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\section{Introduction}
Let $(A,B,C)\in\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times m}\times\mathbb{R}^{p\times m}$ be a realization of a linear, time-invariant system
\begin{align}\label{lti}
\boldsymbol{\Sigma}:\quad \dot x(t)=Ax(t)+Bu(t),\quad x(0)=0,\quad y(t)=Cx(t)
\end{align}
and assume that $A$ is Hurwitz which implies \eqref{lti} is asymptotically stable.
The infinite reachability and observability Gramians
\begin{align*}
P_{\infty}=\int_0^{\infty} \operatorname{e}^{As}BB^T \operatorname{e}^{A^Ts} ds,\quad Q_{\infty}=\int_0^{\infty} \operatorname{e}^{A^Ts}C^TC \operatorname{e}^{As} ds
\end{align*}
of $(A,B,C)$ solve the Lyapunov equations
\begin{align}\label{BT_Lyap}
AP_{\infty}+P_{\infty}A^T+BB^T=0,\quad A^TQ_{\infty}+Q_{\infty}A^T+C^TC=0.
\end{align}
The first ingredient of balanced truncation~\cite{moore} (BT) is to
simultaneously diagonalize both Gramians through congruence transformations $\hat S P_{\infty} \hat S^T=\hat S^{-T}Q_{\infty}\hat
S^{-1}=\Sigma_{\infty}$ which gives a balanced realization $(\hat S A \hat S^{-1}, \hat SB, C\hat S^{-1})$, where $\Sigma_{\infty}$ is diagonal and
contains the Hankel singular values $\sigma_j$ (HSVs),
i.e., the square root of the eigenvalues of $P_{\infty}Q_{\infty}$. In the second step the reduced order model $\boldsymbol{\Sigma}_r$ is obtained by
keeping only the $r\times r$ upper left block of $\hat S A \hat S^{-1}$ and the associated parts
of $\hat SB, C\hat S^{-1}$, i.e., the smallest $n-r$ HSVs are removed from the system. With Cholesky factorizations $P_{\infty}=L_PL_P^T$,
$Q_{\infty}=L_QL_Q^T$, and the singular value decomposition (SVD)
$X\Sigma_{\infty}Y^T=L_Q^TL_P$, the balancing
transformation is given by $\hat S=L_QX\Sigma_{\infty}^{-\tfrac{1}{2}}$ and $\hat S^{-1}=L_PY\Sigma_{\infty}^{-\tfrac{1}{2}}$, see,
e.g.,~\cite{antoulas}. This leads to non increasingly ordered $\sigma_j$. Moreover, the resulting reduced system $\boldsymbol{\Sigma}_r$ is
asymptotically stable and satisfies the $\mathcal H_{\infty}$ error bound~\cite{morGlo84}
\begin{align}\label{BT_errorB}
\|\boldsymbol{\Sigma}-\boldsymbol{\Sigma}_r\|_{\mathcal H_{\infty}}\leq2(\sigma_{r+1}+\ldots+\sigma_n).
\end{align}
Once the SVD is computed,~\eqref{BT_errorB} can be used to adaptively adjust the reduced order $r$. A generalized $\mathcal
H_{\infty}$-error bound for BT has been proved in \cite{bennerdammcruz, dammbennernewansatz}, where linear stochastic system are investigated.\\
The matrix of truncated HSVs
$\Sigma_2=\operatorname{diag}(\sigma_{r+1}, \ldots, \sigma_n)$ can be used to express the $\mathcal{H}_2$ error bound \cite{antoulas}. It is represented by
\begin{align}\label{h2_bound_infty}
\|\boldsymbol{\Sigma}-\boldsymbol{\Sigma}_r\|^2_{\mathcal H_{2}}\leq\operatorname{tr}(\Sigma_{2} (B_2 B_2^T+2 P_{\infty, M, 2} A_{21}^T)),\end{align}
where $B_2$ is the matrix of the last $n-r$ rows of $\hat S B$, $A_{21}$ is the left lower $(n-r)\times r$ block of $\hat S A \hat S^{-1}$ and
$P_{\infty, M, 2}$ are the last $n-r$ rows of the mixed Gramian $P_{\infty, M}=\hat S \int_0^{\infty} \operatorname{e}^{As}BB_1^T \operatorname{e}^{A_{11}^Ts} ds$.
The bound in (\ref{h2_bound_infty}) has already been extended to stochastic systems in a more general form \cite{redmannbenner,
redmannfreitag, BTtyp2EB}.
In~\cite{morGawJ90} Gawronski and Juang restricted balanced truncation to a finite time interval $[0,\bar T]$, $\bar T<\infty$, by
introducing the time-limited reachability and observability Gramians
\begin{align}\label{TLBT_Gram}
P_{\bar T}:=\int_0^{\bar T} \operatorname{e}^{As}BB^T \operatorname{e}^{A^Ts} ds,\quad Q_{\bar T}=\int_0^{\bar T} \operatorname{e}^{A^Ts}C^TC \operatorname{e}^{As} ds.
\end{align}
It is easy to show that $P_{\bar T},~Q_{\bar T}$ solve the Lyapunov equations
\begin{align}\label{TLBT_Lyap}
AP_{\bar T}+P_{\bar T}A^T+BB^T-F_{\bar T}F_{\bar T}^T&=0,\\
A^TQ_{\bar T}+Q_{\bar T}A^T+C^TC-G_{\bar T}^TG_{\bar T}&=0,
\end{align}
where $G_t:=C \operatorname{e}^{At}$ and $F_t:=\operatorname{e}^{At}B$, $t\in[0, {\bar T}]$. Time-limited balanced truncation (TLBT) is then carried out by
using the Cholesky factors of $P_{\bar T}$, $Q_{\bar T}$ instead of $P_{\infty}, Q_{\infty}$ to construct the balancing transformation
which in this case is denoted by $S$.
This transformation simultaneously diagonalizes $P_{\bar T}$, $Q_{\bar T}$, i.e., $S P_{\bar T} S^T=S^{-T}Q_{\bar
T}S^{-1}=\Sigma_{\bar T}$ and is, thus, referred to as time-limited balancing transformation. The values in $\Sigma_{\bar T}$ are referred to as
time-limited singular values and are, similar to the HSVs, invariant under state-space transformations. Because of the altered Gramian definitions,
TLBT does generally not preserve stability and there is no $\mathcal H_{\infty}$ error bound as in unrestricted BT.\smallskip
The main contribution of this paper is a generalized $\mathcal H_2$ error bound for TLBT. It leads to (\ref{h2_bound_infty}) if $\bar
T\rightarrow \infty$. We provide two representations of this bound. The first one can be used for practical computations and is, hence, an important
tool to assess the obtained accuracy. The second representation is not appropriate for computing the bound but it shows
that, similar to BT, the
time-limited singular values deliver an alternative criterion to find a suitable reduced order dimension $r$. We conclude this paper
by conducting several numerical
experiments which indicate that the time-limited $\mathcal H_2$ bound is tight.
\section{$\mathcal H_2$-type Error Bounds for Time-Limited Balanced Truncation}\label{sec2}
Let $S$ be the time-limited balancing transformation. We partition the balanced realization $(S A
S^{-1},
SB, CS^{-1})$ as follows: \begin{align*}
S A S^{-1}= \begin{bmatrix}{A}_{11}&{A}_{12}\\
{A}_{21}&{A}_{22}\end{bmatrix},\;\;\; S B = \begin{bmatrix}{B}_1\\ {B}_2\end{bmatrix},\;\;\;
C S^{-1} = \begin{bmatrix}{C}_1 &{C}_2\end{bmatrix}, \end{align*}
where ${A}_{11}\in\mathbb{R}^{r\times r}$, $B_{1}\in\mathbb{R}^{r\times m}$, $C_{1}\in\mathbb{R}^{p\times r}$ and the other blocks of appropriate dimensions. Furthermore, we
introduce
{\allowdisplaybreaks \begin{align*}
S F_{\bar T}=\begin{bmatrix}
F_{{\bar T}, 1}\\
F_{{\bar T}, 2}\end{bmatrix},\;G_{\bar T} S^{-1}=\begin{bmatrix}
G_{{\bar T}, 1}& G_{{\bar T}, 2}\end{bmatrix},\;\Sigma_{\bar T}=\begin{bmatrix}
\Sigma_{{\bar T}, 1}& \\
& \Sigma_{{\bar T}, 2}
\end{bmatrix}.\end{align*}}
We consider the corresponding Lyapunov equations in partitioned form:
{\allowdisplaybreaks \begin{align}
\label{donotusenewgram}
\left[\begin{smallmatrix}{A}_{11}&{A}_{12}\\
{A}_{21}&{A}_{22}\end{smallmatrix}\right] \left[\begin{smallmatrix}\Sigma_{{\bar T}, 1}&\\
&\Sigma_{{\bar T}, 2}\end{smallmatrix}\right]+\left[\begin{smallmatrix}\Sigma_{{\bar T}, 1} &\\
&\Sigma_{{\bar T}, 2}\end{smallmatrix}\right] \left[\begin{smallmatrix} A^T_{11}&A^T_{21}\\
A^T_{12}& A^T_{22} \end{smallmatrix}\right]=& -\left[\begin{smallmatrix} {B}_1 B_1^T& {B}_1 B_2^T \\
{B}_2 B_1^T& {B}_2 B_2^T \end{smallmatrix}\right]\\\nonumber
&+\left[\begin{smallmatrix} F_{\bar T,1} F_{\bar T,1}^T& F_{\bar T,1} F_{\bar T,2}^T \\
F_{\bar T,2}F_{\bar T,1}^T& F_{\bar T,2}F_{\bar T,2}^T \end{smallmatrix}\right], \\ \label{ersteeqbt2ob}
\left[\begin{smallmatrix}{A}^T_{11}&{A}^T_{21}\\
{A}^T_{12}&{A}^T_{22}\end{smallmatrix}\right]\left[\begin{smallmatrix}\Sigma_{{\bar T}, 1}&\\
&\Sigma_{{\bar T}, 2}\end{smallmatrix}\right]+\left[\begin{smallmatrix}\Sigma_{{\bar T}, 1}&\\
&\Sigma_{{\bar T}, 2}\end{smallmatrix}\right]\left[\begin{smallmatrix}A_{11}&A_{12}\\
A_{21}& A_{22}\end{smallmatrix}\right]=& -\left[\begin{smallmatrix}C_1^TC_1&C_1^TC_2 \\
C^T_2C_1& C^T_2C_2 \end{smallmatrix}\right]\\\nonumber
&+\left[\begin{smallmatrix} G_{\bar T,1}^T G_{\bar T,1}& G_{\bar T,1}^TG_{\bar T,2} \\
G_{\bar T,2}^TG_{\bar T,1}& G_{\bar T,2}^TG_{\bar T,2} \end{smallmatrix}\right].
\end{align}}
The TLBT reduced system that approximates (\ref{lti}) is given by
\begin{align*}
\dot x_r(t)=A_{11}x_r(t)+B_1u(t),\quad x_r(0)=0,\quad y_r(t)=C_1 x_r(t).
\end{align*}
The goal of this section is to find a bound for the error between $y$ and $y_r$. Since we have zero initial conditions for both the reduced and the
full system, we have the following representations for the outputs {\allowdisplaybreaks \begin{align*}
y(t)&=C x(t)=C \int_0^t \operatorname{e}^{A(t-s)} B u(s) ds, \\
y_r(t)&=C_1 x_r(t)=C_1 \int_0^t \operatorname{e}^{A_{11}(t-s)} B_1 u(s) ds,
\end{align*}}
where $t\in [0, {\bar T}]$. To find a first representation for the error bound, arguments from \cite{redmannbenner,
redmannfreitag, BTtyp2EB} are used. There a generalized $\mathcal H_2$ error bound for stochastic systems has been derived. Some easy rearrangements
yield a first error estimate {\allowdisplaybreaks \begin{align*}
&\left\|y(t)- y_r(t)\right\|_2 \\&=\left\|C \int_0^t \operatorname{e}^{A(t-s)} B u(s) ds- C_1 \int_0^t \operatorname{e}^{A_{11}(t-s)} B_1 u(s) ds\right\|_2\\&\leq
\int_0^t \left\|\left(C \operatorname{e}^{A(t-s)} B - C_1 \operatorname{e}^{A_{11}(t-s)} B_1\right) u(s)\right\|_2 ds\\&\leq \int_0^t \left\| C
\operatorname{e}^{A(t-s)} B - C_1 \operatorname{e}^{A_{11}(t-s)} B_1\right\|_F \left\|u(s)\right\|_2 ds.
\end{align*}}
By the Cauchy Schwarz inequality it holds that{\allowdisplaybreaks \begin{align*}
&\left\|y(t)- y_r(t)\right\|_2\\&\leq
\left(\int_0^t \left\|C \operatorname{e}^{A(t-s)} B - C_1 \operatorname{e}^{A_{11}(t-s)} B_1\right\|_F^2 ds\right)^{\frac{1}{2}} \left(\int_0^t
\left\|u(s)\right\|_2^2 ds\right)^{\frac{1}{2}}.
\end{align*}}
Using substitution, the definition of the Frobenius norm and the linearity of the integral, we obtain
{\allowdisplaybreaks \begin{align*}
& \int_0^t \left\| C \operatorname{e}^{A(t-s)} B - C_1 \operatorname{e}^{A_{11}(t-s)} B_1\right\|_F^2 ds\\&=\int_0^t \left\|C \operatorname{e}^{As} B - C_1 \operatorname{e}^{A_{11}s}
B_1\right\|_F^2 ds \\&\leq \int_0^{\bar T} \left\|C \operatorname{e}^{As} B - C_1 \operatorname{e}^{A_{11}s} B_1\right\|_F^2 ds\\&=\int_0^{\bar T} \operatorname{tr}\left(C
\operatorname{e}^{As}BB^T
\operatorname{e}^{A^Ts} C^T\right)ds\\&\quad +\int_0^{\bar T}\operatorname{tr}\left(C_1 \operatorname{e}^{A_{11}s}B_1B_1^T\operatorname{e}^{A_{11}^Ts} C_1^T\right)ds\\&\quad
-2\int_0^{\bar T}\operatorname{tr}\left(C
\operatorname{e}^{As}BB_1^T \operatorname{e}^{A_{11}^Ts} C_1^T\right) ds\\&=\operatorname{tr}\left(C P_{\bar T} C^T\right)+\operatorname{tr}\left(C_1 P_{{\bar T}, r}
C_1^T\right)-2\;\operatorname{tr}\left(C P_{{\bar T}, M} C_1^T\right),
\end{align*}}
where $P_{\bar T}:=\int_0^{\bar T} \operatorname{e}^{As}BB^T \operatorname{e}^{A^Ts}ds$, $P_{{\bar T}, r}:=\int_0^{\bar T} \operatorname{e}^{A_{11}s}B_1B_1^T\operatorname{e}^{A_{11}^Ts} ds$
and
$P_{{\bar T},M}:=\int_0^{\bar T} \operatorname{e}^{As}BB_1^T \operatorname{e}^{A_{11}^Ts} ds$.
Matrix-valued integrals of this form can under some conditions be expressed as unique solutions of matrix equations.
\begin{lem}\label{timeint_mateq}
Let $A_1\in\mathbb{R}^{n\times n},~A_2\in\mathbb{R}^{r\times r}$ with $\Lambda(A_1)\cap-\Lambda(A_2)=\emptyset$ and
$B_1\in\mathbb{R}^{n\times m}$, $B_2\in\mathbb{R}^{r\times m}$. Then,
\begin{align*}
X=\int_{0}^{\bar T}\operatorname{e}^{A_1s}B_1B_2^T \operatorname{e}^{A_2^Ts}ds
\end{align*}
solves the Sylvester equation
\begin{align*}
A_{1} X+X A_{2}^T &=-B_1 B_2^T+\operatorname{e}^{A_1{\bar T}}B_1B_2^T\operatorname{e}^{A_2^T{\bar T}}.
\end{align*}
\end{lem}
\begin{proof}
The integral is equivalent to
{\allowdisplaybreaks \begin{align*}
\operatorname{vec} X&=\int_{0}^{\bar T}\operatorname{vec}\operatorname{e}^{A_1s}B_1B_2^T \operatorname{e}^{A_2^Ts}ds\\
&=\int_{0}^{\bar T}\operatorname{e}^{A_2s}\otimes \operatorname{e}^{A_1s}ds\operatorname{vec} B_1B_2^T\\
&=\int_{0}^{\bar T}\operatorname{e}^{\left(I_r\otimes A_1 +A_2\otimes I_n \right)s}ds \operatorname{vec} B_1B_2^T,
\end{align*}}
where we used \cite[Theorem~10.9]{Hig08}. The matrix $\mathcal{A}:=I_r\otimes A_1 +A_2\otimes I_n$ is nonsingular and it holds that
\begin{align*}
\operatorname{vec} X&=\mathcal{A}^{-1}\left(\operatorname{e}^{\mathcal{A}{\bar T}}-I_{nr}\right)\operatorname{vec} B_1B_2^T\\
\Leftrightarrow\quad \mathcal{A}\operatorname{vec} X&=\left(\operatorname{e}^{\mathcal{A}{\bar T}}-I_{nr}\right)\operatorname{vec} B_1B_2^T
\end{align*}
and the claim follows after de-vectorization.
\end{proof}
\begin{bem}
The result of the above Lemma is also a consequence of the product rule. Setting $g_1(t):=\operatorname{e}^{A_1 t}B_1$ and $g_2(t):=B_2^T \operatorname{e}^{A_2^T t}$, it
holds
that \begin{align*}
&g_1(\bar T)g_2(\bar T)-g_1(0)g_2(0)=\int_{0}^{\bar T}g_1(s)dg_2(s)+\int_{0}^{\bar T}d g_1(s) g_2(s)\\&= \int_{0}^{\bar T}g_1(s)g_2(s)ds\;A_2^T
+A_1 \int_{0}^{\bar T} g_1(s) g_2(s) ds,
\end{align*}
since $dg_2(s)=g_2(s) A_2^T ds$ and $dg_1(s)=A_1 g_1(s) ds$.
The time-limited Gramians~\eqref{TLBT_Gram} also exists for unstable systems. Therefore, it is, e.g. in~\cite[Section 7.6.5]{antoulas}, discussed to
use TLBT to
reduce unstable systems. The above Lemma further reveals that in this situation and if
$\Lambda(A)\cap-\Lambda(A)=\emptyset$, the time-limited Gramians can still be obtained by solving the time-limited
Lyapunov equations~\eqref{TLBT_Lyap} which is important from a numerical point of view.
In this work, however, we will not pursue the reduction of unstable systems further.
\end{bem}
From now on we assume that $\Lambda(A_{11})\cap-\Lambda(A_{11})=\emptyset$ and $\Lambda(A)\cap-\Lambda(A_{11})=\emptyset$, implying by
Lemma~\ref{timeint_mateq} that
the matrices $P_{{\bar T}, r}$ and $P_{{\bar T}, M}$ are the unique solutions of
{\allowdisplaybreaks
\begin{subequations}\label{lyapeq_btbound}
\begin{align}\label{nonbalregramp}
A_{11} P_{{\bar T}, r}+P_{{\bar T},r} A_{11}^T &=-B_1 B_1^T+F_{{\bar T}, r} F_{{\bar T}, r}^T,\\
\label{mixeqbt2blsa}
A P_{{\bar T}, M}+ P_{{\bar T}, M} A_{11}^T &=-B B_1^T+F_{{\bar T}} F_{{\bar T}, r}^T,
\end{align}
\end{subequations}
}
where $F_{{\bar T}, r}:=\operatorname{e}^{A_{11}T}B_1$. We have, thus, established the following result.
\begin{thm}\label{thm_basic}
Let $\Lambda(A_{11})\cap-\Lambda(A_{11})=\emptyset$ and $\Lambda(A)\cap-\Lambda(A_{11})=\emptyset$. Then the following error bound holds for the
reduced system
$\boldsymbol{\Sigma}_r$ generated by TLBT
{\allowdisplaybreaks \begin{align}\label{implicdeptilp}
\begin{split}
&\max_{t\in [0, {\bar T}]}\left\|y(t)- y_r(t)\right\|_2\leq\epsilon\left\|u\right\|_{L^2_{\bar T}},\\
&\epsilon:=\left(\operatorname{tr} \left(C P_{\bar T} C^T\right)+\operatorname{tr} \left(C_1 P_{{\bar T}, r}
C_1^T\right)-2\operatorname{tr} \left(C P_{{\bar T},M} C_1^T\right)\right)^{\frac{1}{2}}.
