de sch\'emas), pour des
syst\`emes de coefficients constructibles, qui est celle dont on aura besoin.
\begin{theo}[Th\'eor\`eme 3.13 de \cite{S2}]\label{th:comp2}
Soit $C$ comme
dans le th\'eor\`eme \ref{th:comp1}, soit $X$ l'espace adique associ\'e \`a
un sch\'ema propre sur $C$, et soit $\mathbb{L}$ l'image inverse
sur $X_{\et}$ d'un $\mathbb{F}_p$-faisceau \'etale constructible sur ce sch\'ema.
Alors on a des presque isomorphismes
\[H^i(X_{\et},\mathbb{L})\otimes\mathcal{O}_C^a/p\simeq H^i(X_{\et},\mathbb{L}\otimes
\mathcal{O}^{+a}_X/p).\]
\end{theo}
Comme pour le th\'eor\`eme \ref{th:comp1}, on peut remplacer $\mathcal{O}_C$ par un
$C^+\subset C$ plus g\'en\'eral, et on a une version relative.
La preuve du th\'eor\`eme utilise, outre le th\'eor\`eme \ref{th:comp1} et la
r\'esolution des singularit\'es, le lemme simple suivant.
\begin{lemm}[\cite{S1}, lemme 3.14]
Soit $X$ un espace adique localement noeth\'erien sur
$\Spa(\mathbb{Q}_p,\mathbb{Z}_p)$, et soit $i:Z\longrightarrow X$ un sous-espace ferm\'e de $X$. Alors
le morphisme $i^*\mathcal{O}_X^+/p\longrightarrow\mathcal{O}_Z^+/p$ est un isomorphisme (de faisceaux sur
$X_{\et}$).
\end{lemm}
Pour l'application aux vari\'et\'es de Shimura, il est plus naturel d'introduire
d'abord la vari\'et\'e de Shimura perfecto\"ide.
\section{Vari{\'e}t{\'e} de Shimura perfecto{\"i}de et morphisme de Hodge-Tate}
\label{HT}
On utilise \`a nouveau les notations de la section \ref{1.4}, sauf que
l'on prend ici
\mbox{${\bf{G}}={\bf{GSp}}_{2n}\subset{\bf{GL}}_{2n}$} (le groupe g\'en\'eral symplectique).
Pour tout
sous-groupe ouvert compact $K$ de ${\bf{G}}({\mathbb{A}_f})$ (assez petit), l'espace
localement sym\'etrique $X_K=X_K^{{\bf{G}}}$ est l'ensemble des points complexes d'une
vari\'et\'e quasi-projective lisse sur $\mathbb{Q}$, que l'on notera encore $X_K$.
Cette vari\'et\'e n'est pas projective (sauf si $n=0$), mais elle admet une
compactification canonique $X^*_K$, qui est une vari\'et\'e projective normale
sur $\mathbb{Q}$, appel\'ee compactification minimale, de Baily-Borel ou
de Satake-Baily-Borel (voir l'article \cite{BB} de Baily-Borel pour la
construction sur $\mathbb{C}$, et le livre \cite{CF} de Chai et Faltings pour la
construction sur $\mathbb{Q}$\footnote{Et m\^eme sur $\mathbb{Z}_p$ si $K={\bf{G}}(\mathbb{Z}_p) K^p$ avec
$K^p\subset{\bf{G}}({\mathbb{A}_f}^p)$.}). Notons que $X^*_K$ est muni d'un
fibr\'e en droites ample canonique (sur $X_K$, c'est le d\'eterminant du
faisceau des formes diff\'erentielles
invariantes de degr\'e~$1$ sur le sch\'ema
ab\'elien universel sur $X_K$).
On fixe un nombre premier $p$, et
on note $\mathcal{X}_K$ et $\mathcal{X}^*_K$ les espaces
adiques sur $\Spa(\mathbb{Q}_p,\mathbb{Z}_p)$ associ\'es aux sch\'emas $X_K$ et $X^*_K$.
On note encore $\omega$ le fibr\'e en droites sur $\mathcal{X}^*_K$ associ\'e au
fibr\'e en droites $\omega$ sur $X^*_K$.
D'autre part, soit $V=\mathbb{Q}^{2n}$, muni de la forme symplectique d\'efinie dans la
section \ref{1.4}. On note $\mathrm{Fl}$ la vari\'et\'e (sur $\mathbb{Q}$) des sous-espaces
totalement isotropes $W$ de dimension~$n$ de $V$, qui est munie d'un fibr\'e
en droites ample tautologique $\omega_\mathrm{Fl}=(\bigwedge^n W)^*$.
On note ${\mathscr{F}\!\ell}$ l'espace adique
sur $\Spa(\mathbb{Q}_p,\mathbb{Z}_p)$ associ\'e \`a $\mathrm{Fl}$, et $\omega_{\mathscr{F}\!\ell}$ le fibr\'e en droite
sur ${\mathscr{F}\!\ell}$ correspondant \`a $\omega_\mathrm{Fl}$.
Enfin, on note $\mathbb{Q}_p^{\rm cycl}$ le compl\'et\'e de $\mathbb{Q}_p(\mu_{p^\infty})$ (qui est
l'extension de $\mathbb{Q}_p$ engendr\'ee par toutes les racines de $1$ d'ordre une
puissance de $p$), et $\mathbb{Z}_p^{\rm cycl}$ son anneau des entiers.
Notons que $\mathbb{Q}_p^{\rm cycl}$ est un corps perfecto\"ide.