\end{split}
\end{align}}
\end{thm}
The representation (\ref{implicdeptilp}) of the error bound has the same structure as the one computed in the stochastic framework
\cite{redmannbenner, redmannfreitag, BTtyp2EB} but it is clearly different since solutions of different matrix equations enter in the time-limited
case. The bound in (\ref{implicdeptilp}) can be used for practical computations. It only requires to solve the matrix equations in
(\ref{lyapeq_btbound}) since
$P_{\bar T}$ is already known from the balancing procedure. The matrix equations (\ref{lyapeq_btbound}) are not expensive since
$P_{{\bar T}, r}$ usually is a small matrix and $P_{{\bar T},M}$ only has a few columns.\smallskip
The next theorem provides an alternative representation of
this bound. It can be expressed with the help of $\Sigma_{{\bar T}, 2}=\operatorname{diag}(\sigma_{{\bar T}, r+1}, \ldots, \sigma_{{\bar T}, n})$ which is the
matrix of truncated time-limited singular values. In \cite{redmannbenner, redmannfreitag, BTtyp2EB} representations of
generalized $\mathcal{H}_2$ error bounds have been shown using the truncated HSVs of the underlying stochastic system. However, the matrix equations
(\ref{TLBT_Lyap}) and (\ref{lyapeq_btbound}) have a very different structure than the generalized equations for stochastic system. Therefore, we
need to apply other techniques in order to obtain the result below. This result also shows essential differences in its structure compared to the
stochastic case.
\begin{thm}\label{thmmaimn}
Using the coefficients of the balanced realization of the system, the error bound in (\ref{implicdeptilp}) can be expressed as
follows:{\allowdisplaybreaks \begin{align*}
& \operatorname{tr}\left(C P_{\bar T} C^T+ C_1 P_{{\bar T}, r} C_1^T-2 C P_{{\bar T}, M} C_1^T\right)\\=&\operatorname{tr}(\Sigma_{{\bar T}, 2} (B_2 B_2^T+2 P_{{\bar T},
M,
2} A_{21}^T))-2\operatorname{tr}(G^T_{{\bar T}, 1}
G_{\bar T} P_{{\bar T}, M})\\&+\operatorname{tr}(G^T_{{\bar T}, 1} G_{{\bar T}, 1} P_{{\bar T}, r})+\operatorname{tr}(F_{{\bar T}, 1} F^T_{{\bar T}, 1} \Sigma_{{\bar T},
1})\\&-\operatorname{tr}((F_{{\bar T},
1}-F_{{\bar T}, r}) (F_{{\bar T}, 1}-F_{{\bar T}, r})^T \Sigma_{{\bar T}, 1}),\end{align*}}
where $P_{{\bar T}, M, 2}$ are the last $n-r$ rows of $S P_{{\bar T}, M}$ with $S$ being the balancing transformation.
\begin{proof}
By selecting the left and right upper block of (\ref{ersteeqbt2ob}), we
have{\allowdisplaybreaks \begin{align}\label{firstc1c2bt2jk}
A_{11}^T \Sigma_{{\bar T}, 1}+\Sigma_{{\bar T}, 1} A_{11} &=-C_1^TC_1+G_{{\bar T}, 1}^T G_{{\bar T},1}\\ \label{c1c2eqbt2}
A_{21}^T \Sigma_{{\bar T}, 2}+\Sigma_{{\bar T}, 1} A_{12} &=-C_1^T C_2+G_{{\bar T}, 1}^TG_{{\bar T},2}.
\end{align}}
We introduce the reduced order system observability Gramian $Q_{{\bar T}, r}:=\int_0^{\bar T} \operatorname{e}^{A^T_{11}s}C_1^T C_1\operatorname{e}^{A_{11}s} ds$ which
satisfies
{\allowdisplaybreaks \begin{align}\label{redgramobsbt2fzu}
A_{11}^T Q_{{\bar T}, r}+Q_{{\bar T}, r} A_{11} =-C_1^T C_1+G_{{\bar T}, r}^T G_{{\bar T}, r}
\end{align}}
with $G_{{\bar T}, r}:=C_1 \operatorname{e}^{A_{11}{\bar T}}$.
We make use of the integral representations of $P_{\bar T}$ and $Q_{\bar T}$ and apply properties of the trace. Hence, we have
\begin{align*}
&\operatorname{tr}(C P_{\bar T} C^T)=\int_0^{\bar T} \operatorname{tr}(C \operatorname{e}^{As}BB^T \operatorname{e}^{A^Ts} C^T)ds\\&=\int_0^{\bar T} \operatorname{tr}(B^T \operatorname{e}^{A^Ts} C^TC
\operatorname{e}^{As}B)ds=\operatorname{tr}(B^T
Q_{\bar T} B).
\end{align*}
Using the balancing transformation $S$ and the partition of $SB$, we obtain \begin{align*}
\operatorname{tr}(B^TQ_{\bar T} B)&=\operatorname{tr}(B^TS^T
S^{-T}Q_{\bar T}S^{-1}SB)=\operatorname{tr}(B^TS^T \Sigma_{\bar T}SB)\\&=\operatorname{tr}(B_1^T\Sigma_{\bar T, 1}B_1)+\operatorname{tr}(B_2^T \Sigma_{\bar T, 2} B_2).
\end{align*}
The partition of $CS^{-1}$ and $S P_{{\bar T}, M}=\left[\begin{smallmatrix}
P_{{\bar T}, M, 1} \\
P_{{\bar T}, M, 2}
\end{smallmatrix}\right]$ yield
\begin{align*}
\operatorname{tr}(C P_{\bar T,M}C_1^T)&=\operatorname{tr}(C S^{-1} S P_{\bar T,M} C_1^T)\\&=\operatorname{tr}(C_1 P_{\bar T,M, 1} C_1^T)+\operatorname{tr}(C_2 P_{\bar T,M, 2}
C_1^T).
\end{align*}
For $\epsilon$ in \eqref{implicdeptilp} this leads to
\begin{align}\label{insertforebbt2fh}
\epsilon^2=&\operatorname{tr}(B_1^T \Sigma_{{\bar T},1} B_1)+\operatorname{tr}(B_2^T \Sigma_{{\bar T}, 2} B_2)+
\operatorname{tr}(C_1 P_{{\bar T}, r} C_1^T)\\ \nonumber&-2\operatorname{tr}(C_1 P_{{\bar T}, M, 1} C_1^T)-2\operatorname{tr}(C_2 P_{{\bar T}, M, 2} C_1^T).
\end{align}
We insert equation (\ref{c1c2eqbt2}) which yields
{\allowdisplaybreaks \begin{align*}
\operatorname{tr}(C_2 P_{{\bar T}, M, 2} C_1^T)&=\operatorname{tr}(P_{{\bar T},M, 2} C_1^T C_2 )\\&=-\operatorname{tr}(P_{{\bar T}, M, 2} (A_{21}^T
\Sigma_{{\bar T}, 2}+\Sigma_{{\bar T}, 1} A_{12}))\\&\quad+\operatorname{tr}(P_{{\bar T},M, 2}G_{{\bar T}, 1}^TG_{{\bar T},2} )\\&=-\operatorname{tr}(\Sigma_{{\bar T}, 2}
P_{{\bar T}, M, 2}
A_{21}^T)-\operatorname{tr}(\Sigma_{{\bar T}, 1}
A_{12} P_{{\bar T}, M, 2})\\&\quad +\operatorname{tr}(G_{{\bar T}, 1}^TG_{{\bar T},2} P_{{\bar T},M, 2}).
\end{align*}}
We multiply (\ref{mixeqbt2blsa}) with $S$ from the left and evaluate the resulting upper block of the equation: \begin{align*}
-A_{12} P_{{\bar T}, M, 2}=A_{11} P_{{\bar T}, M, 1}+P_{{\bar T}, M, 1} A_{11}^T+B_1 B_1^T-F_{{\bar T}, 1}F_{{\bar T}, r}^T.\\
\end{align*}
Hence, we have {\allowdisplaybreaks \begin{align*}
&-2\operatorname{tr}(C_2 P_{{\bar T}, M, 2} C_1^T)=\\&2[\operatorname{tr}( \Sigma_{{\bar T}, 1}F_{{\bar T}, 1}F_{{\bar T}, r}^T)-\operatorname{tr}( \Sigma_{{\bar T}, 1} (B_1
B_1^T+A_{11} P_{{\bar T},
M, 1}+ P_{{\bar T}, M, 1}
A_{11}^T))]\\&+2[\operatorname{tr}(\Sigma_2 P_{{\bar T}, M, 2} A_{21}^T)-\operatorname{tr}(G_{{\bar T}, 1}^TG_{{\bar T},2} P_{{\bar T},M, 2})].
\end{align*}}
Using equation (\ref{firstc1c2bt2jk}), we obtain {\allowdisplaybreaks\begin{align*}
\operatorname{tr}(\Sigma_{{\bar T},1} (A_{11} P_{{\bar T}, M, 1}+P_{{\bar T}, M, 1} A_{11}^T))&=\operatorname{tr}(P_{{\bar T}, M, 1}(\Sigma_{{\bar T}, 1} A_{11}+ A_{11}^T
\Sigma_{{\bar T},
1}))\\
&=\operatorname{tr}(P_{{\bar T}, M, 1}(G_{{\bar T}, 1}^T G_{{\bar T}, 1}- C_1^T C_1)),
\end{align*}}
so that{\allowdisplaybreaks\begin{align*}
&-2\operatorname{tr}(C_2 P_{{\bar T},M, 2} C_1^T)\\&=2[\operatorname{tr}(\Sigma_{{\bar T},2} P_{{\bar T}, M, 2} A_{21}^T)-\operatorname{tr}(B_1^T\Sigma_{{\bar T}, 1} B_1)+\operatorname{tr}(C_1
P_{{\bar T}, M,
1} C_1^T)]\\&\quad +2[\operatorname{tr}(\Sigma_{{\bar T}, 1}F_{{\bar T}, 1}F_{{\bar T}, r}^T)-\operatorname{tr}(G_{{\bar T}, 1}^TG_{{\bar T}} P_{{\bar T},M})].
\end{align*}}
Inserting this result into equation (\ref{insertforebbt2fh}) provides{\allowdisplaybreaks\begin{align*}
\epsilon^2=&\operatorname{tr}(\Sigma_{{\bar T}, 2} (B_2 B_2^T+2 P_{{\bar T}, M, 2} A_{21}^T))\\&+2[\operatorname{tr}(\Sigma_{{\bar T}, 1}F_{{\bar T}, 1}F_{{\bar T},
r}^T)-\operatorname{tr}(G_{{\bar T}, 1}^TG_{{\bar T}}
P_{{\bar T},M})]\\&+\operatorname{tr}(C_1 P_{{\bar T}, r} C_1^T)-\operatorname{tr}(B_1^T \Sigma_{{\bar T},1} B_1).
\end{align*}}
With the integral representations of $P_{{\bar T}, r}$ and $Q_{{\bar T}, r}$ it holds that
\begin{align*}
&\operatorname{tr}(C_1 P_{{\bar T}, r} C_1^T)=\int_0^{\bar T} \operatorname{tr}(C_1 \operatorname{e}^{A_{11}s}B_1B_1^T \operatorname{e}^{A_{11}^Ts} C_1^T)ds\\&=\int_0^{\bar T} \operatorname{tr}(B_1^T
\operatorname{e}^{A_{11}^Ts}
C_1^TC_1 \operatorname{e}^{A_{11}s}B_1)ds=\operatorname{tr}(B_1^T Q_{{\bar T}, r} B_1).
\end{align*}
So, we have {\allowdisplaybreaks \begin{align*}
\operatorname{tr}(C_1 P_{{\bar T}, r} C_1^T)-\operatorname{tr}(B_1^T \Sigma_{{\bar T}, 1} B_1)=\operatorname{tr}(B_1 B_1^T(Q_{{\bar T}, r}-\Sigma_{{\bar T}, 1})).
\end{align*}}
Combining equations (\ref{firstc1c2bt2jk}) and (\ref{redgramobsbt2fzu}), we have
\begin{align}\label{difQRSig}
A_{11}^T (Q_{{\bar T}, r}-\Sigma_{{\bar T}, 1})+(Q_{{\bar T}, r}-\Sigma_{{\bar T}, 1}) A_{11} =G_{{\bar T}, r}^T G_{{\bar T}, r}-G_{{\bar T},
1}^T
G_{{\bar T},1}.
\end{align}
Inserting (\ref{nonbalregramp}) and (\ref{difQRSig}) gives {\allowdisplaybreaks\begin{align*}
&\operatorname{tr}(C_1 P_{{\bar T}, r} C_1^T)-\operatorname{tr}(B_1^T \Sigma_{{\bar T}, 1} B_1)\\&=-\operatorname{tr}((A_{11} P_{{\bar T}, r}+P_{{\bar T}, r} A_{11}^T-F_{{\bar T}, r}
F_{{\bar T},
r}^T)(Q_{{\bar T},
r}-\Sigma_{{\bar T}, 1}))\\&=-\operatorname{tr}(P_{{\bar T}, r}((Q_{{\bar T}, r}-\Sigma_{{\bar T}, 1})A_{11}+A_{11}^T(Q_{{\bar T}, r}-\Sigma_{{\bar
T},1})))\\&\quad +\operatorname{tr}(F_{{\bar T}, r} F_{{\bar T},
r}^T(Q_{{\bar T}, r}-\Sigma_{{\bar T}, 1}))\\&=\operatorname{tr}(P_{{\bar T}, r}(G_{{\bar T}, 1}^T G_{{\bar T},1}-G_{{\bar T}, r}^T G_{{\bar T},
r}))+\operatorname{tr}(F_{{\bar T}, r} F_{{\bar T},
r}^T(Q_{{\bar T}, r}-\Sigma_{{\bar T}, 1})).
\end{align*}}
Using again the integral representations of $P_{{\bar T}, r}$ and $Q_{{\bar T}, r}$, we see that {\allowdisplaybreaks \begin{align*}
\operatorname{tr}(P_{{\bar T}, r}G_{{\bar T}, r}^T G_{{\bar T},r})&=\int_0^{\bar T} \operatorname{tr}(\operatorname{e}^{A_{11}s}B_1B_1^T \operatorname{e}^{A_{11}^Ts}
\operatorname{e}^{A^T_{11}{\bar T}}C_1C_1^T\operatorname{e}^{A_{11}{\bar T}})ds
\\& =\int_0^{\bar T} \operatorname{tr}(C_1^T\operatorname{e}^{A_{11}s}\operatorname{e}^{A_{11}{\bar T}}B_1B_1^T \operatorname{e}^{A^T_{11}{\bar T}}\operatorname{e}^{A_{11}^Ts}C_1)ds \\& =\int_0^{\bar T}
\operatorname{tr}(B_1^T \operatorname{e}^{A^T_{11}{\bar T}}\operatorname{e}^{A_{11}^Ts}C_1 C_1^T\operatorname{e}^{A_{11}s}\operatorname{e}^{A_{11}{\bar T}}B_1)ds\\&=\operatorname{tr}(F_{{\bar T}, r}^TQ_{{\bar T}, r}
F_{{\bar T},
r})=\operatorname{tr}(F_{{\bar T}, r} F_{{\bar T}, r}^TQ_{{\bar T}, r}). \end{align*}}
Hence, we have {\allowdisplaybreaks \begin{align*}
&\operatorname{tr}(C_1 P_{{\bar T}, r} C_1^T)-\operatorname{tr}(B_1^T \Sigma_{{\bar T}, 1} B_1)=\operatorname{tr}(P_{{\bar T}, r}G_{{\bar T}, 1}^T G_{{\bar T},1})-\operatorname{tr}(F_{{\bar T},
r} F_{{\bar T},
r}^T\Sigma_{{\bar T}, 1}).
\end{align*}}
The error bound $\epsilon^2$ then is{\allowdisplaybreaks\begin{align*}
\epsilon^2=&\operatorname{tr}(\Sigma_{{\bar T}, 2} (B_2 B_2^T+2 P_{{\bar T}, M, 2} A_{21}^T))\\&+2[\operatorname{tr}(\Sigma_{{\bar T}, 1}F_{{\bar T}, 1}F_{{\bar T},
r}^T)-\operatorname{tr}(G_{{\bar T}, 1}^TG_{{\bar T}}
P_{{\bar T},M})]\\&+\operatorname{tr}(P_{{\bar T}, r}G_{{\bar T}, 1}^T G_{{\bar T},1})-\operatorname{tr}(F_{{\bar T}, r} F_{{\bar T}, r}^T\Sigma_{{\bar T}, 1}).
\end{align*}}
Since {\allowdisplaybreaks \begin{align*}
& 2\operatorname{tr}(\Sigma_{{\bar T}, 1}F_{{\bar T}, 1}F_{{\bar T}, r}^T)=2\left\langle \Sigma_{{\bar T}, 1}^{\frac{1}{2}} F_{{\bar T}, r}, \Sigma_{{\bar
T}, 1}^{\frac{1}{2}}F_{{\bar T},
1}\right\rangle_F\\=&\left\|\Sigma_{{\bar T}, 1}^{\frac{1}{2}} F_{{\bar T}, r}\right\|^2_F+\left\|\Sigma_{{\bar T}, 1}^{\frac{1}{2}} F_{{\bar T},
1}\right\|^2_F-\left\|\Sigma_{{\bar T}, 1}^{\frac{1}{2}} (F_{{\bar T}, 1}-F_{{\bar T}, r})\right\|^2_F,
\end{align*}}
we obtain {\allowdisplaybreaks\begin{align*}
\epsilon^2=&\operatorname{tr}(\Sigma_{{\bar T}, 2} (B_2 B_2^T+2 P_{{\bar T}, M, 2} A_{21}^T))\\&+\operatorname{tr}(\Sigma_{{\bar T}, 1}F_{{\bar T}, 1}F_{{\bar T},
1}^T))-2\operatorname{tr}(P_{{\bar T},M} G_{{\bar T}, 1}^TG_{{\bar T}}
)+\operatorname{tr}(P_{{\bar T}, r}G_{{\bar T}, 1}^T G_{{\bar T},1})\\&-\operatorname{tr}(\Sigma_{{\bar T}, 1} (F_{{\bar T}, 1}-F_{{\bar T}, r})(F_{{\bar T}, 1}-F_{{\bar
T}, r})^T)
\end{align*}}
which is the claimed result.
\end{proof}
\end{thm}
We now discuss the impact of the remainder term $R_{\bar T}:=-2\operatorname{tr}(G^T_{{\bar T}, 1} G_{\bar T} P_{{\bar T}, M})+\operatorname{tr}(G^T_{{\bar T}, 1} G_{{\bar
T}, 1} P_{{\bar T},
r})+\operatorname{tr}(F_{{\bar T}, 1} F^T_{{\bar T}, 1}
\Sigma_{{\bar T}, 1})$ of the error bound in Theorem \ref{thmmaimn}. Every summand of $R_{\bar T}$ can be bounded from above as follows:
\begin{align*}
\operatorname{tr}(G^T_{{\bar T}, 1} G_{\bar T} P_{{\bar T}, M})&\leq \left\| G_{{\bar T}, 1}\right\|_F \left\| G_{{\bar T}}\right\|_F \left\| P_{{\bar T},
M}\right\|_F, \\
\operatorname{tr}(F_{{\bar T}, 1} F^T_{{\bar T}, 1} \Sigma_{{\bar T}, 1})&= \left\|\Sigma_{{\bar T}, 1}^{\frac{1}{2}} F_{{\bar T}, 1}\right\|^2_F\leq \left\|
F_{{\bar T},
1}\right\|^2_F
\operatorname{tr}(\Sigma_{{\bar T},1}),\\
\operatorname{tr}(G^T_{{\bar T}, 1} G_{{\bar T}, 1} P_{{\bar T}, r})&= \left\|P_{{\bar T}, r}^{\frac{1}{2}} G^T_{{\bar T}, 1}\right\|^2_F\leq \left\| G_{{\bar
T}, 1}\right\|^2_F
\operatorname{tr}(P_{{\bar T},r}).
\end{align*}
If $A$ is asymptotically stable, then the norms $\left\| F_{{\bar T}, 1}\right\|_F$,$\left\| G_{{\bar T}, 1}\right\|_F$ and $\left\| G_{{\bar
T}}\right\|_F$
decay exponentially fast, i.e., they are bounded by $c_1\operatorname{e}^{-c_2{\bar T}}$, where $c_1, c_2>0$ are suitable constants.\smallskip
Now, if the terminal time $\bar T$ is sufficiently large, the term $R_{\bar T}$ is small and hence it can be neglected in the error bound. For very
stable systems ($c_2$ is large), $\bar T$ can be chosen small and for slowly decaying systems (small constant $c_2$), $\bar T$ needs to be large in
order to have a sufficiently small $R_{\bar T}$. If the remainder term $R_{\bar T}$ is small, it can be concluded from Theorem \ref{thmmaimn} that
TLBT works well if the truncated time-limited singular values $\sigma_{{\bar T}, r+1}, \ldots, \sigma_{{\bar T}, n}$ are small.\smallskip
For non-stable systems the remainder term $R_{\bar T}$ in the error bound is expected to be large (exponential growth) which might be an
indicator for a large error when applying TLBT to these systems.
\begin{bem}
The representation in Theorem \ref{thmmaimn} is not appropriate to determine the error bound since $B_2$ and $A_{21}$ are never computed in practice.
However, for asymptotically stable systems \eqref{lti} ($R_{\bar T}$ is expected to be small) we know that the reduced order dimension $r$ has to be
chosen such that $\sigma_{{\bar T}, r+1}, \ldots, \sigma_{{\bar T}, n}$ are small in order to guarantee a good approximation. Consequently, looking
at the time-limited singular values instead of computing the error bound~\eqref{implicdeptilp} provides an alternative way to find a suitable reduced
order dimension.
\end{bem}
\section{Practical Considerations}
Here we review the practical execution of TLBT for large-scale systems and evaluate the usefulness of the error bound~\eqref{implicdeptilp} in actual
computations. Directly solving the Lyapunov equations~\eqref{BT_Lyap}, \eqref{TLBT_Lyap} is infeasible for large dimensions.
Therefore, for large-scale systems it has become common practice to approximate the Gramians by low-rank factorizations, e.g.,
$P_{\infty}\approx Z_{\infty}Z_{\infty}^T$ with low-rank factors $Z_{\infty}\in\mathbb{R}^{n\times h}$, rank$(Z_{\infty})=h\ll n$, and similarly for
the other
Gramians. This is justified by the often observed and proven fast singular value decay of solutions of Lyapunov equations~\cite{Gra04}, especially if
$p,m\ll
n$. For this situation there exist efficient algorithms~\cite{BenS13,Sim16} employing techniques from sparse numerical linear algebra for computing
the low-rank
solution factors. For the Lyapunov equations~\eqref{TLBT_Lyap} in TLBT, a rational Krylov subspace method~\cite{DruS11} is proposed in~\cite{morKue17}
that is also able to deal with the arising matrix exponentials. With low-rank approximations $P_{\bar T}\approx Z_{P_{\bar T}}Z_{P_{\bar T}}^T$,
$Q_{\bar
T}\approx
Z_{Q_{\bar T}}Z_{Q_{\bar T}}^T$, one computes the SVD $X\Sigma Y^T=Z_{P_{\bar T}}^TZ_{Q_{\bar T}}$ and projection matrices
$V=Z_{P_{\bar T}} Y_1\Sigma_1^{-\tfrac{1}{2}}$ and $W:=Z_{Q_{\bar T}}X_1\Sigma_1^{-\tfrac{1}{2}}$, where $\Sigma_1$ contains the largest $r$ singular
values and
$X_1,Y_1$ the associated singular vectors.
The reduced order model $\boldsymbol{\Sigma}_r$ is obtained via $A_{11}:=W^TAV$, $B_1:=W^TB$, $C_1:=CV$ which makes it clear that some of the
quantities of the
bound in Theorem~\ref{thmmaimn} are not accessible in practical computations.
However, we may nevertheless acquire an approximation of~\eqref{implicdeptilp}. For this $\operatorname{tr} \left(C P_{\bar T}C^T\right)$ can be approximated by
$\operatorname{tr}
\left(CZ_{P_{\bar T}}^T Z_{P_{\bar T}}C^T\right)$,
$\operatorname{tr} \left(C_1 P_{\bar T,r}C_1^T\right)$ requires solving the $r$ dimensional Lyapunov equation~\eqref{nonbalregramp}, and $\operatorname{tr} \left(C P_{\bar
T,M}C_1^T\right)$
requires the solution of the Sylvester equation~\eqref{mixeqbt2blsa}, which amounts to solve $r$ linear systems of equations defined by $A-\alpha I$,
$\alpha\in\Lambda(A_{11})$ see, e.g.,~\cite[Algorithm~7.6.2]{GolV13}.
Unlike the error bound in BT~\eqref{BT_errorB}, the TLBT bound~\eqref{implicdeptilp} cannot be easily used to adjust the reduced order because when
changing $r$
to, say, $r+d$, $d\geq 1$,
the solutions of~\eqref{lyapeq_btbound} have to be computed entirely from scratch. Especially because of the Sylvester equation~\eqref{mixeqbt2blsa},
this would
be increasingly expensive.
TLBT can with minor adjustments be applied to generalized state-space systems
\begin{align}\label{glti}
\boldsymbol{\Sigma}:\quad E\dot x(t)=Ax(t)+Bu(t),\quad x(0)=0,\quad y(t)=Cx(t)
\end{align}
with $E$ nonsingular. In that case the time-limited Gramians are $P_{\bar T}$, $E^TQ_{\bar T}E$, where $P_{\bar T}$, $Q_{\bar T}$ solve the
generalized Lyapunov equations
\begin{align}\label{TLBT_GLyap}
\begin{split}
AP_{\bar T}E^T+EP_{\bar T}A^T+BB^T-F^E_{\bar T}(F^E_{\bar T})^T&=0,\\
A^TQ_{\bar T}E+E^TQ_{\bar T}A^T+C^TC-(G^E_{\bar T})^TG^E_{\bar T}&=0
\end{split}
\end{align}
with $F^E_{t}:=E\operatorname{e}^{E^{-1}A t}E^{-1}B$ and $G^E_{t}:=C\operatorname{e}^{E^{-1}At}$, see~\cite{morKue17}. Hence, the derivations of Section~\ref{sec2} can be
carried out
as before by using the quantities in \eqref{TLBT_GLyap}. In particular, in the constant in the bound~\eqref{implicdeptilp},
$P_{\bar T,M}$ has to be replaced by the solution $P^E_{\bar T,M}$ of
\begin{align*}
AP^E_{\bar T,M}+EP^E_{\bar T,M}A_{11}+B\tilde B_1-F^E_{\bar T}(F^E_{{\bar T}, r})^T=0,
\end{align*}
where $SE^{-1}B=\begin{bmatrix}\tilde{B}_1\\ \tilde{B}_2\end{bmatrix}$, $F^E_{{\bar T}, r}:=\operatorname{e}^{A_{11}\bar T}\tilde B_1$. Here we employed that the
mass
matrix $E$ is transformed to the identity in (TL)BT. The transformation matrices $V,W$ for TLBT are constructed as before but using the SVD $X\Sigma
Y^T=Z_{P_{\bar T}}^TEZ_{Q_{\bar T}}$, where $Z_{P_{\bar T}},~Z_{Q_{\bar T}}$ are low-rank solution factors of~\eqref{TLBT_GLyap}.
\section{Numerical Experiments}
All following computations are
carried out in MATLAB\textsuperscript{\textregistered} ~8.0.0.783 on a Intel\textsuperscript{\textregistered}Xeon\textsuperscript{\textregistered} CPU X5650 (2.67GHz, 48 GB RAM).
We use the rail model from the Oberwolfach benchmark collection\footnotemark[1] which represents a finite element discretization of a cooling process
of a steel
rail. \footnotetext[1]{http://portal.uni-freiburg.de/imteksimulation/downloads/benchmark}
It provides symmetric positive and negative definite matrices $M$ and, respectively, $A$, as well as $B\in\mathbb{R}^{n\times 7}$,
$C\in\mathbb{R}^{6\times n}$.
We begin
with the coarsest discretization level with $n=1357$ which still allows to compute the matrix exponentials and Lyapunov solutions by direct methods.
The final time is $\bar T=100$, the input chosen as $u(t)=50\mathbf{1}_7$ ($\mathbf{1}_h:=[1,\ldots,1]^T\in\mathbb{R}^h$), and the time integration is
carried
out using an implicit midpoint rule until $T=400$ with a fixed time step $\delta t=0.04$. We generate reduced order models of dimension $r=40$ by both
BT and
TLBT. Figure~\ref{fig:rail1_error} shows the obtained errors $\|y(t)-y_r(t)\|_2$ and the bound~\eqref{implicdeptilp}, clearly indicating that the
proposed
bound is valid. Of course, after leaving $[0,\bar T]$, \eqref{implicdeptilp} is no longer valid and
$\|y(t)-y_r(t)\|_2>\epsilon\left\|u\right\|_{L^2_{\bar T}}$
for some $t>\bar T$.
We also see that ordinary BT provides less accurate reduced order models. It is important to point out that almost identical results were obtained
if low-rank Gramian approximations computed by rational Krylov subspace methods~\cite{DruS11,morKue17} are used. In particular, running the method for
the
restricted Gramians with the same settings as in~\cite{morKue17} led to $\vert \epsilon^{\text{approx.}}-\epsilon^{\text{exact}}\vert\approx 1.6\cdot
10^{-9}$
and visually indistinguishable error norms $\|y(t)-y_r(t)\|_2$.
\begin{figure}[t]
\centering
\includegraphics{H2ErrorBound-figure0.pdf}
\caption{Results obtained by BT and TLBT for small rail model ($n=1357$, $\bar T=100$, $u(t)=50\mathbf{1}_7$, $r=40$).}\label{fig:rail1_error}
\end{figure}
We continue by investigating the influence of the final time $\bar T$ and the reduced order
$r$ to $\max\limits_{t\in [0, {\bar T}]}\|y(t)-y_r(t)\|_2$ and~\eqref{implicdeptilp}.
The results are visualized in Figure~\ref{fig:rail1_errvs_rT}. For the top plot we fixed $\bar T=100$ and varied the reduced order
$r=10,\ldots,100$. Apparently, TLBT achieves smaller
errors than BT for increasing $r$. After some value of $r$, the bound~\eqref{implicdeptilp} appears to stagnate and fails to capture the decreasing
behavior of
the error.
The bottom plot shows the results for a fixed $r=50$ but different final times $\bar T=50,\ldots,300$ which for TLBT requires, naturally, computing
(approximations of) the matrix exponentials and $P_{\bar T},~Q_{\bar T}$ for each value of $\bar T$. The results indicate that increasing $\bar T$
also
increases the achieved error and the bound~\eqref{implicdeptilp} appears to capture this behavior.
As investigated for TLBT in~\cite{morKue17}, for even larger final times $\bar T$, TLBT will at some point produce errors which are very close to
those of BT.
\begin{figure}[t]
\centering
\includegraphics{H2ErrorBound-figure1.pdf}
\\
\quad\includegraphics{H2ErrorBound-figure2.pdf
\caption{Influence of $r$ (top) and $\bar T$ (bottom) for small rail model.}
\label{fig:rail1_errvs_rT}
\end{figure}
Next we experiment with a larger version of the rail model with $n=79841$. This size requires using low-rank solution factors of the Gramians.
We set $u(t)=u_*(t):=[\sin(4t\pi/100),\cos(t\pi/100),3,\operatorname{e}^{-2t},\cos(t/100)\operatorname{e}^{-t},\tfrac{1}{1+t^2},\tfrac{1}{1+\sqrt{t}}]^T$ and $\bar T=150$.
Motivated by Theorem~\ref{thmmaimn}, we experiment with an automatic determination of the reduced order $r$ s.t. $\sum_{i=r+1}^{\hat n}\sigma_{i,\bar
T}\leq \tau$ for some specified tolerance $0<\tau\ll 1$ and $\hat n:=\min(\text{rank}(Z_{P_{\bar T}}),\text{rank}(Z_{Q_{\bar T}}))$, i.e., similar as
in
unrestricted BT. The obtained reduced orders $r$ in BT and TLBT, as well as the largest errors in $[0,\bar T]$ and~\eqref{implicdeptilp} are shown in
Figure~\ref{fig:rail79_errvstau} against different values $\tau=10^{-7},\ldots,10^{-2}$.
TLBT again achieves smaller errors than BT and approximately two orders of magnitude smaller than $\tau$. Note that the obtained reduced orders $r$
of TLBT are
for $\tau=10^{-4},10^{-3},10^{-2}$ slightly larger than those of BT.
This experiment nevertheless suggests that choosing the order $r$ in TLBT automatically by looking at the time-limited singular values is as reliable
as in BT.
\begin{figure}[t]
\centering
\includegraphics{H2ErrorBound-figure3.pdf}
\caption{Automatically adjusted orders $r$, maximum errors, bound ~\eqref{implicdeptilp} against tolerances $\tau$ for the larger rail model
($n=79841$,
$\bar T=150$, $u(t)=u_*(t)$).}
\label{fig:rail79_errvstau}
\end{figure}
\section{Conclusion}
In this paper, we have studied time-limited balanced truncation, an alternative to conventional balanced truncation. This scheme can outperform the
conventional ansatz when seeking for a good reduced order model on a certain finite time interval but, so far, no theory on error bounds has
been established. Therefore, we proved an $\mathcal H_2$ error bound in this work. We provided two different representations for the bound. One is
appropriate for practical computations, whereas the other one shows that the time-limited singular values can be used as well in order to
determine a suitable reduced order dimension. This paper also contains numerical experiments in which we presented the performance of the error bound.
\section*{Acknowledgements}
The authors thank the organizers of the LMS-EPSRC Durham Symposium on Model Order Reduction. The stimulating atmosphere during this meeting has
resulted in the development of the ideas behind this paper. Moreover, the authors thank Peter Benner for his helpful comments.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,528 |
Q: SQL Server Connection Issues with Heroku I wrote my Python app on a windows machine and now I am trying to deploy it to Heroku. At first, my commit kept failing because it couldn't install pyodbc. To fix that, I added a Heroku build pack to preinstall a required apt package first.
I included in the Aptfile:
unixodbc
unixodbc-dev
python-pyodbc
libsqliteodbc
I was able to successfully commit but now I am receiving the following error when trying to run the app:
pyodbc.Error: ('01000', "[01000] [unixODBC][Driver Manager]Can't open lib 'SQL Server' : file not found (0) (SQLDriverConnect)")
Here is the SQL connection syntax in my code that is working just fine on windows:
cnxn = pyodbc.connect(driver='{SQL Server}', host='Company.database.windows.net', database='MyDatabse',trusted_connection='no', user='###', password='###')
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 518 |
Q: Blocking on Asynchronous Calls in Worker Thread in Javascript I have created a Worker in javascript that needs to loop indefinitely. Inside the loop, it needs to make several asynchronous calls and wait for them to complete before continuing on.
There are two immediate ways to do this that I can see that both seem to be fatally flawed:
1)
while(true) {
var promise = async_call();
while(!promise.isResolved()) { }
}
2)
var func = function() {
var promise = async_call();
promise.done(func);
}
func();
For #1, it falls apart because the inner while loop potentially burns up a lot of cpu; and if the thread is not interruptable, (are javascript threads interruptable? I don't think they are), then the async_call will never actually get a chance to complete, so we just get stuck in a loop.
For #2, it would work well if there was tail-call optimization, but I don't think any javascript implementations employ this, so it would quickly result in a stack overflow or other recursion limit.
Lastly, I need a way to signal to both loops when they should terminate. I can easily do this by putting a boolean stop variable into the code, but, again, that relies on the worker thread being interruptable, such that I can set stop to true.
Is there a design pattern that I have overlooked or am otherwise unfamiliar with here? How can I get my worker to execute as fast as possible without resorting to something like setInterval paired with polling isResolved?
A: Pattern 2 is perfectly fine. It will not end up in a stack overflow if the callback is really asynchronous (some Promise libraries enforce that), as the event will be executed with a completely new stack. Only if you would synchronously call func you'd get the overflow, but you just synchronously register it as a callback.
A: Assuming async_call is a XHR request or setTimeout callback, you won't overflow the stack. Event callbacks (such as XHR completion) are called on a new stack; it's not really recursion.
As far as signaling the worker to stop, use message passing.
var worker = new Worker('worker.js');
// when you need to stop it:
worker.postMessage('stop');
worker.js
onmessage = function(e) {
if (e.data == 'stop') {
// stop
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,263 |
From Wikivoyage
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Germany's capital was at the heart of the Cold War and still bears the scars of wars and partition, but it's quickly reclaiming its place as one of Europe's — if not the world's — top cities.
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This travel guide page was last edited at 15:13, on 11 July 2019 by Wikivoyage user Ground Zero. Based on work by Wikivoyage users ThunderingTyphoons!, AndreCarrotflower, SelfieCity, ARR8, Traveler100, Mx. Granger, Wrh2, Torty3 and Nurg and others.
About Wikivoyage | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,957 |
21 state units accept Lodha reforms
With Supreme Court verdict already out, 21 state units have already confirmed in writing to Lodha Panel about implementing all reforms even as former presidents N Srinivasan and Anurag Thakur joined forces in Bengaluru.
It was learnt that 24 administrators of various state units who are either disqualified or going into compulsory cooling off met in the Garden City to discuss their future.
"Today 21 state units have already written to BCCI that they are adopting Lodha Panel reforms. So if 24 individuals, who are no longer qualified officials meet anywhere in India, no one should be bothered. They are individuals who went on their own capacity. These officials who went stand disqualified as per SC verdict," a source close to Lodha Panel told PTI today.
Asked about reports that some associations may not allow matches to happen in their stadiums, the source, himself a legal expert of repute said: "It is strange that an individual no longer associated with BCCI can make such claims about a facility built on government land and with BCCI. It's not an individual property."
In Bengaluru today, apart from Srinivasan and Thakur, former secretary Ajay Shirke, sacked by Apex Court along with joint secretary Amitabh Chaudhary and treasurer Aniruddh Chaudhary attended the meeting. IPL chairman Rajeev Shukla was another prominent member.
All these officials are now Persona Non Grata after the SC verdict on January 2.
Mumbai Cricket Association's PV Shetty was also present in the meeting.
The six associations that did not come for the meeting are institutional bodies Railways, Services, Universities along with National Cricket Club (NCC), Vidarbha CA and DDCA.
Putting aside their differences, Thakur and Srinivasan, who had taken potshots at each other not so long ago exchanged opinions on way forward in light of the Apex Court verdict.
"Yes, it was an informal meeting. Thakur and Srinivasan were very cordial with each other. Obviously the current situation was discussed. Srinivasan asked if we are all together in it or not. Even Thakur understands, he needs Srinivasan by his side now. Out of 24, at least 18 are still Srini loyalists," a state association official told PTI on conditions of anonymity.
Asked if there was any discussion of state associations preventing the new set of observers from hosting matches at cricket stadiums, he said: "That's absurd. Nothing of that sort has been discussed. At least it won't happen in my association," said the official from one of the eastern state units.
Was there any discussions on forming parallel body, he kept cards close to his chest.
"Today's meeting was about checking unity and we will again have a meeting in two weeks time. Let the January 19 pass and we can take it on from there. As of now, nothing concrete has been discussed. There is a Supreme Court verdict and we can't do anything about it now," he added.
Courtesy: Press Trust of India
Mumbai Cricket Association's PV Shetty was also present in the meeting.The six associations that did not come for the meeting are institutional bodies Railways, Services, Universities along with National Cricket Club (NCC), Vidarbha CA and DDCA. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,623 |
Киммо Ханну Тапио Хакола (; 27 июля 1958, Йювяскюля, Финляндия) — финский композитор.
Биография
Родился 27 июля 1958 года в Йювяскюля, в Финляндии.
Начал своё музыкальное образование беря частные уроки композиции у известного финского композитора Эйноюхани Раутаваара.
В 1981 году поступил в Академию Сибелиуса, где обучался у Магнуса Линдберга и .
Среди произведений композитора несколько концертов в том числе фортепианный концерт (1996) и концерт для кларнета (2001), несколько струнных концертов и концертов для хора и оркестра. Композитор является также автором ряда опер — «Marsin mestarilaulajat» и «Sinapinsiemen».
6 июля 2012 года на международном оперном фестивале в Савонлинна состоялась мировая премьера его трагикомической оперы-буфф «La Fenice» (либретто Юха-Пекка Хотинен, автор перевода либретто на итальянский — Никола Райно).
Творчество
1996 — Фортепианный концерт
2001 — Концерт для кларнета
Опера — «Marsin mestarilaulajat»
Опера — «Sinapinsiemen»
2012 — опера «La Fenice»
Примечания
Ссылки
Kimmo Hakola на Fennica Gehrman
Оперные композиторы Финляндии | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,388 |
Q: Appending a list to a list of lists In the following code, I am trying to append a list called a to a list of lists b.
a = [5,4]
b = [[4],[3],[8]]
b[2].append(a)
Python outputs
[[4], [3], [8, [5, 4]]]
However, I want the elements to be appended as integers not as a list so b should be [[4], [3], [8, 5, 4]] and then I want to merge the lists so b would be [4, 3, 8, 5, 4]. I want to be able to do this so I can use the sum function to find the sum of b's elements. Does anyone have suggestions regarding how this can be done?
A: First, you are trying to extend a list, not append to it. Specifically, you want to do
b[2].extend(a)
append() adds a single element to a list. extend() adds many elements to a list. extend() accepts any iterable object, not just lists. But it's most common to pass it a list.
Once you have your desired list-of-lists, e.g.
[[4], [3], [8, 5, 4]]
then you need to concatenate those lists to get a flat list of ints. You can use sum() for that -- adding lists is not that different from adding ints.
b = sum(b, [])
The trick here is that you have to pass the initial (empty) value to sum(), otherwise it tries to add the lists in b as though they were numbers.
Finally, you can sum the flattened list as you intended:
sum(b)
A: My suggestion is to use the chain function taken from the itertools builtin package to achieve your goal in a more concise and pythonic way.
Just one line of code is needed:
sum(chain(a, tuple(chain.from_iterable(b))))
A: If I understood it correctly, you want a list full of integers -not a list with lists of integers-.
Anyways I'll try to give a solution for both of them.
for i in range(len(a)):
b[-1].append(a[i])
Outputs: b = [[4], [3], [8, 5, 4]]
or maybe just every number in one list:
for i in range(len(a)):
b.append([a[i]])
Outputs: b = [[4], [3], [8], [5], [4]]
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 333 |
using System;
using Xamarin.Forms;
namespace ElasticSharp.Mobile
{
public partial class AboutPage : ContentPage
{
public AboutPage()
{
InitializeComponent();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,733 |
Llano Blanco kan syfta på:
Geografi
Mexiko
Llano Blanco, Sonora, ort,
Llano Blanco, Hidalgo, ort, Zimapán,
Llano Blanco, Querétaro Arteaga, ort, Cadereyta de Montes,
Llano Blanco, Guanajuato, ort, Santa Catarina,
Robotskapade Mexikoförgreningar | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,277 |
\section{Motivations}
The Hermitian Jacobi process was introduced in \cite{Dou} as a multidimensional analogue of the real Jacobi process. It is a stationary matrix-valued process whose distribution converges weakly in the large-time limit to the matrix-variate Beta distribution describing the Jacobi unitary ensemble (hereafter JUE). The latter was used in \cite{DFS} as a random matrix-model for a Multi-Input-Multi-Output (MIMO) optical fiber channel. There, numerical evidences for the Shannon capacity and for the outage probability were supplied and support the efficiency of the matrix model. From a general fact about unitarily-invariant matrix models, this capacity may be expressed through the Christoffel-Darboux kernel for Jacobi polynomials which is the one-point correlation function of the underlying eigenvalues process (\cite{Meh}). Yet another expression for it was recently obtained in \cite{DN} relying on a remarkable formula for the moments of the unitary selberg weight (\cite{CDLV}). The strategy employed in \cite{CDLV} was partially adapted in \cite{Del-Dem} to the Hermitian Jacobi process and led to a quite complicated formula for its moments which did not allow to derive their large-size limits. The main ingredients used in \cite{Del-Dem} were the expansion of Newton power sums in the basis of Schur functions, the determinantal form of the symmetric Jacobi polynomials and an integral form of the Cauchy-Binet formula (known as Andreief's identity).
In this paper, we follow another approach to compute the moments of the Hermitian Jacobi process based on a change of basis in the algebra of symmetric functions (with a fixed number of indeterminates). More precisely, we rather express the Newton power sums in the basis of symmetric Jacobi polynomials since the latter are mutually orthogonal with respect to the unitary Selberg weight. Doing so leads to the determinant of an `almost triangular' matrix which we express in a product form using row operations. After a careful rearrangement of the terms, we end up with a considerably simpler moment formula compared to the one obtained in \cite{Del-Dem} (Theorem 1). Actually, the latter involves three nested and alternating sums together with a determinant whose entries are Beta functions. Up to our best knowledge, this determinant has no closed form except in very few special cases. The moment formula obtained in this paper contains only two nested and alternating sums whose summands are ratios of Gamma functions.