Le th\'eor\`eme suivant est l'un des r\'esultats centraux de l'article
\cite{S}.
\begin{theo}[Th\'eor\`emes III.1.2 et III.3.17 de
\cite{S}]\label{th:perf}
On fixe un sous-groupe compact ouvert assez petit $K^p$ de
${\bf{G}}({\mathbb{A}_f}^p)$.
\begin{enumerate}
\item[\rm (i)] Il existe un espace perfecto\"ide $\mathcal{X}^*_{\Gamma(p^\infty)K^p}=
\mathcal{X}^*_{\Gamma(p^\infty)}$\footnote{Voir plus bas pour l'explication du {\og $\Gamma(
p^\infty)$\fg}.}
sur $\mathbb{Q}_p^{cycl}$,
unique \`a isomorphisme unique pr\`es, tel que
\[\mathcal{X}^*_{\Gamma(p^\infty)K^p}\sim\varprojlim_{K_p}\mathcal{X}^*_{K_p K^p},\]
o\`u $K_p$ parcourt l'ensemble des sous-groupes compacts ouverts de ${\bf{G}}(\mathbb{Q}_p)$.
\footnote{Voir la d\'efinition 2.4.1 de \cite{SW} pour $\sim$. En particulier,
d'apr\`es le th\'eor\`eme 2.4.7 de \cite{SW},
ceci implique que le topos \'etale $\mathcal{X}^*_{\Gamma(p^\infty)K^p,\et}$ est la limite projective
des $\mathcal{X}^*_{K_p K^p,\et}$.}
\item[\rm (ii)] On a une application des p\'eriodes de Hodge-Tate
${\bf{G}}(\mathbb{Q}_p)$-\'equivariante $\pi_{HT}:\mathcal{X}^*_{\Gamma(p^\infty)K^p}\longrightarrow{\mathscr{F}\!\ell}$, qui commute avec
tous les op\'erateurs de Hecke hors de $p$,\footnote{C'est-\`a-dire donn\'e par
des $g\in{\bf{G}}({\mathbb{A}_f})$ de composante en $p$ \'egale \`a $1$.} pour l'action triviale
de ces op\'erateurs sur ${\mathscr{F}\!\ell}$.
\item[\rm (iii)] On a un isomorphisme canonique $\omega=\pi_{HT}^*\omega_{\mathscr{F}\!\ell}$.
\item[\rm (iv)] On a un recouvrement de ${\mathscr{F}\!\ell}$ par des ouverts affino\"ides $U$
\footnote{Explicites, ce sont les ${\mathscr{F}\!\ell}_J$ d\'efinis plus bas.} tels que :
\begin{enumerate}
\item $V=\pi_{HT}^{-1}(U)$ est perfecto\"ide affino\"ide;
\item pour tout $K_p\subset{\bf{G}}(\mathbb{Q}_p)$ assez petit, il existe $V_{K_p}\subset
\mathcal{X}^*_{K_p K^p}$ d'image inverse $V$ dans $\mathcal{X}^*_{\Gamma(p^\infty)K^p}$;
\item le morphisme suivant est d'image dense :
\[\varinjlim_{K_p} H^0(V_{K_p},\mathcal{O}_{\mathcal{X}^*_{K_p K^p}})\longrightarrow H^0(V,\mathcal{O}_{\mathcal{X}^*_{K^p}}).\]
\end{enumerate}
\end{enumerate}
\end{theo}
De plus, le th\'eor\`eme ci-dessus s'\'etend \`a toutes les vari\'et\'es
de Shimura de type Hodge (th\'eor\`eme IV.1.1 de \cite{S}), en particulier
aux vari\'et\'es de Shimura de type PEL.
La pr\'esence du bord de $\mathcal{X}^*_K$ cause quelques probl\`emes techniques (dont
la plupart sont trait\'es dans les sections II.2 et II.3 de \cite{S}).
Nous allons
donner une esquisse de preuve qui ignore ces probl\`emes.
On fixe $K^p\subset{\bf{G}}({\mathbb{A}_f}^p)$, et on note, pour tout $m\in\mathbb{N}^*$,
\[\Gamma_0(p^m)=\{\gamma\in{\bf{GSp}}_{2n}(\mathbb{Z}_p)|\gamma=\begin{pmatrix}* & * \\
0 & * \end{pmatrix}\mod p^m\mbox{ et }\det(\gamma)=1\mod p^m\},\]
\[\Gamma(p^m)=\{\gamma\in{\bf{GSp}}_{2n}(\mathbb{Z}_p)|\gamma=\begin{pmatrix}1 & 0 \\
0 & 1 \end{pmatrix}\mod p^m\}.\]
Dans la premi\`ere partie de la preuve, on prouve (i) pour la limite sur
les $K_p=\Gamma_0(p^m)$ et dans un voisinage du lieu anticanonique. Expliquons
ce que ceci signifie.
Soit $A\longrightarrow S$ un sch\'ema ab\'elien sur un sch\'ema $S$ de caract\'eristique $p$. Si $A^{(p)}$ est le changement de base de $A\longrightarrow S$ par le morphisme de
Frobenius absolu de $S$, alors il existe une unique isog\'enie $V:A^{(p)}\longrightarrow
A$ (appel\'ee \emph{Verschiebung}) dont le compos\'e (dans les deux sens)
avec le morphisme de
Frobenius relatif $A\longrightarrow A^{(p)}$ est la multiplication par $p$.