As a potential application of our formula, we propose the Hermitian Jacobi process as a dynamical analogue of the MIMO Jacobi channel studied in \cite{DFS} and compute its Shannon capacity for small power per-antenna at the transmitter. Motivated by free probability theory, we also give some interest in the case when the size of the Hermitian Jacobi process is larger than the moment order. In this respect, our moment formula may be written as a linear combination of terminating balanced ${}_4F_3$-hypergeometric series evaluated at unit argument (\cite{AAR}, Chapter 3).
The paper is organized as follows. In the next section, we briefly review the construction of the Hermitian Jacobi process and recall the semi-group density of its eigenvalues process (when it exists). In the third section, we state our main result in Theorem \ref{teo}below and prove it. For ease of reading, we proceed in several steps until getting the sought moment formula. In the last section, we discuss the application of our main result to optical fibers MIMO channel and to the large-size limit of the moments of the Hermitian Jacobi process.
\section{A review of the Hermitian Jacobi process}
For sake of completeness, we recall the construction of the Hermitian Jacobi process and the expression of the semi-group density of its eigenvalues process. We refer the reader to \cite{Dou} and \cite{Del-Dem} for further details.
Denote $U(d), d \geq 2,$ the group of complex unitary matrices. Let $p, m \leq d$ be two integers and let $Y = (Y_t)_{t \geq 0}$ be a $U(d)$-valued stochastic process. Set :
\begin{align*}
X_t \oplus 0 := PY_tQ, \quad t \geq 0,
\end{align*}
where :
\begin{align*}
P :=\begin{pmatrix}
\textrm{Id}_{m\times m} & 0_{m\times (d-m)}\\
0_{ (d-m)\times m} & 0_{(d-m)\times(d-m)}
\end{pmatrix}, \quad
Q :=\begin{pmatrix}
\textrm{Id}_{p\times p} & 0_{p\times (d-p)}\\
0_{ (d-p)\times p}& 0_{(d-p)\times(d-p)}
\end{pmatrix}.
\end{align*}
are orthogonal projections. In other words, $X$ is the upper left corner of $Y$. Assume now that $Y$ is the Brownian motion on $U(d)$ starting at the identity matrix. Then,
\begin{equation*}
J_t := X_tX_t^* = PY_tQY_t^*P, \quad t \geq 0,
\end{equation*}
is called the Hermitian Jacobi process of size $m \times m$ and of parameters $(p,q)$ where $q=d-p$. As $t \rightarrow +\infty$, $Y_t \rightarrow Y_\infty,$ where $Y_\infty$ is a Haar unitary matrix and the convergence holds in the weak sense. Moreover, it was proved in \cite{Col} that the random matrix
\begin{equation*}
J_\infty=X_\infty X_\infty^* = PY_{\infty}QY_{\infty}^*P
\end{equation*}
has the same distribution drawn from JUE with suitable parameters.
For any $n \geq 1$, define the $n$-th moment of $J_t$ by:
\begin{align*}
M_{n, p, m, d}(t) :=\mathbb{E} \left(\textrm{tr}\left(\left(J_t\right)^n\right)\right),
\end{align*}
for fixed time $t \geq 0$ and write simply $M_n(t)$. Since the matrix Jacobi process is Hermitian, then
\begin{align*}
M_n(t) =\mathbb{E} \left(\sum_{k=1}^m (\lambda_k(t))^n\right),
\end{align*}
where $(\lambda_k(t), t \geq 0)_{k=1}^m$ is the eigenvalues process of $(J_t)_{t \geq 0}$ and $\mathbb{E}$ stands for the expectation of the underlying probability space. If
\begin{equation*}
r:= p-m \geq 0, \quad s:= d-p-m = q-m \geq 0,
\end{equation*}
then the distribution of the eigenvalues process is absolutely-continuous with respect to Lebesgue measure in $\mathbb{R}^m$. Besides, its semi-group density is given by a bilinear generating series of symmetric Jacobi polynomials with Jack parameter equals to $1$. More precisely, let
\begin{equation*}
\tau=(\tau_1 \geq \tau_2 \geq ... \geq \tau_m \geq 0)
\end{equation*}
be a partition of length at most $m$ and let $(P_k^{r,s})_{k\geq 0}$ be the sequence of orthonormal Jacobi polynomials with respect to the beta weight:
\begin{equation*}
u^r(1-u)^s {\bf 1}_{[0,1]}(u).
\end{equation*}
These polynomials may be defined through the Gauss hypergeometric function as:
\begin{equation*}
P_k^{r,s}(u) := \left[\frac{(2k+r+s+1)\Gamma(k+r+s+1)k!}{\Gamma(r+k+1)\Gamma(s+k+1)}\right]^{1/2}\frac{(r+1)_k}{k!}{}_2F_1(-k, k+r+s+1, r+1; u).
\end{equation*}
Then the orthonormal symmetric Jacobi polynomial corresponding to $\tau$ is defined by:
\begin{align*}
P_{\tau}^{r,s,m}(x_1,...,x_m) := \frac{\det(P_{\tau_i-i+m}^{r,s}(x_j))_{1\leq i,j \leq m}}{V(x_1,...,x_m)}, \quad V(x_1,...,x_m) := \prod_{1\leq i<j\leq m}(x_i-x_j),
\end{align*}
if the coordinates $(x_1,...,x_m)$ do not overlap and by L'H\^opital's rule otherwise. An expansion of these polynomials in the basis of Schur functions may be found in \cite{Las}. These polynomials are mutually orthonormal with respect to the unitary Selberg weight:
\begin{equation*}
W^{r,s,m}(y_1,\dots, y_m):= [V(y_1, \dots, y_m)]^2\prod_{i=1}^m y_i^r(1-y_i)^s{\bf 1}_{0<y_m<...<y_1<1},
\end{equation*}
in the sense that two elements corresponding to different partitions are orthogonal and the norm of each $P_{\tau}^{r,s,m}$ equals one (see e.g. \cite{BGU}, Theorem 3.1). Moreover, the semi-group density of the eigenvalues process of $J_t$ admits the following absolutely-convergent expansion (\cite{Dem01}):
\begin{align*}
G_t^{r,s,m}(1^m,y) := \sum_{\tau} e^{-\nu_\tau t}P_\tau^{r,s,m}(1^m)P_\tau^{r,s,m}(y) W^{r,s,m}(y_1,\dots, y_m),
\end{align*}
where $1^m := (\underbrace{1, \dots, 1}_{m \, \, \textrm{times}}),$ and
\begin{align*}
\nu_{\tau} :=\sum_{i=1}^m\tau_i(\tau_i+r+s+1+2(m-i)).
\end{align*}
If $(\tilde{P}_{k}^{r,s})_{k \geq 0}$ denotes the sequence of orthogonal Jacobi polynomials:
\begin{equation*}
\tilde{P}_k^{r,s}(u) := \frac{(r+1)_k}{k!}{}_2F_1(-k, k+r+s+1, r+1; u),
\end{equation*}
then $G_t^{r,s,m}(1^m,y)$ may be written as:
\begin{align}\label{SD}
G_t^{r,s,m}(1^m,y) = \sum_{\tau} e^{-\nu_\tau t}\prod_{j=1}^m \frac{1}{(||\tilde{P}_{\tau_j+m-j}^{r,s}||_2)^2}\tilde{P}_\tau^{r,s,m}(1^m)\tilde{P}_\tau^{r,s,m}(y) W^{r,s,m}(y_1,\dots, y_m),
\end{align}
where $(||\tilde{P}_{\tau_j+m-j}^{r,s}||_2)^2$ is the squared $L^2$-norm of the one-variable Jacobi polynomial and
\begin{align*}
\tilde{P}_\tau^{r,s,m}(x_1,...,x_m) := \frac{\det(\tilde{P}_{\tau_i-i+m}^{r,s}(x_j))_{1\leq i,j \leq m}}{V(x_1,...,x_m)}.
\end{align*}
Indeed, Andreief's identity (\cite{DG}, p.37) shows that $(\tilde{P}_\tau^{r,s,m})_{\tau}$ is an orthogonal set with respect the unitary Selberg weight and that the squared $L^2$-norm of $\tilde{P}_\tau^{r,s,m}$ with respect to $W^{r,s,m}$ is nothing else but:
\begin{equation*}
\prod_{j=1}^m (||P_{\tau_j+m-j}^{r,s}||_2)^2.
\end{equation*}
On the other hand, the polynomial set $(\tilde{P}_\tau^{r,s,m})_{\tau}$ may be mapped to the set of symmetric Jacobi polynomials $(Q_{\tau}^{r,s,m})_{\tau}$ considered in \cite{Ols-Osi} by the affine transformation:
\begin{equation*}
(x_1,\dots, x_m) \in [0,1]^m \mapsto (1-2x_1, \dots, 1-2x_m) \in [-1,1]^m.
\end{equation*}
More precisely, one has:
\begin{equation*}
P_{\tau}^{r,s,m}(x_1,\dots,x_m) = (-2)^{m(m-1)/2}Q_{\tau}^{r,s,m}(1-2x_1,\dots,1-2x_m).
\end{equation*}
Moreover, the following mirror property is satisfied by $(Q_{\tau}^{r,s,m})_{\tau}$:
\begin{equation*}
Q_{\tau}^{r,s,m}(-x_1,\dots,-x_m) = (-1)^{|\tau|}Q_{\tau}^{s,r,m}(x_1,\dots,x_m),
\end{equation*}
and is inherited from their one-variable analogues. Indeed, one checks directly this property when $x$ has distinct coordinates using the determinantal form of $Q_{\tau}^{r,s,m}$ then extends it by continuity. In particular:
\begin{equation*}
P_{\tau}^{r,s,m}(1^m) = (-2)^{m(m-1)/2}Q_{\tau}^{r,s,m}(\underbrace{-1, \dots, -1}_{m \, \textrm{times}}) = (-1)^{|\tau|}(-2)^{m(m-1)/2}Q_{\tau}^{s,r,m}(1^m).
\end{equation*}
But Proposition 7.1 in \cite{Ols-Osi} gives:
\begin{multline*}
Q_{\tau}^{s,r,m}(1^m) = \prod_{1 \leq i < j \leq m}(\tau_i+\tau_j+2m-i-j+r+s+1)(\tau_i-\tau_j + j-i)
\\ \prod_{j=1}^m \frac{\Gamma(\tau_j+m-j+s+1)2^{-(m-j)}}{\Gamma(\tau_j+m-j+1)\Gamma(m-j+s+1)\Gamma(m-j+1)}.
\end{multline*}
As a result, we get the special value:
\begin{multline}\label{SV}
\tilde{P}_\tau^{r,s,m}(1^m) = (-1)^{|\tau|+m(m-1)/2} \prod_{1 \leq i < j \leq m}(\tau_i+\tau_j+2m-i-j+r+s+1)(\tau_i-\tau_j + j-i)
\\ \prod_{j=1}^m \frac{\Gamma(\tau_j+m-j+s+1)}{\Gamma(\tau_j+m-j+1)\Gamma(m-j+s+1)\Gamma(m-j+1)},
\end{multline}
which will be used in our forthcoming computations below.
\section{Main result: The moment formula}
Let $n \geq 1$ and recall that a hook $\alpha$ of weight $|\alpha| = n$ is a partition of the form:
\begin{equation*}
\alpha = (n-k, 1^k).
\end{equation*}
For partitions $\alpha, \tau$, recall the order induced by the containment of their Young diagrams: $\tau \subseteq \alpha$ if and only if $\tau_i \leq \alpha_i$ for any $1 \leq i \leq l(\tau) \leq l(\alpha)$, where the length $l(\tau)$ is the number of non-zero components of $\tau$.
On the other hand, the $n$-th moment of the stationary distribution $J_{\infty}$ is given by the normalized integral\footnote{As for fixed $t > 0$, we omit the dependence of the stationary moments on $(r,s,m)$.}:
\begin{align*}
M_n(\infty) &:= \frac{1}{Z^{r,s,m}}\int \left( \sum_{i=1}^m y_i^n \right) W^{r,s,m}(y)dy
\\& = \frac{1}{m!Z^{r,s,m}}\int_{[0,1]^m} \left( \sum_{i=1}^m y_i^n \right) [V(y_1, \dots, y_m)]^2\prod_{i=1}^m y_i^r(1-y_i)^s dy,
\end{align*}
where
\begin{equation*}
Z^{r,s,m}:= \int W^{r,s,m}(y)dy,
\end{equation*}
is the Selberg integral. The explicit expression of $M_n(\infty)$ may be read off Corollary 2.3 in \cite{CDLV}. With these notations, our main result is stated as follows:
\begin{teo}\label{teo}
The $n$-th moment of the Hermitian Jacobi process is given by:
\begin{equation}\label{FinalForm}
M_n(t) = M_n(\infty) + \sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} \sum_{\substack{\tau \subseteq \alpha \\ \tau \neq \emptyset}}
\frac{e^{-\nu_\tau t} \, \tilde{V}_{\alpha_1, \tau_1}^{r,s,m} \, U_{l(\alpha), l(\tau)}^{r,s,m} }{(r+s+\tau_1+2m-l(\tau)) (\tau_1+l(\tau)-1)}
\end{equation}
where
\begin{align*}
\tilde{V}_{\alpha_1, \tau_1}^{r,s,m}
= \frac{(r+s+2\tau_1+2m-1)\Gamma(\tau_1+2m+r+s)\Gamma(\alpha_1+m)\Gamma(r+\alpha_1+m)\Gamma(\tau_1+m+s)}{(\alpha_1-\tau_1)!(\tau_1-1)!\Gamma(r+s+\alpha_1+\tau_1+2m)\Gamma(r+\tau_1+m)},
\end{align*}
and
\begin{align*}
U_{l(\alpha), l(\tau)}^{r,s,m} : = \frac{(2m+r+s+1-2l(\tau))\Gamma(r+m-l(\tau)+1)\Gamma(r+s+2m-l(\alpha)-l(\tau)+1)}{(l(\alpha)- l(\tau))!(l(\tau)-1)!\Gamma(m-l(\alpha)+1)\Gamma(r+m-l(\alpha)+1)\Gamma(m+s-l(\tau)+1)\Gamma(2m+r+s-l(\tau)+1)}.
\end{align*}
\end{teo}
The rest of this section is devoted to the proof of this result. Due to lengthy computations, we shall proceed in several steps where in each step, we simplify the moment expression obtained in the previous one.
\subsection{The basis change}
We start with performing the change of basis from Schur polynomials to symmetric Jacobi polynomials. Doing so leads to the following formula for $M_n(t)$ :
\begin{pro}
For any $m, n \geq 1, t > 0$, we have:
\begin{multline}\label{MF1}
M_n(t)=\sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} \sum_{\tau \subseteq \alpha}
e^{-\nu_\tau t}\tilde{P}_\tau^{r,s,m}(1)\det \left( \frac{(-(\alpha_i-i+m))_{\tau_j-j+m}}{\Gamma(r+s+\tau_j-j+\alpha_i-i+2m+2)} \right)_{i,j=1}^m
\\ \times \prod_{j=1}^m\frac{(r+s+2(\tau_j-j+m)+1)\Gamma(r+\alpha_j+m-j+1)\Gamma(r+s+\tau_j-j+m+1)}{\Gamma(r+\tau_j-j+m+1)},
\end{multline}
where
\begin{equation*}
(-(\alpha_i-i+m))_{\tau_j-j+m} = (-1)^{\tau_j + m-j} \frac{(\alpha_i+m-i)!}{(\alpha_i-\tau_j + j- i)!}{\bf 1}_{\alpha_i-i \geq \tau_j-j}.
\end{equation*}
\end{pro}
\begin{proof}
Recall the $n$-th Newton power sum (\cite{MD}):
\begin{align*}
p_n(y) := \sum_{i=1}^m y_i^n,
\end{align*}
as well as the Schur polynomials associated to a partition $\tau$ of length $l(\tau) \leq m$ (\cite{MD}):
\begin{equation*}
s_{\tau}(x) := \frac{\det(x_j^{\tau_i-i+m})_{1\leq i,j \leq m}}{V(x_1,...,x_m)}.
\end{equation*}
These symmetric functions are related by the representation-theoretical formula (see e.g. \cite{MD}, p. 48):
\begin{align*}
p_n(y) = \sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} s_{\alpha}(y).
\end{align*}
In order to integrate the Newton sum against the semi-group density \eqref{SD}, we shall further expand the Schur polynomials in the basis of symmetric Jacobi polynomials $(\tilde{P}_\tau^{r,s,m})_{\tau}$. To this end, we appeal to the inversion formula (\cite{KK}):
\begin{align*}
y^j=\Gamma(r+j+1)\sum_{l=0}^j \frac{(-j)_l(r+s+2l+1)\Gamma(r+s+l+1)}{\Gamma(r+l+1)\Gamma(r+s+l+j+2)} \tilde{P}_l^{r,s}(y),
\end{align*}
together with Proposition 3.1 in \cite{Ols} (change of basis formula). Doing so yields:
\begin{align*}
s_\alpha(y) = \prod_{i=1}^m\Gamma(r+\alpha_i+m-i+1)\sum_{\mu \subset \alpha} \det \left(a(\alpha_i-i+m,\mu_j-j+m)\right)_{1\leq i,j\leq m} \tilde{P}_\mu^{r,s,m}(y),
\end{align*}
where we set:
\begin{multline*}
a(\alpha_i-i+m,\mu_j-j+m):= (-(\alpha_i-i+m))_{\mu_j-j+m} \\ \frac{(r+s+2(\mu_j-j+m)+1)\Gamma(r+s+\mu_j-j+m+1)}{\Gamma(r+\mu_j-j+m+1)\Gamma(r+s+\mu_j-j+\alpha_i-i+2m+2)}.
\end{multline*}
Integrating $y \mapsto p_n(y)G_t^{r,s,m}(1^m,y)$ and applying Fubini Theorem, we are led to:
\begin{align*}
\int \left( \sum_{i=1}^m y_i^n \right) \tilde{P}_\tau^{r,s,m}(y) & W^{r,s,m}(y)dy = \frac{1}{m!}\int_{[0,1]^m} \left( \sum_{i=1}^m y_i^n \right) \tilde{P}_\tau^{r,s,m}(y) [V(y_1, \dots, y_m)]^2\prod_{i=1}^m y_i^r(1-y_i)^s dy
\\& = \sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} \sum_{\mu \subset \alpha} \det \left(a(\alpha_i-i+m,\mu_j-j+m)\right)_{1\leq i,j\leq m}
\\& \frac{1}{m!} \int_{[0,1]^m} \tilde{P}_\mu^{r,s}(y) \tilde{P}_\tau^{r,s,m}(y) [V(y_1, \dots, y_m)]^2\prod_{i=1}^m y_i^r(1-y_i)^s dy
\\& =\sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} \sum_{\tau \subset \alpha} \prod_{j=1}^m (||P_{\tau_j+m-j}^{r,s}||_2)^2 \det \left(a(\alpha_i-i+m,\tau_j-j+m)\right)_{1\leq i,j\leq m},
\end{align*}
where the last equality follows again from Andreief's identity. Keeping in mind the series expansion \eqref{SD}, the stated moment formula follows.
\end{proof}
\subsection{An almost upper-triangular matrix}
For sake of simplicity, we introduce the following notations :
\begin{align*}
n_i=\alpha_i+m-i, \quad m_i=\tau_i +m-i.
\end{align*}
Using \eqref{SV}, the moment formula \eqref{MF1} is written more explicitly as:
\begin{multline}\label{MF2}
M_n(t) = \sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} \sum_{\tau \subseteq \alpha} e^{-\nu_\tau t} \prod_{1 \leq i < j \leq m}(m_i+m_j+r+s+1)(m_i-m_j)
\\ (-1)^{|\tau|+m(m-1)/2} \prod_{j=1}^m\frac{(r+s+2m_j+1)\Gamma(r+n_j+1)\Gamma(m_j+s+1)\Gamma(r+s+m_j+1)}{\Gamma(r+m_j+1)\Gamma(m_j+1)\Gamma(m-j+s+1)\Gamma(m-j+1)}
\\ \det \left( \frac{(-n_i)_{m_j}}{\Gamma(r+s+n_i+m_j+2)} \right)_{i,j=1}^m.
\end{multline}
Since $\alpha_i = \tau_j$ for $i, j > l(\alpha), 2 \leq i,j \leq l(\tau),$ then $n_i < m_j$ provided that $i > j$. Similarly, $1=\alpha_i > \tau_j=0, l(\tau)+1 \leq i,j \leq l(\alpha)-1,$ implies the same conclusion when $i > j+1$. These elementary observations show that the matrix above is `almost upper-triangular'.
\begin{lem}\label{Upper}
For any hook $\alpha$ of weight $n \geq 1$ and length $l(\alpha) \leq m$, and any $\tau \subset \alpha$, set:
\begin{equation*}
b_{\alpha, \tau}(i,j) := \frac{(-n_i)_{m_j}}{\Gamma(r+s+n_i+m_j+2)}, \quad B_{\alpha, \tau} := \left(b_{\alpha,\tau}(i,j)\right)_{i,j=1}^m.
\end{equation*}
Then $b_{\alpha, \tau}(i,j) = 0$ for $i \geq j+2,$ and:
\begin{itemize}
\item If $l(\alpha) < l(\tau) + 2$ then $B_{\alpha, \tau}$ is upper triangular.
\item Otherwise, $b_{\alpha, \tau}(j+1,j) = 0$ for $j \geq l(\alpha)$ or $j \leq l(\tau)$ while
\begin{equation*}
b_{\alpha, \tau}(j+1,j) = \frac{(-1)^{m-j}(m-j)!}{\Gamma(r+s+2m-2j+2)}, \quad l(\tau)+1 \leq j \leq l(\alpha)-1, \quad \tau \neq \emptyset.
\end{equation*}
\end{itemize}
\end{lem}
\begin{proof}
Take $i > j \geq 1$. Then, $\alpha_i-i \leq 1-i$ while $1-i <\tau_j-j$ except when $\tau_j=0$ and $i=j+1$. Consequently, $(-(\alpha_i-i+m))_{\tau_j-j+m} = 0$ in the following three cases:
\begin{enumerate}
\item $\alpha_i=0$.