Cette isog\'enie induit un morphisme $V^*:\omega_{A/S}\longrightarrow\omega_{A^{(p)}/S}=
\omega_{A/S}^{\otimes p}$,\footnote{$\omega_{A/S}=\wedge^{\dim(A/S)}(\Omega^1_{A/S})$.}
c'est-\`a-dire une section globale $\Ha(A/S)$
de $\omega_{A/S}^{\otimes(p-1)}$, appel\'ee \emph{invariant de Hasse de $A$}.
Rappelons que $\Ha(A/S)$ est inversible si et seulement si $A$ est ordinaire
\footnote{C'est-\`a-dire que $A[p](x)$ a exactement $p^{\dim(A/S)}$ \'el\'ements
pour tout point g\'eom\'etrique $x$ de $S$.} (lemme III.2.5 de \cite{S}).
L'invariant de Hasse permet donc de mesurer \`a quel point $A$ est loin
d'\^etre ordinaire.
Dans la suite, si on \'ecrit $p^\varepsilon$,
avec $\varepsilon\in[0,1[$, on supposera
toujours que $\varepsilon$ est dans l'image de $\mathbb{Z}_p^{\rm cycl}$ par la valuation
$p$-adique, et $p^\varepsilon$ sera un \'el\'ement (quelconque) de $\mathbb{Z}_p^{\rm cycl}$
de valuation $\varepsilon$.
On note $\mathcal{X}=\mathcal{X}_{{\bf{G}}(\mathbb{Z}_p)K^p}$ (la vari\'et\'e de Shimura de niveau trivial en $p$).
Pour $\varepsilon$ comme ci-dessus, on d\'efinit un ouvert $\mathcal{X}(\varepsilon)$
par la condition $|\Ha|\geq |p^\varepsilon|$ (o\`u on prend l'invariant de Hasse
de la vari\'et\'e ab\'elienne universelle). Pour que cela ait un sens, on doit
d'abord d\'efinir $\mathcal{X}(\varepsilon)$ sur un mod\`ele entier de $\mathcal{X}$ (qui existe
car on a pris le niveau trivial en $p$) en utilisant un rel\`evement local
de l'invariant de Hasse, et montrer que le r\'esultat ne d\'epend pas de
ce rel\`evement; voir la d\'efinition III.2.11 et le lemme III.2.12
de \cite{S}. On note aussi $\Abf\longrightarrow\mathcal{X}$ l'espace adique associ\'e au
sch\'ema ab\'elien universel,
et $\Abf(\varepsilon)=\Abf\times_\mathcal{X}\X(\varepsilon)$. Si $K_p\subset{\bf{G}}(\mathbb{Q}_p)$ est
quelconque, on note $\mathcal{X}_{K_p K^p}(\varepsilon)=\mathcal{X}_{K_p K^p}\times_\mathcal{X}\X(\varepsilon)$.
On fixe $\varepsilon$ et $m\in\mathbb{N}$. Alors, sur les mod\`eles entiers, le
sch\'ema ab\'elien universel $\Abf(p^{-m}\varepsilon)\longrightarrow\mathcal{X}(p^{-m}\varepsilon)$
admet un sous-groupe canonique de niveau $m$, c'est-\`a-dire un sous-groupe
$C_m$ du groupe des points de $p^m$-torsion qui est \'egal \`a $\Ker(F^m)$
modulo $p^{1-\varepsilon}$ ($F$ est le Frobenius absolu); voir le (ii) du
th\'eor\`eme III.2.14 de \cite{S}. Comme $X_{\Gamma_0(p^m)\mathrm{K}^p}$ param\`etre les
vari\'et\'es ab\'eliennes (principalement polaris\'ees) $A$
munies d'une structure de niveau $K^p$ et d'un sous-groupe totalement
isotrope de
$A[p^m]$, l'existence de $C_m$ donne un morphisme $\mathcal{X}(p^{-m}\varepsilon)
\longrightarrow\mathcal{X}_{\Gamma_0(p^m)}$. Scholze prouve alors les r\'esultats suivants (th\'eor\`eme
III.2.14(iii),(iv) et lemme III.2.16 de \cite{S}) :
\begin{enumerate}
\item[(a)] Pour tout $m\geq 1$, on a un diagramme cart\'esien
\[\xymatrix{\mathcal{X}(p^{-m-1}\varepsilon)\ar[r]\ar[d] & \mathcal{X}_{\Gamma_0(p^{m+1})K^p}\ar[d] \\
\mathcal{X}(p^{-m}\varepsilon)\ar[r] & \mathcal{X}_{\Gamma_0(p^{m})K^p}}\]
o\`u les fl\`eches horizontales sont celles d\'efinies ci-dessus, la fl\`eche
verticale de droite est la projection canonique et la fl\`eche verticale de
gauche est un rel\`evement du Frobenius.
\item[(b)] L'image $X_{\Gamma_0(p^m)K^p}(\varepsilon)_a$ du morphisme
$\mathcal{X}(p^{-m}\varepsilon)\longrightarrow\mathcal{X}_{\Gamma_0(p^m)}$ est un sous-espace ouvert et ferm\'e
de $\mathcal{X}_{\Gamma^0(p^m)K^p}(\varepsilon)$, d\'efini par la condition que
$D[p]\cap C_1=\{0\}$, o\`u $D$ est le sous-groupe totalement isotrope donn\'e
par la structure de niveau $\Gamma_0(p^m)$.\footnote{Le {\og a\fg} signifie
{\og anticanonique\fg}.}
De plus,
$X_{\Gamma_0(p^m)K^p}(\varepsilon)_a$ est affino\"ide pour $m$ assez grand.
\end{enumerate}
On d\'eduit des r\'esultats ci-dessus qu'il existe un espace perfecto\"ide
affino\"ide $\mathcal{X}_{\Gamma_0(p^\infty)}(\varepsilon)_a$ tel que
\[\mathcal{X}_{\Gamma_0(p^\infty)}(\varepsilon)_a\sim\varprojlim_m\mathcal{X}_{\Gamma_0(p^m)K^p}
(\varepsilon)_a.\]
La deuxi\`eme \'etape consiste \`a passer au niveau $\Gamma(p^m)$, afin
d'obtenir un espace affino\"ide perfecto\"ide $\mathcal{X}_{\Gamma(p^\infty)}
(\varepsilon)_a$ tel que
\[\mathcal{X}_{\Gamma(p^\infty)}(\varepsilon)_a\sim\varprojlim_m\mathcal{X}_{\Gamma(p^m)K^p}
(\varepsilon)_a.\]
La seule difficult\'e vient du fait que les morphismes $\mathcal{X}^*_{\Gamma(p^m)K^p}\longrightarrow
\mathcal{X}^*_{\Gamma_0(p^m)K^p}$ ne sont pas \'etales au bord. Voir la section III.2.5
de \cite{S} pour les d\'etails.
Pour tout espace adique $\mathcal{Y}$, on note $|\mathcal{Y}|$ l'espace topologique sous-jacent.
Soit
\[|X_{\Gamma(p^\infty)}|=\varprojlim_m|\mathcal{X}_{\Gamma(p^m)K^p}|.\]
On a trouv\'e un ouvert $|\mathcal{X}_{\Gamma(p^\infty)}(\varepsilon)_a|$
de cet espace dont chaque point admet un voisinage
(venant d'un) perfecto\"ide affino\"ide. Notons que le groupe ${\bf{G}}(\mathbb{Q}_p)$ agit
de mani\`ere continue
sur $|\mathcal{X}_{\Gamma(p^\infty)}|$, et que la condition {\og avoir un voisinage
perfecto\"ide affino\"ide\fg} est stable par cette action. L'id\'ee est maintenant
de montrer que les ${\bf{G}}(\mathbb{Q}_p)$-translat\'es de $|\mathcal{X}_{\Gamma(p^\infty)}(\varepsilon)_a|$ recouvrent $|\mathcal{X}_{\Gamma(p^\infty)}|$. Pour cela, Scholze utilise le morphisme
des p\'eriodes de Hodge-Tate.
Si $A$ est une vari\'et\'e ab\'elienne sur une extension compl\`ete et
alg\'ebriquement close $C$ de $\mathbb{Q}_p$, la suite spectrale de Hodge-Tate (voir par exemple
le th\'eor\`eme 3.20 de \cite{S2}) donne une suite exacte courte
\[0\longrightarrow (\Lie A)(1)\longrightarrow T_p A\otimes_{\mathbb{Z}_p} C\longrightarrow (\Lie A^*)^*\longrightarrow 0,\]
o\`u $T_p A$ est le module de Tate $p$-adique de $A$. Si $A$ vient d'un point
de $|\mathcal{X}_{\Gamma(p^\infty)}|$, alors on a un isomorphisme $T_p A\simeq\mathbb{Z}_p^{2n}$
(donn\'e par la structure de niveau infini en $p$), donc la suite exacte
ci-dessus d\'efinit un point de ${\mathscr{F}\!\ell}(C)$. On obtient ainsi une application
${\bf{G}}(\mathbb{Q}_p)$-\'equivariante $|\pi_{HT}|:|\mathcal{X}_{\Gamma(p^\infty)}|\longrightarrow|{\mathscr{F}\!\ell}|$, dont on
montre qu'elle est continue en regardant ce qui se passe sur des voisinages
pro-\'etales perfecto\"ides des points (lemme III.3.4 de \cite{S}). En effet,
la trivialisation du module de Tate $p$-adique $T_p A$ donn\'ee par la
structure de niveau infinie en $p$ existe en fait sur des voisinages
pro-\'etales des points de la vari\'et\'e de Shimura perfecto\"ide,
\footnote{Il est ici important de disposer de la topologie pro-\'etale. En
effet, sur un voisinage \'etale, on ne pourrait obtenir que des
trivialisations des groupes de points de torsion $A[p^N]$.}
ce qui permet, sur un tel voisinage,
de d\'efinir une filtration de Hodge-Tate relative, et donc de montrer que
le morphisme de Hodge-Tate est un morphisme d'espaces adiques (et en particulier
continu).
On a le plongement de Pl\"ucker ${\mathscr{F}\!\ell}\subset\mathbb{P}^{\left(\substack{2n \\ n}\right)-1}$
(d\'efini en envoyant $W$ sur $\wedge^n W$). On note $s_J$ les coordonn\'ees
homog\`enes sur le but de ce plongement, o\`u $J$ parcourt les sous-ensembles
de $\{1,\dots,2n\}$ de cardinal $n$. Si on fixe
un tel $J$, on a un ouvert affino\"ide
${\mathscr{F}\!\ell}_J$ de ${\mathscr{F}\!\ell}$, d\'efini par la condition $|s_{J'}|\leq |s_J|$, pour tout
$J'$.