\item $i \geq j+2$.
\item $i=j+1$ and $\tau_{j} \geq 1$.
\end{enumerate}
In particular, $B_{\alpha,\tau}$ is upper triangular if $l(\alpha) = l(\tau)$ or $l(\alpha) = l(\tau)+1$ since then $\alpha_i \leq \tau_j$. Otherwise, if $l(\alpha) \geq l(\tau)+2$ then $b(j+1,j)$ vanishes except for $l(\tau)+1 \leq j \leq l(\alpha)-1$ in which case
\begin{equation*}
\alpha_{j+1}=1 > \tau_j = 0 \quad \Rightarrow \quad n_{j+1} = m_j = m-j.
\end{equation*}
\end{proof}
\subsection{Further simplifications}
According to Lemma \ref{Upper}, \eqref{MF2} is expanded as:
\begin{multline}\label{MF3}
M_n(t)=\sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} \sum_{\tau \subseteq \alpha}\frac{e^{-\nu_\tau t}}{(\alpha_1-\tau_1)!}\prod_{1 \leq i < j \leq m}(m_i+m_j+r+s+1)(m_i-m_j)
\\ \prod_{j=1}^{l(\tau)} \frac{n_j!}{\Gamma(r+s+n_j+m_j+2)} \det \left( \frac{n_i!}{(n_i-m_j)!\Gamma(r+s+n_i+m_j+2)}\right)_{\substack{j=l(\tau)+1 \dots l(\alpha), \\ i \leq j+1}}
\\ \prod_{j=l(\alpha)+1}^m \frac{n_j!}{\Gamma(r+s+2m_j+2)}
\prod_{j=1}^m\frac{(r+s+2m_j+1)\Gamma(r+n_j+1)\Gamma(m_j+s+1)\Gamma(r+s+m_j+1)}{\Gamma(r+m_j+1)\Gamma(m_j+1)\Gamma(m-j+s+1)\Gamma(m-j+1)},
\end{multline}
where an empty determinant or product equals one. This expression can be considerably simplified into the one below where we prove that the factors corresponding to indices $l(\alpha) + 1 \leq i,j \leq m$ cancel. To this end, we find it convenient to single out the contribution of the empty partition which corresponds to the stationary regime $t \rightarrow +\infty$.
\begin{cor}
The moment formula \eqref{MF3} reduces to:
\begin{multline}\label{MF4}
M_n(t) = M_n(\infty) + \sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} \sum_{\substack{\tau \subseteq \alpha \\ \tau \neq \emptyset}}
\frac{e^{-\nu_\tau t}}{(r+s+\tau_1+2m-l(\tau))(\tau_1+l(\tau)-1)}V_{\alpha_1, \tau_1}^{r,s,m} \\ \prod_{i=2}^{l(\tau)} \frac{(m-i+1)(m-i+s+1)}{(2m-i-l(\tau)+r+s+2)(1+l(\tau)-i)}\prod_{j=l(\tau)+1}^{l(\alpha)}(m-j+1)(r+m-j+1)
\\ \det \left(\frac{\Gamma(r+s+2(m-j)+2)}{(n_i-m_j)!\Gamma(r+s+n_i+m_j+2)}\right)_{\substack{j=l(\tau)+1 \dots l(\alpha), \\ i \leq j+1}},
\end{multline}
where for a non empty hook $\tau \subseteq \alpha$, we set:
\begin{equation}\label{Contrib1}
V_{\alpha_1, \tau_1}^{r,s,m} := \frac{(r+s+2\tau_1+2m-1)\Gamma(\tau_1+2m+r+s)\Gamma(\alpha_1+m)\Gamma(r+\alpha_1+m)\Gamma(\tau_1+m+s)}{(\alpha_1-\tau_1)!(\tau_1-1)!\Gamma(r+s+\alpha_1+\tau_1+2m)\Gamma(m)\Gamma(r+\tau_1+m)\Gamma(m+s)}.
\end{equation}
\end{cor}
\begin{proof}
We only consider hooks $\tau$ such that $l(\tau) \geq 1$ and we proceed in three steps. In the first one, we work out the product:
\begin{align*}
\prod_{l(\alpha)+1\leq i < j \leq m}(m_i+m_j+r+s+1)(m_i-m_j) &= \prod_{i=l(\alpha)+1}^{m-1}\prod_{j=i+1}^m(2m-i-j+r+s+1)(j-i)
\\& = \prod_{i=l(\alpha)+1}^{m-1}\Gamma(m-i+1)(m-i+r+s+1)_{m-i}
\\& = \prod_{i=l(\alpha)+1}^{m}\frac{\Gamma(m-i+1)\Gamma(2m-2i+r+s+1)}{\Gamma(m-i+r+s+1)}.
\end{align*}
Since $n_j = m_j = (m-j)$ for $j \geq l(\alpha)+1$, then
\begin{multline*}
\prod_{l(\alpha)+1\leq i < j \leq m}(m_i+m_j+r+s+1)(m_i-m_j) \\ \prod_{j=l(\alpha)+1}^m \frac{n_j!}{\Gamma(r+s+2m_j+2)}
\frac{(r+s+2m_j+1)\Gamma(r+n_j+1)\Gamma(m_j+s+1)\Gamma(r+s+m_j+1)}{\Gamma(r+m_j+1)\Gamma(m_j+1)\Gamma(m-j+s+1)\Gamma(m-j+1)}
\end{multline*}
equals one. In the second step, we split the product
\begin{equation*}
\prod_{\substack{1 \leq i \leq l(\alpha) \\ i+1 \leq j \leq m}}(m_i+m_j+r+s+1)(m_i-m_j)
\end{equation*}
into
\begin{equation*}
\prod_{i=l(\tau)+1}^{l(\alpha)} \prod_{j= i+1}^m (m_i+m_j+r+s+1)(m_i-m_j)
\end{equation*}
and
\begin{equation*}
\prod_{i=1}^{l(\tau)} \prod_{j= i+1}^m (m_i+m_j+r+s+1)(m_i-m_j).
\end{equation*}
The first product is expressed as:
\begin{equation}\label{FirstProd}
\prod_{i=l(\tau)+1}^{l(\alpha)}\prod_{j= i+1}^m (2m-i-j+r+s+1)(j-i) = \prod_{i=l(\tau)+1}^{l(\alpha)} \frac{\Gamma(m-i+1)\Gamma(2m-2i+r+s+1)}{\Gamma(m-i+r+s+1)}.
\end{equation}
As to the second, it splits in turn into:
\begin{eqnarray*}
\prod_{1 \leq i < j \leq l(\tau)} (m_i+m_j+r+s+1)(m_i-m_j) & = & \prod_{i=1}^{l(\tau)} \frac{\Gamma(\tau_i +2m-2i+ r+s+2)\Gamma(\tau_i + l(\tau)-i)}{\Gamma(\tau_i+2m-i-l(\tau)+r+s+2)\Gamma(\tau_i)}\\
\prod_{i=1}^{l(\tau)} \prod_{j= l(\tau)+1}^m (m_i+m_j+r+s+1)(m_i-m_j) & = & \prod_{i=1}^{l(\tau)} \frac{\Gamma(\tau_i+2m-i-l(\tau)+r+s+1)\Gamma(\tau_i+m-i+1)}{\Gamma(\tau_i+m-i+r+s+1)\Gamma(\tau_i+l(\tau)-i+1)}
\end{eqnarray*}
yielding
\begin{multline*}
\prod_{i=1}^{l(\tau)} \prod_{j= i+1}^m (m_i+m_j+r+s+1)(m_i-m_j) = \\
\prod_{i=1}^{l(\tau)}\frac{\Gamma(\tau_i+2m-2i+r+s+2)\Gamma(m_i+1)}{(\tau_i+2m-i-l(\tau)+r+s+1)(\tau_i+l(\tau) -i)\Gamma(\tau_i)\Gamma(m_i+r+s+1)}.
\end{multline*}
Since $n_j = 1+m-j= m_j+1$ for $l(\tau)+1 \leq j \leq l(\alpha)$, then \eqref{FirstProd} implies that
\begin{multline*}
\left[\prod_{i=l(\tau)+1}^{l(\alpha)}\prod_{j= i+1}^m (2m-i-j+r+s+1)(j-i)\right]
\\ \prod_{j=l(\tau)+1}^{l(\alpha)}\frac{(r+s+2m_j+1)\Gamma(n_j+1)\Gamma(r+n_j+1)\Gamma(m_j+s+1)\Gamma(r+s+m_j+1)}{\Gamma(r+m_j+1)\Gamma(m_j+1)\Gamma(m-j+s+1)\Gamma(m-j+1)},
\end{multline*}
reduces to
\begin{equation}\label{Prod2}
\prod_{j=l(\tau)+1}^{l(\alpha)}\Gamma(r+s+2(m-j)+2)(m-j+1)(r+m-j+1).
\end{equation}
Moreover, $n_j = m_j = 1+m-j$ when $2 \leq j \leq l(\tau)$ whence
\begin{multline}\label{Prod1}
\left[\prod_{i=2}^{l(\tau)} \prod_{j= i+1}^m (m_i+m_j+r+s+1)(m_i-m_j)\right] \times \\ \prod_{i=2}^{l(\tau)} \frac{(r+s+2m_j+1)\Gamma(r+n_j+1)\Gamma(n_j+1)\Gamma(m_j+s+1)\Gamma(r+s+m_j+1)}{\Gamma(r+s+n_j+m_j+2)\Gamma(r+m_j+1)\Gamma(m_j+1)\Gamma(m-j+s+1)\Gamma(m-j+1)} \\ = \prod_{i=2}^{l(\tau)} \frac{(m-i+1)(m-i+s+1)}{(2m-i-l(\tau)+r+s+2)(1+l(\tau)-i)}.
\end{multline}
Finally, the contribution of the terms corresponding to $(\alpha_1, \tau_1)$ is given by:
\begin{equation}\label{FirstRow}
\frac{(r+s+2\tau_1+2m-1)\Gamma(\alpha_1+m)\Gamma(r+\alpha_1+m)\Gamma(\tau_1+m+s)\Gamma(\tau_1+2m+r+s)}{(r+s+\tau_1+2m-l(\tau))\Gamma(r+s+\alpha_1+\tau_1+2m)\Gamma(m)\Gamma(r+\tau_1+m)\Gamma(m+s)\Gamma(\tau_1)(\tau_1+l(\tau)-1)}.
\end{equation}
Gathering \eqref{Prod2}, \eqref{Prod1}, \eqref{FirstRow} and keeping in mind \eqref{MF3}, we are done.
\end{proof}
\subsection{An auxiliary determinant: end of the proof}
In this paragraph, we end the proof of Theorem \ref{teo} after expressing the determinant of the submatrix
\begin{equation*}
(b_{\alpha, \tau}(i,j))_{i,j = l(\tau)+1}^{l(\alpha)},
\end{equation*}
when it is not empty, in a product form. This expression is stated in the following lemma:
\begin{lem}\label{Determinant}
Let $\tau$ be a hook of length $l(\tau) \geq 1$ and let $\alpha \supset \tau$ be a hook such that $l(\alpha) \geq l(\tau) + 1$. Then
\begin{equation*}
\det \left(\frac{\Gamma(r+s+2(m-j)+2)}{(n_i-m_j)!\Gamma(r+s+n_i+m_j+2)}\right)_{\substack{j=l(\tau)+1 \dots l(\alpha), \\ i \leq j+1}}
= \frac{1}{(l(\alpha)- l(\tau))!}\prod_{j= l(\tau)+1}^{l(\alpha)}\frac{1}{ r + s + 2m - l(\tau)+1 -j}
\end{equation*}
\end{lem}
\begin{proof}
When $l(\alpha) \geq l(\tau)+1 \geq 2$, then $\alpha_i = 1, n_i = m-i+1 = m_i+1$ so that
\begin{multline*}
\det \left(\frac{\Gamma(r+s+2(m-j)+2)}{(n_i-m_j)!\Gamma(r+s+n_i+m_j+2)}\right)_{\substack{j=l(\tau)+1 \dots l(\alpha), \\ i \leq j+1}} = \\
\det \left(\frac{\Gamma(r+s+2(m-j)+2)}{(j-i+1)!\Gamma(r+s+2m-i-j+3)}\right)_{\substack{j=l(\tau)+1 \dots l(\alpha), \\ i \leq j+1}}.
\end{multline*}
Set
\begin{equation*}
N := r+s+ 2m+2, \qquad L:= l(\alpha)- l(\tau),
\end{equation*}
and for every $i,j\in \{l(\tau)+1, \cdots, l(\alpha)\}$ set also:
$$
a_{i- l(\tau), j- l(\tau)} := \frac{\Gamma(N-2j)}{(j-i+1)! \Gamma(N-i-j+1)}{\bf 1}_{\{i\le j+1\}}.
$$
Then, the determinant we need to compute is
$$
\det[a_{k,l}]_{k, l = 1}^L=
\left|
\begin{array}{ccccc}
a_{11} & a_{12} & \cdots & & a_{1L} \\
1 & a_{22} & a_{23} & \cdots & a_{2L} \\
0 & 1 & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & \\
0 & \cdots & 0 & 1 & a_{L L} \\
\end{array}
\right|,
$$
where
$$
a_{kk} = \frac{1}{N-2k -2l(\tau)}; \quad k \in \{ 1,\cdots, L\}.
$$
Using the row operation
\begin{equation*}
R_2 \longrightarrow R_2 - \frac{1}{a_{11}}R_1,
\end{equation*}
one gets
$$
\det[a_{k,l}]_{k, l = 1}^L = \frac{1}{N -2l(\tau)-2} \left|
\begin{array}{ccccc}
a^{'}_{22} & a^{'}_{23} & \cdots & & a^{'}_{2L} \\
1 & a^{'}_{33} & \ddots & \ddots & \vdots \\
0 & \ddots & \ddots & \ddots & \\
\vdots & \ddots & \ddots & \ddots & \\
0 &\cdots & 0 & 1 & a^{'}_{L L} \\
\end{array}
\right|$$
where
\begin{equation*}
a^{'}_{kl} = \left\{
\begin{array}{ll}
a_{kl}- \displaystyle \frac{a_{(k-1) l}}{a_{11}}, & \hbox{if $k = 2$;} \\
a_{kl}, & \hbox{otherwise.}
\end{array}
\right.
\end{equation*}
More explicity,
\begin{equation*}
a^{'}_{2l} = \frac{\Gamma(N-2l(\tau)-2l)}{l(N-2l(\tau)-l-1)(l-2)!\Gamma(N-2l(\tau)-l-2)} = \frac{1}{l(N-2l(\tau)-l-1)}a_{3l}, \quad l \geq 2.
\end{equation*}
In particular,
\begin{equation*}
a^{'}_{22} = \frac{1}{2( N-2l(\tau)-3)}, \quad a^{'}_{23} = \frac{1}{3(N-2l(\tau)-4)(N-2l(\tau)-6)}.
\end{equation*}
Now, the second row operation:
\begin{equation*}
R_3 \longrightarrow R_3 - \frac{1}{a_{11}}R_2,
\end{equation*}
transforms the matrix third row into:
\begin{equation*}
\frac{\Gamma(N-2l(\tau)-2l)!}{l(N-2l(\tau)-l-1)(l-3)!\Gamma(N-2l(\tau)-l-3)} = \frac{1}{l(N-2l(\tau)-l-1)}a_{4l}, \quad l \geq 3.
\end{equation*}
Iterating these row operations, one gets:
\begin{align*}
\det[a_{k,l}]_{k, l = 1}^L = \prod_{k=1}^{L}\frac{1}{k(N- 2l(\tau)-k-1)} = \frac{1}{(l(\alpha)- l(\tau))!}\prod_{k= l(\tau)+1}^{l(\alpha)}\frac{1}{ r + s + 2m - l(\tau)+1 -k},
\end{align*}
as claimed.
\end{proof}
\begin{rem}
The determinant computed in the previous lemma may be written as:
\begin{equation*}
\prod_{j=1}^{l(\alpha)-l(\tau)} \Gamma(r+s+2m-2\l(\tau)-2j+2)\det \left(\frac{1}{(j-i+1)!\Gamma(r+s+2m-2l(\tau)+2 -i-j)!}\right)_{i,j=1}^{l(\alpha)-l(\tau)},
\end{equation*}
with the convention that $(j-i+1)! = \infty$ when $j-i+1 < 0$.
On the other hand, if $A$ and $L_i, 1 \leq i \leq L,$ are indeterminates then the following identity holds (take $B=2$ in \cite{Kra}, Theorem 26, eq. (3.13)):
\begin{equation*}
\det\left(\frac{1}{(L_i+j)!(A+L_i-j)!}\right)_{i,j=1}^L = \prod_{1 \leq i < j \leq L}(L_i-L_j) \prod_{i=1}^L\frac{(A-2i+1)_{i-1}}{(L_i+L)!(L_i+A-1)!}.
\end{equation*}
Choosing $L_i = -i+1, A = r+s+2m-2l(\tau)+1$, and recalling $L = l(\alpha)-l(\tau)$, one gets another proof of the previous lemma after some simplifications. The authors thank the anonymous referee for this hint.
\end{rem}
\begin{proof}[End of the proof of the main result]
With the help of lemma \ref{Determinant}, the formula \eqref{MF4} is written as:
\begin{multline*}
M_n(t) = M_n(\infty) + \sum_{\substack{\alpha \, \, \textrm{hook} \\ |\alpha| = n, l(\alpha) \leq m}} (-1)^{n-\alpha_1} \sum_{\substack{\tau \subseteq \alpha \\ \tau \neq \emptyset}}
\frac{e^{-\nu_\tau t}V_{\alpha_1, \tau_1}^{r,s,m}}{(r+s+\tau_1+2m-l(\tau)) (\tau_1+l(\tau)-1)(l(\alpha)- l(\tau))!(l(\tau)-1)!}
\\ \prod_{i=2}^{l(\tau)} \frac{(m-i+1)(m-i+s+1)}{(2m-i-l(\tau)+r+s+2)} \prod_{j=l(\tau)+1}^{l(\alpha)}\frac{(m-j+1)(r+m-j+1)}{ r + s + 2m - l(\tau)+1 -j}.
\end{multline*}
Now, the products appearing in the RHS may be expressed through the Gamma function as:
\begin{equation*}
\prod_{i=2}^{l(\alpha)}(m-i+1) = \frac{\Gamma(m)}{\Gamma(m-l(\alpha)+1)},
\end{equation*}
\begin{equation*}
\prod_{i=2}^{l(\tau)}\frac{(m-i+s+1)}{(2m-i-l(\tau)+r+s+2)} = \frac{\Gamma(m+s)\Gamma(2m+r+s+2-2l(\tau))}{\Gamma(m+s+1-l(\tau))\Gamma(2m+r+s+1-l(\tau))},
\end{equation*}
\begin{equation*}
\prod_{j=l(\tau)+1}^{l(\alpha)}\frac{(r+m-j+1)}{ r + s + 2m - l(\tau)+1 -j} = \frac{\Gamma(m+r+1-l(\tau)) \Gamma(2m+r+s+1-l(\tau)-l(\alpha))}{\Gamma(m+r+1-l(\alpha))\Gamma(2m+r+s+1-2l(\tau))}.
\end{equation*}
Finally, we appeal to \eqref{Contrib1} to obtain \eqref{FinalForm}. Theorem \ref{teo} is proved.
\end{proof}
\begin{rem}
Recall the notations:
\begin{equation*}
r = p-m, \quad s = d-p-m = q-m.
\end{equation*}
Then
\begin{align*}
\tilde{V}_{\alpha_1, \tau_1}^{r,s,m} = \frac{(d+2\tau_1-1)\Gamma(d+\tau_1)\Gamma(\alpha_1+m)\Gamma(p+\alpha_1)\Gamma(q+\tau_1)}{(\alpha_1-\tau_1)!\Gamma(d+\alpha_1+\tau_1)\Gamma(p+\tau_1)\Gamma(\tau_1)},
\end{align*}
and
\begin{align*}
U_{l(\alpha), l(\tau)}^{r,s,m} : = \frac{(d+1-2l(\tau))\Gamma(d-l(\alpha)-l(\tau)+1)\Gamma(p-l(\tau)+1)}{(l(\alpha)- l(\tau))!(l(\tau)-1)!\Gamma(m-l(\alpha)+1)\Gamma(p-l(\alpha)+1)\Gamma(q-l(\tau)+1)\Gamma(d-l(\tau)+1)}.
\end{align*}
\end{rem}
\section{Further perspectives}
So far, we computed the moments of the Hermitian Jacobi process. In this section, we discuss two perspectives which we think they worth being developed in future research works. The first perspective is concerned with a possible application to optical fibers MIMO channel if one takes into account the time variation of the communication system. The second one is rather motivated by the random-matrix approach to free probability theory. More precisely, the marginal of the Hermitian Jacobi process at any fixed time $t > 0$ converges strongly as $m \rightarrow \infty$ to the so-called free Jacobi process (\cite{CDK}) and the moments of the spectral measure of the latter were determined for equal projections in \cite{Dem-Ham}. As a matter of fact, it would be quite interesting to determine the large $m$-limit of $M_n(t)$ (after rescaling the parameters $r = r(m), s= s(m), d= d(m)$ in order to get a non trivial limit) in order to generalize and to give another proof of the expression derived in \cite{Dem-Ham}. In this respect, we shall assume $m \geq n$ (or $m$ large enough) and write the moment formula as a linear combination of terminating and balanced ${}_4F_3$-series evaluated at unit argument. Though this hypergeometric series obeys Whipple's transformation (see e.g. \cite{AAR}, Theorem 3.3.3), we do not succeed to derive a closed formula for it which would certainly open the way to investigate the large $m$-limit of $M_n(t)$.