En utilisant les r\'esultats de son article \cite{SW} avec Weinstein (plus
pr\'ecis\'ement, le th\'eor\`eme B de cet article), Scholze montre que
l'image inverse de ${\mathscr{F}\!\ell}(\mathbb{Q}_p)$ par $|\pi_{HT}|$ est le lieu ordinaire
(lemme III.3.6 de \cite{S}), puis qu'il existe un voisinage ouvert $U$ du
point $x=0^n\oplus\mathbb{Q}_p^n$ de ${\mathscr{F}\!\ell}$ tel que $|\pi_{HT}|^{-1}(U)\subset
|\mathcal{X}_{\Gamma_0(p^\infty)}(\varepsilon)_a|$ (pour $\varepsilon>0$ convenable).
Or, si $\gamma\in{\bf{G}}(\mathbb{Q}_p)$
est l'\'el\'ement diagonal $(p,\dots,p,1,\dots,1)$, alors $\gamma^N
{\mathscr{F}\!\ell}_{\{n+1,\dots,2n\}}$ pour $N$ assez grand. Comme ${\bf{G}}(\mathbb{Z}_p).{\mathscr{F}\!\ell}_{\{n+1,\dots,2n\}}=
{\mathscr{F}\!\ell}$, on en d\'eduit que ${\bf{G}}(\mathbb{Q}_p).U={\mathscr{F}\!\ell}$, ce qui permet de construire
un {\og atlas perfecto\"ide\fg} de $|\mathcal{X}_{\Gamma(p^\infty)}|$, donc d'obtenir l'espace
perfecto\"ide $\mathcal{X}_{\Gamma(p^\infty)}$ (corollaire III.3.11 de \cite{S}). Une
fois que l'on a $\mathcal{X}_{\Gamma(p^\infty)}$, il est assez facile de montrer que
$|\pi_{HT}|$ vient d'un morphisme d'espaces adiques
$\pi_{HT}:\mathcal{X}_{\Gamma(p^\infty)}\longrightarrow{\mathscr{F}\!\ell}$ (et un peu moins facile de montrer que ce
morphisme s'\'etend au bord, voir les corollaires III.3.12 et III.3.16 de
\cite{S}).
\section{Cohomologie compl\'et\'ee et faux invariants de Hasse}
\subsection{Cohomologie compl\'et\'ee}\label{cc}
On utilise les notations de la section pr\'ec\'edente, et on fixe
toujours un sous-groupe ouvert compact (assez petit) $K^p$ de ${\bf{G}}({\mathbb{A}_f}^p)$.
Rappelons que la cohomologie compl\'et\'ee de la vari\'et\'e de Shimura de
${\bf{G}}$ en niveau mod\'er\'e, dont la d\'efinition est due
\`a Calegari et Emerton (\cite{CE}), est donn\'ee par la formule :
\[\widetilde{H}^i_{c,K^p}=\varprojlim_m\varinjlim_{K_p} H^i_c(X^{{\bf{G}}}_{K_p K^p},
\mathbb{Z}/p^m\mathbb{Z}).\]
C'est un $\mathbb{Z}_p$-module $p$-adiquement complet, qui admet une action
de ${\bf{G}}(\mathbb{Q}_p)\times\mathcal{H}^S$, pour tout ensemble fini de nombres premiers $S$
tel que $K^p$ soit hypersp\'ecial en dehors de $S$.\footnote{Bien s\^ur,
$H^i_{c,K^p}$ admet en fait une action de toute l'alg\`ebre de Hecke mod\'er\'ee
de niveau $K^p$.} On fixe un tel $S$.
\begin{rema} La cohomologie compl\'et\'ee permet de d\'efinir
une notion assez g\'en\'erale
de {\og repr\'esentation automorphe $p$-adique\fg} pour un groupe r\'eductif
quelconque (c'\'etait d'ailleurs l'une des motivations de Calegari et Emerton).
\end{rema}
On note aussi, pour tout $m\in\mathbb{N}$,
\[\widetilde{H}^i_{c,K^p}(\mathbb{Z}/p^m\mathbb{Z})=\varinjlim_{K_p} H^i_c(X^{{\bf{G}}}_{K_p K^p},
\mathbb{Z}/p^m\mathbb{Z}).\]
Les r\'esultats des sections \ref{comp} et \ref{HT} permettent de montrer sans
trop de peine le r\'esultat suivant :
\begin{theo}[Th\'eor\`eme IV.2.1 de \cite{S}]\label{th:C} Soit $\Ical\subset
\mathcal{O}_{\mathcal{X}^*_{\Gamma(p^\infty)K^p}}$ l'id\'eal du bord, et $\Ical^+=\Ical\cap
\mathcal{O}_{\mathcal{X}^*_{\Gamma(p^\infty)K^p}}^+$. Soit $C$ une extension compl\`ete et
alg\'ebriquement close de $\mathbb{Q}_p$.
Alors on a des presque isomorphismes naturels
\[\widetilde{H}^i_{c,K^p}(\mathbb{Z}/p^m\mathbb{Z})\otimes_{\mathbb{Z}/p^m\mathbb{Z}}\mathcal{O}_C^a/p^m\simeq
H^i(\mathcal{X}^*_{\Gamma(p^\infty)K^p},\Ical^{+a}/p^m),\]
compatibles avec l'action de $\mathcal{H}^S$.\footnote{Voir l'\'enonc\'e du th\'eor\`eme
IV.2.1 de \cite{S} pour la d\'efinition de cette action.}.
\end{theo}
\begin{rema} En particulier, on peut voir la cohomologie compl\'et\'ee comme
la cohomologie \'etale de la vari\'et\'e de Shimura perfecto\"ide, ce qui
en donne une interpr\'etation naturelle.