\subsection{Application to Optical fibers MIMO channels}
In \cite{DFS}, the authors used the JUE to model an optical fibers MIMO channel. Actually, the transfer matrix of this model is a truncation of a Haar unitary matrix and reflects the situation when only a part of the modes in the fiber is used. If we further take into account the time variation of the transfer matrix, then a natural dynamical candidate for modeling an optical fibers MIMO channel with $m$ antennas at the receiver and $p$ antennas at the transmitter would be a $m \times p$ truncation of a $d \times d$ unitary Brownian motion. In this case, the statistical behavior of the channel is governed by the eigenvalues of the Hermitian Jacobi process. In particular, the unitary invariance of $J_t$ for fixed time $t$ implies that the Shannon capacity of the channel is given by (we assume that the Gaussian noise is centered and have identity covariance matrix, \cite{Tel}):
\begin{equation*}
C_t(m, p, d, \rho) :=\mathbb{E}\left[\log\det\left(\textrm{Id}_{m \times m} + \frac{\mathcal{P}}{p}J_t \right)\right]
\end{equation*}
where $\mathcal{P}$ is the total power at the transmitter and $\rho:= \mathcal{P}/p$. If $\rho \leq 1$ then the capacity is expanded as:
\begin{align*}
C_t(m, p, d, \rho) &:= \mathbb{E}\left[\sum_{k=1}^m \log\left(1+\rho \lambda_k\right)\right] = \sum_{n=1}^{\infty}\frac{(-\rho)^n}{n}M_n(t)
\\& = C_{\infty} (m, p, d, \rho) + \sum_{\substack{\alpha \, \textrm{hook} \\ 1 \leq l(\alpha) \leq m}} \sum_{\substack{\tau \subseteq \alpha \\ \tau \neq \emptyset}} \frac{(-1)^{\alpha_1}\rho^{|\alpha|}}{|\alpha|}
\frac{e^{-\nu_\tau t} \, \tilde{V}_{\alpha_1, \tau_1}^{r,s,m} \, U_{l(\alpha), l(\tau)}^{r,s,m} }{(r+s+\tau_1+2m-l(\tau)) (\tau_1+l(\tau)-1)}
\end{align*}
where $C_{\infty} (m, p, d, \rho)$ is the capacity of a channel drawn from the JUE (\cite{DFS}, \cite{DN}). Of course, the condition $\rho \leq 1$ is redundant since only needed to expand the logarithm into power series. In a future research work, we shall work out the expression of $C_t(m, p, d, \rho)$ and get rid of this condition.
\subsection{The large $m$-limit}
Reversing the summation order in \eqref{FinalForm}, we shall fix a hook
\begin{equation*}
\tau = (h-j, 1^j), \quad 0 \leq j \leq h-1,
\end{equation*}
of weight $1 \leq h=|\tau| \leq n$ then sum over hooks
\begin{equation*}
\alpha = (n-k, 1^k), \quad j \leq k \leq j+n-h,
\end{equation*}
of weight $n$ and containing $\tau$. Doing so and extracting the terms depending only on $\alpha$, we are led lead to the following alternating sum:
\begin{equation*}
\sum_{k = j}^{j+n-h} \frac{(-1)^{k}\Gamma(n-k+m)\Gamma(p+n-k)\Gamma(d-k-j-1)}{(n-h+j-k)!(k- j)!\Gamma(m-k)\Gamma(p-k)\Gamma(d+n-k+h-j)}.
\end{equation*}
Performing the index change $k \mapsto n-h+j-k$ there, we transform this sum into:
\begin{equation*}
(-1)^{n-h+j}\sum_{k=0}^{n-h}\frac{(-1)^{k}\Gamma(h-j+m+k)\Gamma(p+h-j+k)\Gamma(d-n+h-2j-1+k)}{k! (n-h-k)!\Gamma(m+h-n-j+k)\Gamma(p+h-n-j+k)\Gamma(d+k+2h-2j)}.
\end{equation*}
Up to Gamma factors which do not depend on $k$, this sum may be expressed as a terminating ${}_4F_3$ hypergeometric series at unit argument:
\begin{multline*}
\frac{(-1)^{n-h+j}\Gamma(h-j+m)\Gamma(p+h-j)\Gamma(d-n+h-2j-1)}{(n-h)!\Gamma(d+2h-2j)\Gamma(m+h-n-j)\Gamma(p+h-n-j))} \\
{}_4F_3\left(\begin{matrix} -(n-h), m+h-j, p+h-j, d-n+h-2j-1; \\ m-n+h-j, p-n+h-j, d+2h-2j; \end{matrix}, 1\right),
\end{multline*}
which is balanced (one plus the sum of the upper parameters equal to the sum of the lower ones).
\section*{Acknowledgments}
The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the financial support for this research under the number (cba-2019-2-2-I-5394) during the academic year 1440 AH / 2019 AD.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,955 |
Merkel cites 'very, very slow' progress on Ukraine peace deal
By Reuters Staff
German Chancellor Angela Merkel speaks during a news conference to present the outcome of the G20 leaders summit in Hamburg, Germany July 8, 2017. REUTERS/Axel Schmidt
HAMBURG (Reuters) - German Chancellor Angela Merkel said on Saturday there was no glossing over the fact that there had been "very, very slow" progress in implementing the Minsk peace accords aimed at ending years of violence in eastern Ukraine.
Merkel said she would hold four-way telephone talks on next steps soon with the leaders of Russia, Ukraine and France following a more procedural conversation with French President Emmanuel Macron and Russian President Vladimir Putin on the sidelines of a G20 summit in Hamburg.
"We agreed to continue the process. But we also observed that progress had been very, very slow - with stagnation in some cases, relapses in others. We didn't gloss over the situation," she said. "We will stay in touch, we'll stick with the format. We don't have any other basis."
Reporting by Andrea Shalal; Editing by Noah Barkin | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 88 |
\section{Introduction}
The physical content of our theories of the fundamental interactions
is profoundly affected by the gauge symmetries that lie at their heart.
Such theories are in
fact examples of systems with constraints and it is
well known that a consequence of this for QED is that only two of the
four initial $A^\mu$ potentials are actually physical. The implications
of the gauge symmetries of QED for charged fields (such as electrons)
are less well understood and for non-abelian theories, such as QCD, the
extraction of the physical degrees of freedom has not been performed.
This talk reports recent progress in understanding these
fundamental issues. In particular a gauge invariant
description of charged fields in electrodynamics and a
physical interpretation is provided. This leads to predictions which are
then tested in perturbation theory. The gauge structure of scalar QED
is so similar to that of standard QED that exactly
the same predictions may be made for it. They are also verified
here.
Any description of a physical charge must be gauge invariant and Gau\ss's law implies
an intimate link between charges and a chromo-(electro-)magnetic cloud. Such a
description in terms of a so-called dressed field, $\psi_f$, was proposed by Dirac in the 50's:
$
\psi_{f}(x)=\exp\left\{ ie\int d^4zf_\mu(z,x)A^\mu(z)\right\}
\psi(x)
$. This can be seen to be gauge invariant if $f_\mu(z,x)$ satisfies
$\partial_\mu^z f^\mu(z,x)=
\delta^{(4)}(z-x)$.
Not all gauge invariant descriptions are, however,
physically relevant. In Fig.\ref{fig} two such
possible clouds are shown.
\begin{figure}
\centerline{\hbox{\epsfysize=5cm
\epsfbox{fig4.ps}
}}
\caption{Two gauge invariant configurations of a static field surrounded by a
electromagnetic field. A stronger field is represented by a whiter shading.
The very singular string-like configuration in (a) can be shown to decay into the
Coulombic one in (b) which is stable~\protect{\cite{www}}.
\label{fig}}
\end{figure}
Clearly, (a) is not stable in QED and for a static charge it will decay into the Coulomb
cloud in~(b). Our claim~\cite{LaMcMu96a,slow,fast} is that the latter,
and a more general version corresponding to a charge moving with velocity
$\vec v$, are suitable for constructing physical asymptotic states. This in either QED or scalar
QED since the magnetic field associated with the magnetic moment of an electron falls off rapidly
away from the charge and is thus infra-red safe.
To be explicit, for an electron moving with velocity $\vec v$ we
dress the fermion as follows
\begin{equation}
\psi_v=\exp\left\{
ie{g^{\mu\nu}-(\eta+ v )^\mu(\eta- v)^\nu\over\partial^2-
(\eta\cdot\partial)^2+(
v\cdot\partial)^2}\partial_\nu A_\mu
\right\}
\psi ,
\label{eq:boos}
\end{equation}
where $v=(0,\vec v)$, and $\eta=(1,\vec 0)$.
Rather than giving the
arguments underlying this statement, we now report a perturbative calculation~\cite{slow,fast}
which verifies the following.
Recall that the usual gauge-dependent fermion propagator in QED or QCD is plagued with infra-red
divergences in an on-shell scheme.
These reflect the fact that the fermion is not a good physical state ---the
chromo-(electro-)magnetic field it generates is missing. If our dressing has a physical
significance, we should be able to perform an {\em infra-red finite}
on-shell renormalization for the propagator of the dressed charge defined by~(\ref{eq:boos}).
For this non covariant description, we find~\cite{fast}
a multiplicative {\em matrix} renormalization
\begin{equation}
\psi^{({\rm bare})}_v=\sqrt{Z_2} \exp\left\{
-i { Z'\over Z_2}\sigma^{\mu\nu}\eta_\mu v_\nu \right\}\psi_v
\quad\mbox{\rm and}\quad m^{({\rm bare})}=m-
\delta m
\label{eq:Zs}
\end{equation}
necessary.
This is reminiscent of a naive Lorentz boost upon a fermion. The mass shift renormalization
from demanding that the pole is at $m$
yields the standard gauge-invariant result found in any textbook. The residue renormalization
condition is where the infra-red divergences are usually found and we find here for our
non covariant case three equations for only two unknowns, $Z'$ and $Z_2$. It is highly
gratifying that at the expected physical momentum, $p=m\gamma(1,\vec v)$, we can consistently
solve these three equations and further that {\em the renormalization constants are gauge-invariant
and infra-red finite}.
The matrix renormalization~(\ref{eq:Zs}) is forced upon us by the fermion structure of QED. We
have checked that in scalar QED where such a scheme is not possible, a straightforward
multiplicative renormalization also yields infra-red finite results --- this
despite exactly the same non-covariant dressing also being used in the scalar theory.
The next step is the non-abelian theory. Quarks and gluons are believed to be confined inside
colourless hadrons and yet the success of the constituent quark model
and the jet structures observed in experiments show that it must be
possible to attach some physical meaning to quarks and gluons. The
Lagrangian fields are, however, gauge dependent ---like their QED
counterparts--- and we need to find the physical degrees of
freedom of such a non-abelian gauge theory. We note here that the action of the gauge dependent
colour charge operator is only gauge invariant on locally gauge invariant objects and so the
colour statistics of the constituent quark
model require a gauge invariant description~\cite{LaMcMu96a}.
Perturbative calculations are possible here, but, it can be shown that a
non-perturbative obstruction, the Gribov ambiguity, prevents quarks and gluons from being true
observables. This is because the QCD equivalent of the dressing function $f_\mu(z,x)$
can be used to fix the gauge. This obstruction
sets the fundamental hadronic scale and the limits of the quark model.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,198 |
How Can I Build a Holistic Health and Wellness Practice Part-time?
Do you dream of ditching your day job so you can work full time as a holistic health and wellness coach or consultant?
Wouldn't it be great if Oprah showed up at your door tomorrow with $500,000 to invest in your brand new yoga and holistic healing center? You'd quit your day job today and open up shop tomorrow, right?
If that's your master plan for starting your own health and wellness practice, in the words of Aerosmith "Dream On!" But if you are serious about making turning your passion for helping people into a profitable business, here are my top tips for making it happen.
There's a lot of talk online about making money doing what you love. I am all for doing work that you are passionate about, but if you are going to actually make money from something you are passionate about, you've got to choose a sustainable and profitable niche. In most cases this involves taking a creative approach to matching what you love or are interested in with a problem that needs to be solved. More specifically, a problem that people will pay you to solve. Otherwise you end up with a hobby or pseudo-business. A pseudo-business has all the looks of a real business (business cards, website, social networking accounts) but no paying customers or clients.
Instead of just focusing on energy healing, focus on using your services on women who want to recover from trauma.
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Lots of people have crossed the digital divide by joining the masses on Facebook, using smart phones, or even starting a personal blog. But there's a difference in using technology as a consumer and learning to use social media and other online tools like an entrepreneur. Social networking is a game changer for new service-based businesses. When used strategically it's a gateway to access the information, ideas, and people that can help you build a profitable practice. Online and mobile tools can help you automate tasks and give you the ability to check-in with projects, clients, and vendors anytime fro anywhere. The right technology and tools means you'll work smarter, not harder to grow your practice.
Process invoices and handle billing.
Send clients and prospects a newsletter.
Automate social posting on Facebook, LinkedIn, Twitter, and Google.
Stay on top of trends and events in your niche.
As a former employedpreneur (a full-time employee with a side business), I am a huge advocate of launching while working. If it was good enough for Sarah Blakely – the Forbes listed self-made billionaire and founder of Spanx, it's good enough for you. There are several benefits to launching while working, especially if you have a family to support. On this path to entrepreneurship, your day job funds your practice. And depending on the type of company you work for, your day job can help prepare you for being your own boss. Want to learn how to market your business? Look for or create opportunities to work with your company's marketing department. Need to sharpen your web design or blogging skills? Then volunteer for projects that will let you build on those skills at work.
Bottom line: It is 100% possible to build a profitable holistic health and wellness practice without quitting your day job.
Previous Post You and Your Clients Win When You Find the Right Niche (FREE Download) Next Post Coaches and Consultants: Are You Having This Dangerous Conversation? | {
"redpajama_set_name": "RedPajamaC4"
} | 3,392 |
10 April 2014, Two companies have been appointed to take on management of Buckinghamshire County Council's property services in contracts worth £1.8 million. The deal, signed last week, is expected to save the County Council more than half a million pounds a year.
The maintenance of council properties will be managed by Bilfinger Europa, the national facilities management specialists, while the estates and asset management aspects of the property portfolio will be overseen by estate management specialists Carter Jonas. The partnership with Bilfinger Europa keeps maintenance work under one roof, enabling users of council property - such as schools - to get repairs and maintenance done through a one-stop-shop. As part of the contract, Bilfinger Europa will continue the council's practice of using local suppliers as sub-contractors wherever possible.
Carter Jonas provides the council with a strategic partner with long experience in estates and asset management to help get best value from its property portfolio and make the most efficient and effective use of its buildings.
Since the new partnerships began, the County Council has been working with providers to set up a 24/7 maintenance helpdesk which Europa will manage. | {
"redpajama_set_name": "RedPajamaC4"
} | 416 |
Title: 1C Company
Subject: List of games with DirectX 10 support, List of games using Steam authentication, List of Impulse Reactor and Goo games, List of Games for Windows titles, List of space flight simulator games
Collection: 1991 Establishments in Russia, Companies Based in Moscow, Companies Established in 1991, Erp Software Companies, Software Companies of Russia, Technology Companies Established in 1991, Video Game Companies of Russia, Video Game Development Companies, Video Game Publishers
Boris Nuraliev (President)
Computer software, Video games
www.1c.ru
1C Company (Russian: Фирма "1С") is one of the largest independent Russian software developers and publishers. Its headquarters are in Moscow, Russia. It is best known outside of Russia as a video game developer whose presence has come to dominate events like the KRI, and whose products have begun to show up for international platforms like the Xbox.[2] On the internal Russian market, 1C is considered a leader in business software as well, however game software represents 98% of the 13 tons of software they have been reported as shipping daily.[2]
1C's comprehensive business software suite 1C:Предприятие (1C:Predpriyatiye, 1C:Enterprise[3]) has held an overwhelming market share for over a decade. 1C is also a leader in localizing and publishing Russian-language versions of international software. For instance, more than half of popular Western video games are licensed and published by 1C. The company has over 700 employees, 10,000 business partners, 4,500 authorized retailers, 1,200 training centers, 200 authorized certification locations and over 280 stores in 100 cities.[4]
Business software 2
Entertainment software 3
A 1C advertisement on an envelope
1C was founded in 1991[1] by Boris Nuraliev in Moscow, Russia. The company goal was to provide a software solution to business owners giving them a system that required little or no accounting and development experience.
Both Akella's Sea Dog studio[5] and 1C:Ino-Co (founded as Ino-Co) were acquired in 2007, and 1C Company began to take substantial steps aimed at expansion into the North American video gaming market with distribution deals with Atari,[6] and an appearance at E3 2007.[7] In 2008 the company signed deals for exclusive rights to distribute PC and console games in Russia and CIS countries with titles by Electronic Arts[8][9] and Gamecock Media Group,[10] and began to offer games for online distribution.[11] By 2009, 1C owned and operated 280 stores with another 4000 franchises based on the 1C license in 600 locations across the former Soviet bloc,[2] and in February of the same year 1C merged with SoftClub (the largest Russian console game publisher and distibutor).[12]
In October 2011, 1C Company secured $200m from Baring Vostok Capital Fund[13] and launched an international service for developers entitled 1C:Developer Network (1C:DN).[14]
Buka Entertainment was acquired by Russian developer and publisher 1C Company in 2008.[15]
While virtually unknown in the West, 1C:Enterprise is a market leader in Russia. "1C:Enterprise 8" technology consists of two parts: 1C:Enterprise platform and 1C applied solutions. 1C:Enterprise platform is proprietary and developing only by 1C Company.[16] Current version 1C:Enterprise platform is 8.3.5.[17] Applied solution includes Accounting, Contact Management, Inventory Management, Document Management, and more, developing by 1C Company and a variety of their partners. Applied solutions are open for modification by any developers. The applied solutions is used by tens of thousands of Russian corporations. 1C was one of the first Russian private companies to be awarded with the Russian Federation Government Award in Science in 2002 for development and implementation of 1C:Enterprise.
1C:Enterprise platform allow developers very easy and fast creating business oriented application. Developed applied solutions allow work in thick, thin and web-client.[18] Also supporting creating mobile applications for Android and iOS in the same environment using 1C programming language.[19]
Some international configurations based on the platform 1C:Enterprise are available. Among them are: 1C:Accounting Suite,[20] 1C:Small Business.[21]
Primarily considered a Pacific Fighters (2004), 1946 (2006), Birds of Prey (2009, for consoles), Cliffs of Dover (2011) and Battle of Stalingrad (2013).[23] 1C has also developed Theatre Of War, a World War II land-based game published in the West by Battlefront.com.[24]
1C has a long history of in-house development, as well as funding independent developers. 1C works as a publisher and producer with over 30 independent development studios and has produced over 100 projects for PC and consoles including titles like Hard Truck 2, King of the Road, Rig'n'Roll, Space Rangers,[25] Soldiers: Heroes of World War II, Faces of War, Men of War series,[26] Perimeter, Fantasy Wars, Death to Spies, King's Bounty: The Legend and King's Bounty: Armored Princess.
The company continues to distribute titles, with a 2010-2011 release portfolio that included such titles as Men of War: Assault Squad[27] and Men of War: Vietnam,[28] Captain Blood, and Red Orchestra 2: Heroes of Stalingrad.
^ a b About the 1C:Developer Network. 1C:DN. Retrieved 26 April 2012.
^ a b c Rossignol, Jim. Gaming in the Russian Cosmos, Part 1. Rock, Paper, Shotgun. 5 January 2009.
^ What is 1C:Enterprise? 1C:DN.
^ Фирма 1С. 1C.ru. Retrieved 26 April 2012. (Russian)
^ 1C Company Acquires Akella In-House Development Studio . Gamers Hell. 26 November 2007.
^ Remo, Chris. 1C Makes Major North American Push with Atari. Shacknews. 23 April 2007.
^ Remo, Chris. E3 2007 Exhibitors Revealed (Updated). Shacknews. 23 April 2007.
^ Jenkins, David. 1C Company To Distribute EA Games In Russia. Gamasutra. 18 September 2008.
^ Electronic Arts and 1C Company Sign Deal . Gamers Hell. 18 September 2008.
^ Gamecock Media Group and 1C Company Sign Deal. Gamers Hell. 31 July 2008.
^ 1C Company Launches Online Distribution. Gamers Hell. 18 January 2008.
^ Joint press-release of «1C» and «Soft Club» 1C.ru. 24 February 2009. (Russian)
^ Butcher, Mike. Russian software house secures $200m from Baring Vostok Capital. TechCrunch. 3 October 2011.
^ 1C Company launches an international service for developers - 1C:Developer Network.. 1C:DN. 27 October 2011.
^ http://www.centernetworks.com/buka-acquired
^ "1C:Enterprise 8" architecture as a product of engineering idea (S.G. Nuraliev) YellowERP. 2004.
^ 1C Company released a final version 8.3.5 of "1С:Enterprise" platform. YellowERP. 23 July 2014.
^ Cross-platform design of "1С:Enterprise" platform. 1C:DN. 2013.
^ Working on mobile devices 1C:DN. 2013.
^ 1C Company released a new version of 1C:Accounting Suite, an accounting and inventory control application. 1C:DN. 26 January 2012.
^ 1C Company releases 1C:Small Business, an international business application for small enterprises. 1C:DN. 3 April 2012.
^ O'Connor, Alice. Origin signs up 11 more publishers. Shacknews. 24 January 2012.
^ 1C Company. GameSpy. Retrieved 26 April 2012.
^ Goldstein, Maartin. Theatre of War Q&A. Shacknews 29 July 2006.
^ O'Connor, Alice. Space Rangers HD announced for PC. Shacknews. 18 April 2012.
^ Faylor, Chris. Men of War Expansion Brings Red Tide This Fall. Shacknews. 25 June 2009.
^ Faylor, Chris. Men of War: Assault Squad Due This Year. Shacknews. 27 May 2010.
^ Faylor, Chris. Men of War: Vietnam Revealed. Shacknews. 4 June 2010.
(English) 1C Company
Articles with Russian-language external links
Companies based in Moscow
ERP software companies
Video game companies of Russia
Software companies of Russia
Video game development companies
Video game publishers
Ukraine, India, China, Turkey, United Kingdom
Moscow International Business Center, Moscow Oblast, Russian language, Soviet Union, Red Square
Cold War, Battle of Stalingrad, Nazi Germany, Battle of the Atlantic, Second Sino-Japanese War
BioWare, The Sims, Samsung Electronics, Sony, Apple Inc.