\end{rema}
\subsection{Faux invariants de Hasse}\label{faux}
On a presque fini la preuve du th\'eor\`eme \ref{th:princ}, ou de sa version
plus pr\'ecise, le th\'eor\`eme IV.3.1 de \cite{S}, qui dit, en gros,
que, si on fixe $m\geq 1$, alors
tout caract\`ere de $\mathcal{H}^S$ qui appara\^it dans un $\widetilde{H}^i_{c,
\mathrm{K}^p}(\mathbb{Z}/p^m\mathbb{Z})$ appara\^it aussi dans un $H^0(\mathcal{X}^*_{K_p K^p},\omega^{mk}\otimes
\Ical)$, pour un $K_p\subset{\bf{G}}(\mathbb{Q}_p)$ compact ouvert (variable) et
un entier $k\geq 1$ (variable aussi). Ici, $\Ical$ d\'esigne comme en niveau
infini l'id\'eal du bord.
Pour finir la preuve, il faut pouvoir passer des groupes
$H^i(\mathcal{X}^*_{\Gamma(p^\infty)K^p},\Ical^+/p^m)$ (qui sont en niveau infini et en
degr\'e cohomologique quelconque) aux groupes
\mbox{$H^0(\mathcal{X}^*_{K_p K^p},\omega^{mk}\otimes\Ical)$.}
On utilise le recouvrement de $\mathcal{X}^*_{\Gamma(p^\infty)K^p}$ par les ouverts
$\mathcal{V}_J:=(\pi_{HT})^{-1}({\mathscr{F}\!\ell}_J)$ donn\'e par le (iv)
du th\'eor\`eme \ref{th:princ}. Comme les $\mathcal{V}_J$ sont
affino\"ides perfecto\"ides (et gr\^ace \`a une propri\'et\'e technique
du bord, voir le (ii) du th\'eor\`eme IV.1.1 de \cite{S}), la cohomologie
de $\Ical^+/p^m$ sur ces ouverts est presque concentr\'ee en degr\'e $0$.
De plus, comme $\pi_{HT}$ est \'equivariant pour les op\'erateurs de Hecke
en dehors de $p$ (agissant trivialement sur ${\mathscr{F}\!\ell}$), les ouverts $\mathcal{V}_J$
sont stables par ces op\'erateurs, donc $\mathcal{H}^S$ agit encore sur les
$H^i(\mathcal{V}_J,\Ical^+/p^m)$.
On utilise la deuxi\`eme partie du point (iv) du th\'eor\`eme \ref{th:princ}
pour montrer que tout caract\`ere de $\mathcal{H}^S$ apparaissant dans un
$\H^0(\mathcal{V}_J,\Ical^+/p^m)$ appara\^it en fait dans un
$\H^0(\mathcal{V}_{J,K_p},\Ical^+/p^m)$, o\`u $K_p\subset{\bf{G}}(\mathbb{Q}_p)$ est assez petit et
$\mathcal{V}_{J,K_p}\subset\mathcal{X}^*_{K_p K^p}$ est un ouvert affino\"ide d'image inverse
$\mathcal{V}_J$ dans $\mathcal{X}^*_{\Gamma(p^\infty)K^p}$.
Il faut encore montrer comment \'etendre les sections de $\Ical^+/p^m$ sur
$\mathcal{V}_{J,K_p}$ qui sont vecteurs propres pour $\mathcal{H}^S$
\`a $\mathcal{X}^*_{K_p K^p}$ tout entier sans changer les valeurs
propres. La m\'ethode classique consiste \`a multiplier ces sections
propres par une puissance assez grande de l'invariant de Hasse. Ici,
on utilise plut\^ot les {\og faux invariants de Hasse\fg}, qui sont des
\'el\'ements de $H^0(\mathcal{V}_{J,K_p},\omega)$ obtenus par pullback de sections
bien choisies dans $H^0({\mathscr{F}\!\ell}_J,\omega_{\mathscr{F}\!\ell})$ (et par descente \`a un niveau fini
assez petit $K_p$); voir le lemme II.1.1 et la page 72 de \cite{S}.
Le fait que
la multiplication par ces faux invariants de Hasse ne
change pas les valeurs propres de $\mathcal{H}_S$ r\'esulte de la propri\'et\'e
d'\'equivariance de~$\pi_{HT}$.
\section{Quelques autres applications}\label{appl}
Indiquons deux autres applications des r\'esultats de \cite{S}. (Cette liste
d'applications ne se veut en aucun cas exhaustive.)
\subsection{Cohomologie compl\'et\'ee}
Le th\'eor\`eme \ref{th:C} donne une formule pour la cohomologie compl\'et\'ee
de la section \ref{cc}. En utilisant ce th\'eor\`eme et le fait que les
espaces topologiques sous-jacents aux $\mathcal{X}^*_{K_p K^p}$ sont de dimension
cohomologique $\leq d$, o\`u $d=n(n+1)/2$ est la dimension des vari\'et\'es
alg\'ebriques $X_{K_p K^p}$, Scholze en d\'eduit que $\widetilde{H}^i_{c,K^p}=0$
pour $i>d$, puis une grande partie de la conjecture 1.5 de \cite{CE} (corollaire
IV.2.3 de \cite{S}).
\footnote{L\`a encore, les r\'esultats cit\'es sont en fait vrais pour toutes
les vari\'et\'es de Shimura de type Hodge, en particulier celles de type PEL.}
\subsection{Mod\`eles entiers \'etranges}
Les faux invariants de Hasse de la section \ref{faux} permettent de d\'efinir
des mod\`eles entiers jusqu'ici inconnus des vari\'et\'es de Shimura de type
Hodge. Dans le preprint \cite{PSt}, Pilloni et Stroh ont \'etudi\'e ces mod\`eles
entiers et les ont utilis\'es pour construire des repr\'esentations galoisiennes
associ\'ees \`a des repr\'esentations automorphes non n\'ecessairement
cohomologiques\footnote{Mais apparaissant dans la cohomologie coh\'erente
de fibr\'es vectoriels automorphes.}
du groupe ${\bf{G}}$ d\'efinissant les vari\'et\'es de Shimura.
\vskip 4.5cm
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,822 |
One Way Passage is a 1932 American pre-Code romantic film starring William Powell and Kay Francis as star-crossed lovers, directed by Tay Garnett and released by Warner Bros. The screenplay by Robert Lord won the Academy Award for Best Story.
Plot
Dan Hardesty is an escaped murderer, sentenced to hang and on the run. In a Hong Kong bar, he literally bumps into Joan Ames, a terminally ill woman whose friends are wishing her bon voyage. It is love at first sight. In what will become a signature gesture for the couple, they share a Paradise Cocktail, then Dan breaks the bowl of his glass, followed by Joan; they leave the stems crossed on the bar.
San Francisco Police Sergeant Steve Burke captures Dan at gunpoint when he leaves the bar (though out of sight of Joan) and escorts him aboard an ocean liner bound for San Francisco. Dan jumps into the water, dragging Steve with him. He takes the key from Steve's pocket and frees himself. Then he spots Joan among the passengers looking over the rail at them. He rescues floundering non-swimmer Steve rather than escape. Once the ship is underway, he persuades Steve to remove the handcuffs.
Dan and Joan fall in love on the month-long cruise, neither knowing that the other is under the shadow of death.
By chance, two of Dan's friends are also aboard, pickpocket Skippy and con artist "Barrel House Betty", masquerading as "Countess Barilhaus". The countess distracts Steve as much as she can to help Dan. Just before the only stop, at Honolulu, Steve has Dan put in the brig, but Dan gets out with their help and goes ashore to arrange escape on a steamer leaving that night. Joan intercepts him as he leaves the ship, and they spend an idyllic day together. When they drive back to the dock that evening, Dan starts to tell her why he cannot return to the ship, only to see her faint. Dan carries her aboard for medical help and stays by her side, forfeiting his chance at escape. Later, Joan's doctor tells Dan about her condition and that the slightest excitement or shock could be fatal. Dan tells the doctor the truth about himself.
Meanwhile, a romance blooms between Steve and the countess. When they near the end of the voyage, he awkwardly proposes to her. He wants to give up being a cop and live on a chicken ranch he owns. (Earlier in the film, Betty told Skippy that she dreamed of giving it all up and buying herself a chicken ranch.) She starts to tell him her true identity, but her confession is interrupted when a steward delivers a telegram to Steve. It is from his boss, telling him to find notorious con-woman Barrelhouse Betty and bring her in. He says nothing, as he still wants to marry her. They kiss, and Steve throws the telegram overboard.
Steve and Dan get ready to disembark, an overcoat draped over the handcuffs that link them. On an impulse, Joan goes to their cabin, where a steward who overheard the grim truth tells her about it. She frantically searches for Dan, and finds him with Steve. The two lovers part for the last time without letting on that they know each other's secret, and Joan collapses after Dan is out of sight.
They had agreed to meet again a month later, on New Year's Eve, at a bar in Agua Caliente, Mexico. At the appointed time and place, the dance floor is full, but the long bar is empty except for Skippy, standing solemnly at one end, and two bartenders at the other. The bartenders are startled by the sound of glass breaking. They turn to find the crossed stems and shattered pieces of two cocktail glasses lying on the bar. They glisten there for a moment and then vanish.
Cast
William Powell as Dan [Hardesty]
Kay Francis as Joan [Ames]
Aline MacMahon as Betty, aka "Barrel House" Betty and "Countess Barilhaus" and Betty Crowley
Frank McHugh as Skippy
Warren Hymer as Steve [Burke]
Roscoe Karns as Cruise Ship Bartender
Frederick Burton as The Doctor
Mike Donlin as Hong Kong bartender
Production
As the ship draws near San Francisco, Dan and Joan talk about the Golden Gates, remembering the words of a hymn. Dan says that when he was a little boy growing up in San Francisco, he thought the gates in the hymn were the Golden Gate of San Francisco. He is referring to the Golden Gate Strait at the mouth of San Francisco bay, not the Golden Gate Bridge. Construction on the bridge would begin in January 1933.
The film's working title was S.S. Atlantic.
This was the sixth time that Powell and Francis played together, and it was their biggest moneymaker.
Francis' gowns were created by Orry-Kelly, who had just joined Warner Bros. in 1932. He went on to win three Academy Awards for costume design.
James Kendis and Lou Klein are credited as composers. The pervasive love theme was reportedly written by W. Franke Harling, uncredited. Aloha O'e by Hawai'i's Queen Lili'uokalani, also figures prominently in the soundtrack.
Wilson Mizner and Joseph Jackson are credited as screenwriters. Critic Ken Hanke gave credit to Garnett's role in honing the final screenplay: "The comedy content — involving unscrupulous but lovable con artists — has all the earmarks of being the work of noted cynic and part-time con artist Mizner."
Reception
Mordaunt Hall wrote in The New York Times, "In its uncouth, brusque and implausible fashion, 'One Way Passage' ... offers quite a satisfactory entertainment. ... Tay Garnett's direction is clever. He keeps the story on the move with its levity and dashes of far-fetched romance."