Types of business entity
Public limited company, Swedish language, Ltd., Limited partnership, General partnership
List of games with DirectX 10 support
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,226 |
Happy New Year! I added this little freebie pack here to my TPT store this morning and I wanted to share with you. I plan to do a full day of fun New Year's activities with the kiddos when we return to school. I know that first day back after being off might be hard for some of them, so I'm planning some fun New Year's goal setting and other activities, and this little pack will be a fun addition! There are treat bag toppers, bookmarks, and balloon toppers that can be used with pencils, Pixy Stix, or Silly Straws. Double sided tape will work well with the balloons in the file (I will cut off the string) and the treat bag toppers fit on a 4 inch width clear treat bag- just fold and staple to the top! I hope you can use some of what is in the file!
Enjoy the rest of your break if you are off! This second week is flying by for me! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,781 |
Scorpio Sun Capricorn Moon Combinations
Scorpio Sun Capricorn Moon (for ladies)
"Her Illumination: A survivor.
Her dark side: Cold moon.
The position of this lady's sun and moon indicates a basic harmony in self expression and being able to put her ideas across to others. She's be a good communicator or teacher. She could also excel in public administration. Just don't ask her to get too personally involved. Her chilly little Capricorn Moon won't permit it. She's on of the most goal oriented women of the configurations. She has a plan, and if you are an inferior being her interest in you will be minimal. If she's caught in early poverty, she may use you to escape her humble beginnings. But if you don't have your feet planted solidly on the ground she won't be there to celebrate your twentieth anniversary. Whatever her beginnings, they will be light years away from her current dreams and aspirations. In her 20's she may appear frivolous, and may get caught up in some minor flirtations and extravagances. In her 30's she will begin to express her real needs more, and start working toward the stability of a fixed residence, career, and sound financial planting. Her 40's find her find her deeply committed to her life's goals, to acquire, to accomplish and to amass considerable security. This can also be loosely translated as the burning desire for money. You have to admire her consistency, her hard work, her dedication to her goals. This lady is one of the Zodiac's all time champions of survival. As a wife and mother she is not the lighthearted, warm, and tender mother of which fairy tales are made. She's steadfast and will stick by her children through thick and thin. If you're the Capricorn man who can't keep away from her, or the Cancer or Scorpio that wants to know more about her, remember not to have your hopes set on running the house and controlling this woman. She's her own person, and the best you can hope for is that she'll allow you to share her life's plan. Sagittarius brings out her higher side."
(Taken from The Book of Lovers by Carolyn Reynolds.)
Scorpio Sun Capricorn Moon (for guys)
"His Illumination: Climbs to the top.
His dark side: Can be miserly.
I don't care what his early environment was, how poor and pitiful, how lavish and secure, count on it only getting better. He will surmount all odds and make it to the top of the heap. He has one passion, and that is to make lots of money, which he knows buys power. Power is the other commodity he most desires in life. Cold? You bet. Underneath the sexual passions burns a love of money. There also burns a white hot fire to be recognized as a government or public figure. Only when he has achieved his goals will he mellow out and work on relationships. Before that, he is a troublesome husband, father and coworker. He may well be about 50 when his life change occurs. Oh, yes, another thing about his money. It's his. That means he won't lavish it on you. Cavalier in his youth about the use of money, he nearly worships it in old age. If you are into fondling bank deposits and see no value in diamonds and furs and other girls' best friends, he may be the one you want to spend your golden years with. You won't ever have to worry in his earlier years that he will have a lot of job changes, changes of the heart, or residence. There is no floundering around trying to find himself. He knows what he wants early on and if you are a part of his plan, he will be so ardent in his pursuit of you it will make your head spin. You can be his co-partner and watch as he is promoted, and promoted. He will treat you as an inferior person if you don't have a few advancements along the way. If your friends are tacky, well, you'd better set your sights on someone else. He doesn't want to spend his valuable time with lesser beings, unless it has been time specified for some good charity endeavor. Don't ask about his roots or early environment. He has put it all out of his mind. You'd better do the same. You may never meet his mother. Best bets: Capricorn, Cancer or Libra."
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anon on January 23, 2019:
This moon placement is probably the coldest especially for guys.. if you want a fun loving relationship stay away from him.. He's not for you. But if you want someone who will stood by your side broodingly this ones for you. Wait for him to fight off his demons and he's all yours. Except for his money.
:)) on December 23, 2018:
If you're a sagittarius woman stay away from this scorpio man
Stefani on September 28, 2016:
So true! Forgot to mention that if their partner has more monetary gain or advancements quicker than them, they become extremely cold and jealous (although will never admit it).
Sco sun Cap moon on October 24, 2015:
What does cold moon mean?
I the Great Survivor on September 28, 2013:
Very True Thanks.
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www.headofzeus.com
For Susan
And for David Strang, 1938–2012
The dead talk to me in my dreams. When I wake up, I can't remember what they said.
It felt wrong from the start.
The man who sat across from us wore a sleek charcoal suit and a starched white shirt with French cuffs. I made the suit for a Dsquared2 right out of the New York Times Men's Fashion supplement, retail price $1,475. Its perfectly draped cuffs broke over tasseled black loafers that might have cost more than the suit itself. You didn't see that kind of suit in our part of town, much less when it was 108 degrees outside and this was only the first week of May. Yet he didn't sweat.
Still, somehow, $1,475 didn't buy elegance for the wearer, or peace of mind for me.
The suit lacked a tie, which irritated me. I like suits. I am a clotheshorse and they are also handy for concealing my firearm. Today, the rebels wear suits, which are the zenith of great clothing design. Show me a man with stubble and dressed like an adolescent and I'll show you today's version of 1950s conformity. Unfortunately, Phoenix weather only allows me to wear suits six months of the year. I looked at his open collar and thought: here was a suit quietly longing for a smart tie to complete it. The man appeared the same way: incomplete.
He introduced himself as Felix Smith, sat before Peralta's desk, and said he needed our help. We already knew that part. Smith had called the day before, dropped the name of a criminal lawyer who was a friend of Peralta's, and set up this afternoon's meeting. I pulled over the second client's chair and faced him.
"I want you to investigate a suspicious death."
"Let's start with the name of the deceased." Peralta had produced a yellow legal pad and pen.
My partner, who was also not sweating, was in one of his many tan summer suits with a conservative tie. I wore khakis with a long-sleeve linen shirt—this was, after all, Skin Cancer City—but even in the air conditioning, a layer of sweat formed beneath the fabric. In a city where so many people either came to die or, as in the case of illegal immigrants hiking across the desert, died trying to come, I was a native. I was one of the few my age who had stayed or returned. But my body held the DNA of the British Isles and when the temp crept over one hundred five I couldn't stop sweating.
The only cool thing against my body was the Latin cross by the Navajo silversmith Harrison Bitsue that had belonged to Robin, Lindsey's half-sister. Robin and I had walked over to the Heard Museum and she had fallen in love with it. So I bought it for her. I didn't know if Robin was a believer. She would have scoffed at organized religion as she did so many things in the world. But it was all that had come back to me from the medical examiner. I had restrung it on a longer beaded chain and now wore it all the time.
Felix said, "The girl's name was Grace. Grace Hunter."
Peralta asked more questions in his familiar deep voice. Each time, Felix gave a short, precise answer. He held a smart black portfolio but it remained unopened. Grace Hunter was twenty-three. He gave her date of birth and Social Security number, both of which we would need for records searches. She had died on April twenty-second, a little more than two weeks previously. The police had ruled it a suicide.
Peralta took notes. I studied Felix Smith and couldn't shake a feeling of discomfort.
He looked around thirty and his hair was dark and cut short, pushing down on a low forehead. Sitting straight with his hands palms-down on his thighs, his body conveyed strength and self-possession. But he had a nose that looked as if it had been mashed in multiple fights, pocked skin that had ingested too much sun which gave the impression of a flash burn, and the remains of bruises around unsettling, old yellow eyes. Even with his head immobile, those eyes restlessly swept the room.
Joseph Stalin had yellow eyes.
I guessed that his driver's license identified them as hazel.
He wasn't as big as Peralta, but he was plenty big. His head was large, about the same width as the muscled-up neck that held it. His hands were large and hard, with big knuckles, and underneath the suit his plank-like shoulders looked capable of violence. The brawler's face and body didn't go with the tailored suit and the high-shine, pricey shoes. Unless he was somebody's muscle.
But maybe I was being jumpy, paranoid. Peralta kept telling me that.
My agitation kicked up a notch when he said where the girl had died: San Diego. I wanted to start nervously shaking my right leg, playing drums with my hands, or leave the room. After the first jump, I made my leg stay still.
"They say she jumped off a balcony. It was from the nineteenth floor of a condo." His voice was steady, one note above a monotone. Peralta waited several seconds before going on.
"And you don't believe that..."
"No."
Peralta wrote down the address where it happened. It was downtown, near the beautiful Santa Fe railroad station.
"Who is she to you?" It was the first time I had spoken besides the introductions after he walked in the door.
The cat's eyes focused on me. After a pause: "my sister."
"Why not Grace Smith?" I asked.
His eyes narrowed and he assessed me, finding me wanting. "She had a different last name."
I suppose it made sense. Lindsey and Robin had different last names, different fathers. Maybe Felix and Grace's mother remarried. Maybe Grace had been married. I persuaded myself I was being overly suspicious.
I said, "I'm sorry for your loss."
His gaze would have cut me down if it had been a gun.
He produced a photo from his portfolio and slid it across Peralta's immaculate desktop. It was five-by-seven and glossy. The young woman had butterscotch hair with blond streaks, stylishly cut to hang slightly above her shoulders, large brown eyes, very pretty. She didn't look anything like him. Great smile and something more, something magnetic. The camera liked her. She liked the camera.
When the suit sleeve and French cuff rode up with his reach, I saw a multi-colored tattoo on his lower arm and almost unstrapped the gun on my belt. I had recently made enemies in the drug cartels and didn't know if our business was settled.
"This condo." I studied his face. "Was it hers?"
The skin around his eyes tensed. "No."
I waited and after a full two minutes he talked again.
"It belongs to a man named Larry Zisman."
It sounded vaguely familiar but I couldn't place it. Smith sensed it, and continued.
"He was an All-American quarterback for the Sun Devils back in the seventies, then he played pro for ten years before his knees were wrecked."
"Now I remember," I said. "I never read about this in the newspaper."
"Funny about that," Felix replied. "Larry Zisman is a celebrity with a lot of powerful friends."
"Is he married?" I asked.
"Very." Felix adjusted one leg and very slightly winced. It was the first time his face had given away an expression beyond tough.
The next question was logical enough, but Peralta didn't ask it.
"Note?" Peralta could be more taciturn and economical in his language than anyone I had ever met.
"No. She didn't leave a note. Nothing. That's one of the things I don't like."
"What else makes you doubt the police?"
"She was naked and her hands were bound."
Now he had my attention for reasons beyond his appearance. Peralta grunted and I heard his pen scratch along the paper.
"It's a good department," Peralta said. "San Diego. You need to understand that these things are usually what they seem, however much the loved ones want it to be otherwise."
I wasn't sure about that. I had seen botched death investigations, even by good departments.
"I have confidence in you, Sheriff. That's why I'm here."
"I'm the former sheriff." Peralta said it without any emotion, then pulled out the sheet with our fee schedule and handed it across to Felix with his meaty hand.
That was another thing that didn't feel right: "former sheriff."
Until four months ago, Peralta had been the sheriff of Maricopa County for what seemed like forever. Everybody I knew thought he would be sheriff as long as he wanted it, unless he decided to run for governor. I was one of his deputies and the Sheriff's Office historian. It was good work for somebody with a Ph.D. in history in this or any job market.
But those assumptions had been based on another Arizona, before millions of retirees and Midwesterners had collided with the huge wave of illegal immigration before the big housing crash. It was a bad time to be a Hispanic running for office and Peralta lost, even though he was a life-long Republican. There would be no Governor Peralta in today's Arizona. He took the defeat stoically. Instead of moving on to any of the lucrative consulting offers that had come his way, or encouraging the feeler to become San Antonio police chief, he set up shop as a private investigator. And here I was, too, as his partner. He was my oldest friend.
Our office was shabby compared with the places Felix Smith must have been accustomed to, based on his suit. We were on Grand Avenue, the bleakest thoroughfare in a city with abundant competition for the title, in what had once been a little motel, an "auto court." Most of the motel had been bulldozed long ago—Phoenix loved clearing land and leaving it that way. The new mayor was trying to encourage art projects and gardens on vacant lots, but I wondered if the effort would do much good.
Robin had found a 1948 post card of the motel: a charming affair with half a dozen buildings, each with two rooms, a swimming pool, lawns, and palm trees. All that was left was the former front office—a small, square adobe with enough room for our two desks, some file cabinets, and places for clients to sit. Except for our comfortable chairs, the décor was spare. Recently, Peralta had added a black leather sofa.
We were barely moved in. Peralta had sprung for a bookcase for me, but I hadn't put a single volume in it. Boxes of correspondence, all for Peralta, sat behind my desk. Speaking requests for him came almost every day. We really needed a secretary. Behind the office were a bathroom and a storeroom, the latter having been remodeled and fortified by Peralta for gun storage. Robin had named it the Danger Room. We each had a key to it, but it was mostly Peralta's playroom. A super-sized Trane air conditioning system had been installed.
Outside, sixty-year-old asphalt was the best you got for parking. It was as much potholes and the crumbled remains of petroleum products melted and reformed, summer after summer, as it was a parking lot. A carport next to the office was for our vehicles. Mine barely fit thanks to the size of Peralta's extended-cab pickup truck. The only other improvements had been a heavy-grade fence with a section that rolled across the driveway to seal things up tight when we were gone and restoring the old motel sign of a cowboy throwing a lariat. The neon was new and blinked happily into the night. And some well-concealed security cameras. We both had made enemies over the years.
Across the wide, divided avenue stretched railroad tracks and an industrial district. It was two miles and a new lifetime away from our old world: Peralta's palatial suite of offices, and my beloved aerie on the fourth floor of the art deco 1929 county courthouse. No more badge. No more cold cases to solve. After I had left academia—or was I thrown out?—I had taken Peralta's offer of a job in the Sheriff's Office reluctantly. I hadn't even intended to stay in Phoenix. I would be back long enough to sell the house. But I stayed.
It came to seem natural. Deputy David Mapstone.
Then it went away with great suddenness. Much else did, too.
"The point is," Felix said, "I want another opinion. A deeper investigation. I couldn't think of a better person than the former sheriff of Maricopa County. I want the best. Your reputation is very good, too, Doctor Mapstone."
Doctor Mapstone. That had been my grandfather, a dentist. I was merely a guy with too many history degrees. Once I had been mildly proud of the honorific. Now, for reasons I didn't fully understand, it irritated me. Like when somebody other than Lindsey called me "Dave."
He would be getting the best with Peralta, no doubt about that. Perhaps he was pleased that neither of us was awed by Zisman. I recalled now that Zisman's nickname had been "Larry Zip" and he had led many thrilling comebacks when he was at Arizona State. But I wasn't a rabid sports fan or sports historian, and Peralta's passions were golf and baseball. So we wouldn't approach this case as hero worshippers. Still, flattery seemed very out of place coming from this rough-looking, expensively dressed man.
"Was she suicidal?" Peralta asked.
"No."
"Bi-polar? Any mental illness? On any anti-depressants?" Spoken like the former husband of a psychologist. "Did she have a history of emotional problems?"
Felix shook the big head sparingly several times. "She was a sweet girl."
"Did she have enemies?" I asked.
"Of course not." It was the first time his voice had showed anything other than a careful detachment.
I asked other questions. When was the last time he had spoken to her? Two days before her death. How was she? Everything seemed fine. No change in her voice? Nothing new going on in her life? No. No. His voice grew more taut.
Expanding on my winning interpersonal skills, I continued.
"What was she doing in Zisman's condominium?"
"What the hell business is...?" He stopped himself.
"We're going to need to know." This from Peralta's deep, authoritative voice, before which the toughest cops had quailed.
Felix allowed the slightest sigh. "I don't know. I know what you're thinking. I didn't even think she knew the man. She had a boyfriend in San Diego. I can give you his name."
Peralta leaned back and said nothing. Felix rolled his head and knocked out a kink with such force that his neck emitted a sharp pop and I wondered for a second if he might have injured himself.
He was fine. "Money won't be a problem." He floated the fee schedule back across Peralta's desk.
Peralta tapped his pen against the pad. He didn't need a case. Sharon, his ex, had made sure he was set up with money for life and his only vices were guns and beer. I did. I was already digging into savings that had never been plentiful. Lindsey had a good job and her paycheck dropped into our joint checking account every two weeks. But I was reluctant to use her money. I had never thought of her money and my money during our marriage, but that was before the annus horribilis we had gone through. Money would be nice right now, but I wasn't sure I wanted it from this particular case.
"Let me take down a little more information and we'll make a preliminary investigation," Peralta said. "After that, I'll decide if we're going to continue."
"Fair enough," Felix said.
The "little more information" took another forty-five minutes.
Afterward, Felix pushed across a thick envelope. "I hope ten thousand is enough for a retainer."
My greedy heart leapt. My nervous leg didn't celebrate by calming down.
Peralta studied the contents. I could see hundred-dollar bills.
"I didn't want to wait for a check to clear," Felix said. "I'd appreciate it if you can start now."
Maybe Peralta nodded, but the man stood. He handed each of us a card with his name and number. No address.
"What's your line of work, Mister Smith?"
"Between jobs."
Peralta didn't push the question so I let it be.
Felix shook our hands. He gave me a long, vise-like shake. I gave it back as hard as I could and met his stare full on. If he was packing, my peripheral vision wasn't good enough to pick it up.
"I hope you don't mind if I also check you out." Peralta's voice snapped the moment.
"Not a bit."
Felix pivoted and pulled out a platinum money clip. From this, he handed the big man a driver's license. When Peralta had written down what he wanted, he gave it back and thanked him.
Felix let the money clip fall into his pocket. "You can't be too careful."
He turned and walked to the door. As he opened it, a hot gust from the outside caught his left cuff, raising it briefly. Above the pricey loafer on his foot, I saw something that looked like it was out of a Terminator movie. A lower-limb prosthetic, very high-tech, titanium and graphite. He definitely hadn't received it through the average health-care plan. I had read about ones embedded with a microprocessor that were worn by wounded soldiers.
When I looked up again, I saw him watching me watching him. The yellow eyes hated me.
"Feeling guilty?"
I did a little. I walked to the front window and raised the blind. Felix the Cat was sitting in a Mercedes Benz CL, silver, new, insolently bouncing back the sun's glare. The driver's window was down. Who needs air conditioning when it's only 108? He had a cell phone against his head and he was talking animatedly, very different from the stone-like expression he had mostly shown us. He didn't look happy.
"A rig like he had on his leg would only be issued to a disabled veteran." Peralta made more notes as he spoke, his large head and shoulders hunched over the desk.
I let the blind fall and turned back toward him. "The cartel could afford it." I told him about the car, which was not issued by the V.A.
He looked up. "Mapstone, you see Zetas and Sinaloa in your sleep." His tone softened subtly. "Which is understandable, after what you went through."
Yes, I was jumpy. But I saw other things in my sleep.
"I can guarantee you that Chapo Guzman doesn't even know who you are," Peralta went on. Chapo was the boss of the Sinaloa federation. And maybe he didn't. But his lieutenants did.
"Did you catch the tat?" I asked.
He nodded and went back to writing. "Everybody has tattoos now."
"Do you?"
"Maybe." No smile. This passed for raucous Mike Peralta humor. I didn't laugh.
"We shouldn't take this case."
"Why not?"
"Oh, I don't know." I prowled around the small room, absently slid out a file drawer, closed it. "He paid in cash."
Peralta opened the envelope and counted. He peeled off five grand and held it out to me. The bills looked as if they had come out of the U.S. Bureau of Engraving that morning. I made no move to retrieve them. Someday soon I would need to set up an accounting and tax system in the computer if we were actually going to have a PI business.
Peralta gently tapped the Ben Franklins. "Paying clients are nice."
"Cash," I persisted. "Who pays in cash? A criminal."
"That's why you're going to run a background check."
This was a man who until recently had bossed around hundreds of deputies and civilian employees. Now only I was available. I made no move to pick up the phone. "He says his last name is Smith. Smith? Right."
"Some people are actually named Smith." He left my share of the retainer on his desk and slid the envelope containing the remainder into his suit-coat pocket.
"And his sister has a different last name?"
"Families are complicated nowadays. Lindsey and Robin had different last names."
Bile started up my windpipe. Lindsey and Robin. I wanted to curse him. I bit my tongue, literally. It worked. I gained deeper knowledge about the provenance of a clichéd expression. And I said nothing.
Peralta, typically, bulled ahead. "How is Lindsey?"
"Fine." How the hell should I know? She's only my wife, a continent away physically and even further in the geography of the heart.
"When did you talk to her last?"
I told him I called her on Sunday. I called her every Sunday, timing it so I would catch her around noon in D.C.
"She'll get tired of Washington and Homeland Security," he said. "It's a temporary gig, right?"
"I guess."
It was a temporary position that seemed to have no end.
"When she's ready to come home, we could use her here."
I said nothing. Yes, she was the best at cyber crimes. That was the job she did for Peralta when he was sheriff. But the last place my wife wanted to be was back in Phoenix.
I started coughing again. Three wildfires were burning in the forests north and northeast of the city. The previous year had been the worst wildfire season on record and we were off to an ambitious start now. It was the new normal. Yesterday the smoke had combined with the usual smog to obscure the mountains. Somebody flying into Sky Harbor would never know why this was called the Valley of the Sun. The gunk was sending people with asthma to emergency rooms and making me cough. Quite an irony for a place that once claimed clean, dry air that had made it a haven for people with lung ailments.
But that was the least of the reasons why Lindsey didn't want to be here.
Sitting back down, I said again, "We shouldn't take this case."