Leonard Maltin gives the film 3 1/2 out of 4 stars, high praise for a "tender shipboard romance of fugitive Powell and fatally ill Francis, splendidly acted, with good support by MacMahon and McHugh".
Writing in 2013 for the Ashville, N.C., Mountain Express, Ken Hanke described the film as: "The classic doomed lovers/shipboard romance movie... a perfect blend of romantic tragedy and hard-boiled comedy... The two elements perfectly complement each other in a way you find in very few films... A strange and strangely magical film from the very uneven filmmaker Tay Garnett, One Way Passage is a movie that once seen is unlikely to be forgotten... the film's brilliant balance of cynical comedy (provided by Frank McHugh and the wonderful Aline MacMahon) and tragic — ultimately mystical — romance. "
In his autobiography Looking for a Street, Charles Willeford describes seeing the movie as a thirteen-year-old: "One Way Passage" is still my all-time favorite movie, but I have never risked seeing it again. I cried so hard when the movie ended the usher took me out of the lobby and gave me a glass of water.
Box office
According to Warners records, the film earned $791,000 in the US and Canada and $317,000 elsewhere. This success led the studio to remake the film in 1940.
Accolades
Robert Lord won an Oscar for his original story.
The film is recognized by American Film Institute in these lists:
2002: AFI's 100 Years ... 100 Passions – Nominated
Remake
One Way Passage was remade in 1940 as 'Til We Meet Again, featuring Merle Oberon, George Brent, Pat O'Brien, Binnie Barnes and Geraldine Fitzgerald. Although some scenes strongly echo the original, It is not a word for word, shot for shot remake, and there are new characters. Frank McHugh reprises his role as Dan's pickpocket friend; his state of perpetual inebriation is a pose in the later film.
Radio adaptations
"One Way Passage was presented on Warner Brothers Academy Theater April 3, 1938. Ronald Reagan and Gloria Dickson starred in the 30-minute adaptation.
"One Way Passage" was presented on Lux Radio Theatre March 6, 1939. Original stars William Powell and Kay Francis reprised their roles, although Kay Francis filled in for Norma Shearer who bowed out due to illness. The production was 60 minutes in three acts.
One Way Passage was presented on Philip Morris Playhouse September 12, 1941.
One Way Passage was presented on The Screen Guild Theater April 5, 1948 starring Barbara Stanwyck, Robert Taylor, and Ward Bond.
One Way Passage was presented on Hollywood Sound Stage'' February 14, 1952. The 30-minute adaptation starred Ruth Roman and Frank Lovejoy.
References
External links
William Powell: The Life And Films
1932 films
1932 romantic drama films
American romantic drama films
American black-and-white films
1930s English-language films
Films about capital punishment
Films directed by Tay Garnett
Films set on ships
Films that won the Academy Award for Best Story
Films set around New Year
Films produced by Robert Lord (screenwriter)
Films produced by Hal B. Wallis
Warner Bros. films
1930s American films | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,970 |
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COME UP YOUNG MUSIC GROUP
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MIKE BANDZ
http://thisis50.ning.com/profiles/blogs/mike-bandz-drops-highly-anticipated-single-titled-red-bottoms-and?xg_source=activity
http://7thspace.com/headlines/519010/hip_hop_sensation_mike_bandz_teams_up_with_music_magic_box_for_sex_and_drugz.html
Winston-Salem, NC – Mike Bandz, hip-hop artist known for his work with his rap group, 3rd Degree MOB, is entering the world of film and will partner with music promoters musicmagicbox.com to produce a short film based on his new single "Sex and Drugz".
Bandz has been performing for over a decade and put the hip-hop world on notice when he and partner Louie G quickly became the premiere hip-hop artists in and around Winston-Salem, with hits such as "With Me" and "Game On". Bandz is noted for his wordplay and his take-no-prisoners rhymes.
Now Bandz has teamed up with singer Jerry White for "Sex and Drugz", and the short film it has inspired goes into production soon. The film is not a music video.
"To us he is a different type of artist and has a very different personality than most of the rap artists we deal with," says Ryan Knox of musicmagicbox.com.
"So we wanted to take a different approach with him all together, which is where the short film idea came about."
The music promotion company is also planning a major worldwide radio commercial campaign to promote the single in 2016.
LISSA MITCHELL
http://www.devinejams.com/2015/08/i-rise-by-lissa-mitchell.html
"I Rise" – The New Single by Lissa Mitchell – A Gospel Pop Singer
Lissa Mitchell, A Gospel Pop Singer From W-S NC.
Her music has always been an inspiration in her community, but her aim at this point is to give the world a taste of what has always inspired those in her community and around her.
The power and energy she use in her vocals gives meaning to the story line of "I Rise". She describes how Jesus rose from the dead so that she can rise from all her adversities! It's an awesome tempo and the melody is perfect for radio! It's that type of song that should be among the top selection of Christian pop music for DJ's and their rotation.
We hope that you all enjoy and be inspired by Lissa's new single "I Rise". There is more in store for this talented artist and there is more to share on her upcoming new Album, "The Joy".
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Sarceaux es una localidad y comuna de Francia, en la región de Baja Normandía, departamento de Orne, en el distrito de Argentan y cantón de Argentan Oeste.
Su población en el censo de 1999 era de 845 habitantes. Forma parte de la aglomeración urbana de Argentan.
Está integrada en la Communauté de communes du Pays d'Argentan.
Demografía
Enlaces externos
INSEE
Localidades de Orne | {
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