Peralta's obsidian eyes darkened further. "Why?"
"Felix the Cat in his fifteen-hundred-dollar suit, paying you in hundred-dollar bills. He's hiding something. Maybe Zisman had a mistress or not. Maybe Felix is using us for some vendetta against Zisman. The guy's pretty clean from what I remember. He actually came back home to Arizona after making it big and has tried to help out poor kids. Now here's some dude in an expensive suit who wants us to play morals police."
"He only asked us to investigate a suspicious death," he said. "Remember, Felix bridled when you implied Grace was involved with this Zisman."
That was true. Why was I fighting against taking this case?
Peralta swept his arm wide. "Half the bigs in Phoenix stash their mistresses in San Diego condos. Big deal. But we have our first paying client. Have a sense of celebration, Mapstone. This might not lead anywhere. It probably won't. If not, we'll refund most of his money. Bringing the family comfort and closure is a big thing. We can get out of town for a few days, go to a nice, cool place."
I was still about to gasp from Mike Peralta using the word closure. I managed, "You go. I'll hold down the fort. Who knows, we might get another client."
"You're coming with me. You know San Diego."
"It's changed a lot since I lived there."
"Well, you used to live there."
I tried not saying anything.
"You won't see Patty."
I could feel my cheeks warming. "This has nothing to do with Patty."
"I know you," he said.
Yes, he did. He had known me as a young deputy he trained. And then all the years I was away teaching, finally ending up in San Diego. And he had known me when I was married to Patty in San Diego. One marriage dead. Another on life support.
"It's been a long time, Mapstone. She probably doesn't even live there any more."
I stared at the wall. Patty would never part with that house in La Jolla.
The room was still. Only the sound of intermittent traffic on Grand Avenue penetrated the walls. Then a short train rumbled past and the sun started coming through the blinds. Peralta pretended to ignore me.
"Fine. I'll go. Fuck you."
The gunfire put me on the floor.
It was a loud and mechanical sound. One long burst, chucka-chucka-chucka-chucka-chucka. Then two short bursts. I pulled out my heavy Colt Python .357 magnum with a four-inch barrel, rolled away from the door, assumed a firing position, and waited for the shooter to break in. He would be looking at his eye level. I would be below him and put three rounds into his torso before he could take his next breath.
An engine revved and tires screamed against pavement. Then all I heard was silence. The eighty-year-old glass of the windows was untouched. The front door was secure. I wasn't sweating anymore. The ancient linoleum floor was cool. It smelled of old wax and fresh dust.
When I glanced back, Peralta was emerging from the Danger Room. In his hands was the intimidating black form of a Remington 870 Wingmaster shotgun, extended tube magazine, ghost sights.
He racked in a round of double-ought buckshot, producing the international sound of Kiss Your Ass Goodbye.
"That was an AK-47," he said.
"How do you know?"
"Because I was shot at enough by AKs in Vietnam that I'd never forget the sound."
I stood and moved along the wall toward the door.
"Mapstone."
I turned.
"Let's go out the back door."
We stood away from the jamb as Peralta opened the back door. Nobody poured AK rounds through. He tossed a black duffel bag out to draw fire. Nothing. He nodded and I knew what to do.
I stepped outside into the oven and ran along the southeast wall while Peralta went around the other side. It was like the academy so many years ago. The carport was on my side and gave me cover to slide between the cars unseen from Grand Avenue. Felix's Benz was stopped in the closest traffic lane. Nobody else was around. Across the median, a small car zipped by going toward downtown without changing its speed. No traffic was headed in the other direction.
Both hands on the Python, I swept the parking lot and made a slow trot toward the Benz. The sun was in my eyes and the scrunchy pavement was loud under my shoes. Peralta was coming from the other edge of the building in an infantryman's crouch, moving quickly and with a grace that belied his big frame. We reached the car at the same time.
Felix the Cat was very dead.
His face was gone. The nice suit was plastered in blood and bone fragments. More blood, brains, and miscellaneous gore were sprayed across the seat and interior of the car. One bubble of tissue had fallen halfway out of his skull and it took me a few seconds to realize that beneath the blood was an eyeball. His left hand still clutched the cell phone I had seen him holding while he talked in our parking lot. In the passenger seat lay the silver bulk of a Desert Eagle, a nasty semiautomatic pistol. It had done Felix no good. His right hand was in his lap. He had never even been able to reach for the gun. Maybe he had it on the seat when he was still in our parking lot. Or maybe he pulled it out when the other car came beside him.
There was something else: the shooter had been so close and so skilled that no shell casings scattered on the pavement. Not one. I had counted at least nine shots.
I did one more look-around and holstered the .357. Whoever had done the shooting was good. Felix had pulled out onto Grand Avenue when they caught him. His driver-side window was still down; no glass shards were to be found. And only one round had penetrated the fine paint job of the car door. The others went right to target.
I turned to Peralta and asked if he had any evidence gloves.
"I want to see that cell phone and the last number he called."
He shook his head. "Give me your gun."
"What?"
He held out his hand.
I hesitated, and then I slipped off my holster and handed it to him. One, two, three cars sped by.
"I'm going back inside," he said. "You're going to call 911 and sit on the curb. We're not the law any more."
I dialed as he trotted back to our office. The excitement over, my body resumed sweating.
In the distance, I heard sirens.
It was ten p.m. when the Phoenix cops finally cut us loose. Out on Grand, it had been a full response: half a dozen marked cruisers, chopper, news helicopters, Phoenix Fire paramedics, crime scene, and the avenue blocked for hours. Back in the air conditioning of the office, two young detectives had interviewed us. Peralta was cagey. No, he outright lied. Nobody could tell when Peralta was lying, certainly not this pair. He had come back outside and taken control of the narrative and of me.
Here's the way he told it: we were waiting in the office for a potential client when we heard the shots and went out to find the Benz and the dead body. That was when we called the police. Had this potential client given a name? No, Peralta said. It was a man and he didn't give his name. Peralta didn't think to ask for it. We were here and he told him to come on by. Were the detectives thinking this was the man?
They didn't say. They did ask if we knew a subject named Derek Zimmerman.
"Is that the D.B.?" Peralta gave them his best command stare and they responded.
"Maybe, Sheriff."
"Never heard of him. Have you, Mapstone?"
No. I hated him. Why was he lying? I felt all this, forgetting that I had wanted to muck with the investigation by reading the recent calls on our late client's cell phone.
"How about Felix Smith? James Henry Patterson?"
"Who are they?" Peralta asked.
"We're only starting our investigation," one detective offered. "But you might be glad you didn't get this case. The guy was carrying multiple driver's licenses."
I thought about the Desert Eagle on his passenger seat.
They left their cards. If we remembered anything else, please call us at this number...I had done the routine a hundred times myself, when I was on the other side of the badge. Then they left.
"Why did you do that?" I whispered it, as if the detectives were listening at the door.
"I want a trip to San Diego."
"Our client is dead."
"Exactly."
Afterward, I drove east a few blocks on Encanto Boulevard and was enveloped in the trees and grass of the park and the historic districts. The temperature dropped ten degrees. This was a good thing considering that the air conditioning in Lindsey's old Honda Prelude had seen better days. On the north edge of Palmcroft, I sat through the long wait at the Seventh Avenue light, brooding over what had happened. Then I crossed into Willo, past the old fire station, and headed home. A right on Fifth Avenue and a left on Cypress. The street was quiet and most of the houses were dark. Normal people had gone to bed. My house was dark and not inviting. I vowed again to get some lights on timers and drove on.
At the Sonic on McDowell, I ate a foot-long Coney dog and drank a medium Diet Cherry Coke. The bright lights and blaring bubblegum music gave a false sense of protection. The condition of the car gave me no choice but to turn off the engine and open the windows. The climate could thank me later.
An AK-47 was a crappy assassination weapon, so Peralta told me after the cops left. In all but the most expert of hands, it had a tendency to ride up and have bad accuracy. On the other hand, who could miss at that range? Smith/Zimmerman/Patterson had pulled onto Grand and another car came alongside. Did he recognize the car and stop to talk to its occupants? Did the encounter have something to do with the phone call I had seen him making?
Peralta didn't offer any theories. He did say, "Somebody who uses an AK that way, it's his preferred weapon. He likes it."
I ate the wonderful crunchy Sonic ice, marinated in cherry, and took a little comfort that I was the only car in the place. A little comfort.
Oh, I wished Lindsey would call and ask, "How are you, Dave?" and being Lindsey she would know from my voice that I was not fine, nowhere near it, not even in the same state as fine. But my cell phone was silent.
There was nothing to do but go back to the house, which I did after driving around the block four times.
The 1924 Spanish Colonial on Cypress was lovely and forlorn. The old locks and new alarm system were fine, but I still swept through the rooms with my revolver out. The floor-to-ceiling bookshelves that ran the length of the stairway looked at me indifferently. More bookshelves lined the study, books accumulated by my grandparents, by me alone, by Lindsey and me.
Except for the few years I was away, I had lived in this house all my life, and my grandparents before me. Yet the walls silently said, "You are only passing through here. We will remain." The walls didn't care about the tragedies this house had endured.
In the kitchen, I pulled the Beefeater out of the freezer and stirred a martini, the perfect chaser to diet cherry Coke. The only thing I had done to the house lately was to put up new curtains that completely hid the back yard from view when one was standing at the sink. They still did provide privacy, but I couldn't avoid pushing them against the glass as an extra measure of safety.
I have stopped turning on the lights. I have stopped listening to jazz. I have stopped reading books. The outside world holds no appeal, either. I've made myself go to several movies at the AMC downtown at Arizona Center, but I left each one after a few minutes. I couldn't stand any of it. So I sat in silence in the living room, sipping the cold gin, staring out at the street, trying to keep my mind locked down. At least the neighbors had stopped their well-meaning water torture of relentless expressions of sympathy over Robin's death and inquiries about when Lindsey is coming home.
I went to the bedroom and stripped down without turning on any lights. I lay down on Lindsey's side of the bed. It turned into Robin's side, too, bastard that I am. Over on the bedside table sat John Lewis Gaddis' biography of George F. Kennan. I felt all of Kennan's emotional shakiness and had none of his brilliance. My "long telegram" would not be about the Soviet Union but about my own union that was breaking up, if not hopelessly broken. I picked it up and tried to read. Nothing caught the gears of my brain. It wasn't Gaddis' fault. So I tried to sleep. Peralta would be here at seven, packed for San Diego.
Too soon, I found myself on a Central Avenue bus. No, it was an airliner. I didn't recognize anyone around me. But I dropped my cell phone on the floor and it slid backwards. What if Lindsey suddenly called? I got on my knees and found the phone two rows back.
Then I was out on the street. The sidewalk was broken and I had to watch my step. New buildings were going up and others were being restored. Bright paint was being applied. The city had never looked better, with a huge downtown skyline against majestic, snowcapped mountains. I would have to stop criticizing it.
A door was open and I walked in. Instantly, I was in my former office at the old courthouse. The big room was nearly empty and I felt sad, until I saw Robin sitting at the desk. She looked up and gave me that roguish smile. She stood and I took her in my arms, brushing back the long, tousled blond hair and covering her with kisses, sobbing and holding her while she laughed and we talked over each other. She put a finger over my mouth and I was silent, listening to her tell me... Tell me...
Then she was gone.
I was in a hallway painted blood red, looking for Robin. I walked for what seemed twenty-five paces, trying locked doors, and then turned into a narrower passage. I was completely alone. In my pocket, where my cell should have been, was only a wallet. I pulled it out again and it was a pack of Gauloises, the brand of cigarettes Lindsey smoked.
My gut was in full panic gear now but I kept walking, finding new hallways, each one smaller than the one that came before it, turning and turning. Where was I? It seemed as if I was going in circles. There was nobody to ask for directions. My cell phone was gone and my legs moved only with ever-greater effort. I kept going. Behind me was only darkness. Then I could barely make it through the hall; it was so narrow I turned sideways to make it into the next section. Finally, the walls tapered together in a "V" and I was at the end.
I knew by now that I shouldn't push against the drywall, but I did.
I couldn't stop myself.
I always did.
That was when the explosion came.
We were in the back yard on Cypress. Night. Robin was on the ground and I was on my knees, trying to resuscitate her, trying to stop the bleeding. Her blouse was wet from the blood and it was all over my hands.
I looked up and this time the woman with the gun was still standing there.
This time the woman was Lindsey.
Then the dark bedroom greeted me and I was awake, in the dimension where the mountains were low and the city was not reclaiming Central. Where the downtown skyline was still squat, monotonous, and ugly and the only real event of where I had been was Robin's death in the back yard from a single gunshot.
I had this dream nearly every night. I called it my maze dream.
Peralta slid into my driveway at precisely seven a.m. I walked out with my bag and the surly attitude of a non-morning person, stowing my gear in the extended cab of his gigantic Ford F-150. I would leave the argument about his personal contribution to greenhouse gases and climate change for another day. He surprised me with a venti non-fat, no-whip mocha from Starbucks, my usual drink, and one he has disparaged on many occasions as virtually anti-American. He, of course, was drinking black coffee. We backed out, cruised through Willo and Roosevelt, and then slid onto Interstate 10 where it pops out of the deck park by Kenilworth School. It was only ninety-nine degrees. I was in my tan suit with a blue Brooks Brothers polka-dot tie, about to keel over from heat exhaustion.
Neither of us said a word as the suburbs fell away and the truck turned onto Arizona Highway 85 for the short but dangerous connection to Interstate 8. The state was gradually widening what had been a two-lane highway, but people still drove like maniacs and fatalities remained common. Today, the road was nearly empty. If only my head were that way. Jagged bare mountains rose up on either side. I remembered from Boy Scout days that one was called Spring Mountain. I also recalled it was about 355 miles from Phoenix to San Diego. I adjusted the vents again to get the most out of the truck's air conditioning.
When he caught I-8 at Gila Bend, I made my first attempt to breach the battlements of his stubborn personality.
"What about the lawyer Felix mentioned?"
"I called him. He never heard of any of those names."
I asked him if he had given the lawyer a description and he shot me a cutting glance. I thought about Felix sitting there yesterday, so straight and self-possessed in his expensive suit, French cuffs, tattoo, and prosthetic leg. He was not someone to forget.
"So tell me what again we're doing?"
"Driving to San Diego."
Five more miles brought a passing Union Pacific freight train and flat desert.
"You know what I mean." The mocha was finally cool enough to drink.
He declined to answer, so I settled into the seat and watched for more trains. We rode high and mighty along the highway, a steady eighty miles per hour, dwarfed only by semis.
The retiree tract houses and fields of Yuma trickled out to greet us, hotter than hell, and ugly. We went through a McDonald's drive-through and ate on the road like two street cops as we crossed the Colorado River and entered California.
I tried again. "Why did you give a false report to the police, saying Smith, or whatever the hell his name is, was never in our office?"
"It was easier." And that was all he said between mouthfuls of a Quarter Pounder with cheese. Peralta was the most by-the-book hard-ass peace officer I had ever known. I told him this.
"Don't be so quick to judge, Mapstone." Bite, chew, swallow, steer with one hand. "And don't get grease on that arm rest. As I recall, you did a little selective application of the law after Robin was murdered."
That was true, but I wasn't going to let him get me into that dark alley.
"I'm talking about now. We don't owe this guy anything."
"You wanted to break the law yesterday by tampering with evidence."
He was right. I wanted to see the number Felix had called from our parking lot. I was a long way from being a Boy Scout.
Peralta shrugged his big shoulders. "He put us on retainer for ten grand. Our obligation is to the client, and that includes privacy."
Sand dunes loomed up on the south. I knew that a plank road was built here in 1915. I didn't know anything about being a private investigator. Listening to our conversation made me question myself again about joining him in the rough little building on Grand Avenue. Robin had suggested it. She expected to live to see it. I violently shook my head.
"You have a headache?"
"No." I ate the Big Mac and daintily wiped my fingers to protect his fake leather or whatever the hell it was on the armrest. "What would you as sheriff have done to a PI who pulled the stunt you did?"
"Probably prosecute him."
"But now you're applying situational ethics."
"Don't be using your fancy academic language on me, Mapstone." The burger was gone and now his right hand was grabbing French fries. Peralta always ate one fast-food course at a time. "I'm just a simple boy from the barrio."
I cut him off. "You went to Harvard." He knew very well what I meant. I gave up for the moment.
We were now in the Colorado Desert, a very different place from the lush Sonoran Desert that surrounded Phoenix and Tucson: no saguaros or any of the other hundreds of plant and animal species of my home country. It was sun-blasted moonscape, a sea of tranquility: long vistas, distant mountain ranges, few colors beyond off-white, ochre, and brown—and in this spot it declined into a sink that was below sea level.
We finished lunch at eighty-five miles per hour. I policed all the containers and napkins, and then the flat, green fields of the Imperial Valley surrounded us, all irrigated, quite irrationally, by a canal running from the Colorado River. If not for the geology of the Colorado's delta, the Sea of Cortez would go all the way north to Indio. It was an amazing thing to contemplate. North of us was the Salton Sea, accidentally created in 1905 when the Colorado, as it would do before being nearly killed by dams, flooded into crude irrigation canals dug to divert water into what had been the dry Salton Sink. The "sea" became a major bird destination, created its own ecosystem. Now it was dying, helped by the Imperial Valley's toxic runoff. I read the other day that the noxious air from a massive fish kill drifted as far as Los Angeles.
We stopped to relieve ourselves in El Centro. No offense to the local chamber of commerce, but that seemed all it was good for, even though the town now had a Starbucks. The air was hot and hazy and smelled of agricultural chemicals.
Ahead were Plaster City and the startling escarpment of the Laguna Mountains. San Diego had one of the finest natural harbors in the world, but the railroads couldn't easily get there in the nineteenth century because of that mountain range. Instead, they went to Los Angeles and that was that. We passed over one track at Plaster City and it reminded me of the railroad that was finally built to San Diego from the east. If memory served, the sugar baron John Spreckles underwrote it, and the San Diego and Arizona line was one of the most ambitious engineering accomplishments of its day. But it never made money and the land it traversed was so harsh, including a perilous crawl through Carrizo Gorge, that maintaining the railroad was prohibitive.
The wall of mountains and its forbidding canyons beyond did not intimidate Dwight Eisenhower's Interstate Highway System. So I-8 was built in the mid-1960s and San Diego finally had its connection to the east. Every year it brought more Phoenicians to the coast in search of relief from the summer heat.
None of this would have interested Peralta.
We started the serious grind uphill and then we were climbing through terrain strewn with giant boulders, the marble game of the gods. Behind us, the Imperial Valley spread below like a dry seabed. The Interstate twisted and curved, an unwelcome intruder. The sun was on his side, glinting off his thick hair and bringing out the aristocratic profile. I knew this highway well, but the majestic land never ceased its ability to move me.
"So we're going to San Diego," I said. "Do we know if anything this guy told us is correct? He lied about who he was. How do we know this Grace Hunter even existed or killed herself?"
"We don't. We'll find out."
Robert Caro writes about how Lyndon Johnson was a reader of men. Nobody could read Peralta, not even LBJ, and certainly not me. Only occasionally did a "word" reveal itself to a careful observer of his professional mask. As I studied him, I could see an unusual determination in the set of his thick jaw. To be sure, "determination" with Peralta was like saying "deep" about the Grand Canyon. It was always there and spectacular to behold, much less try to hinder. Now, however, the canyon of his tenacity was unusually on display. But I saw something else, too, another word. Concern.
I said, "What if we're being set up?"
"Then it's better to take the initiative."
"What if we're being set up by going to San Diego?"
"I told you, you're not going to see Patty."
When I lapsed into silence and he realized his effort to piss me off had failed, he spoke again.
"Whoever did this would expect us to give a full report to the police and lay low in Phoenix. That would be logical. So we'll do what they don't expect. All you have to do is your history thing."
This was what he used to say when he would barge into my office in the old Court House. It became a longstanding joke. But I blew. "What history?!" It was amazing how his luxurious cab absorbed the sound of my tantrum. "The dead man in front of our office didn't have any history! This isn't a historical case."
As usual, my outburst failed to move him. In as soft a voice as he could manage, "Mapstone, everybody has a history. You need to find it. 'The only new thing in the world is the history you don't know.' Saint Paul said that."
"Harry Truman said that."
"Same difference."
I resisted the familiar urge to reach over and try to strangle him, even if I would lose the fight and he would never even swerve out of his lane.
He said, "We have a name, D.O.B., Social Security Number and photo..."
"Right, and she's a sweet girl who went to Chaparral High in Scottsdale, was a student at San Diego State, worked part-time at the Nordstrom perfume counter at Horton Plaza. She had a boyfriend and somehow she ended up at Larry Zisman's condo on the night of April twenty-second."
"See how much you know?" He lazily draped his arms over the steering wheel despite the tangled road we climbed.
I slumped in my seat. "If any of it is true."
"David." He never called me David. "You have a gift for history. I never thought you'd be happy as a professor. You're a cop down in your bones. But you're a historian, too. You look at a case the same way a historian studies the secondary sources and published material on a subject. You talk to the primary sources, read their recollections. Then you apply a historian's skepticism and diligence, come up with new interpretations, dig out fresh facts, add context, shine the light in a different direction seeking the truth. It's what you do."
It shut me up. Even made me feel better about myself for the moment.
When we crested Laguna Summit, he spoke again.
"Here's something else to consider, Mapstone. We don't know why Felix was shot to hell. Maybe it was because of this girl, or something else in his life that made him carry that Desert Eagle that was on the car seat."
He had been there so briefly, it surprised me he had time to notice and identify the gun on the passenger seat of the Benz. But that was Peralta.
"But," he went on, "he might have been killed because he came to see us. And I don't want the word on the street to be that you can kill our clients. It's bad for business. And it might encourage the wrong kind of people to reach out and touch us."
I let him alone. It was a reminder that there were three ways to do things: the right way, the wrong way, and Peralta's way. Soon we would begin the long descent to San Diego, through the lovely little meadows of eastern San Diego County that looked as if they hadn't changed for a hundred years, back up to Alpine at the edge of the Cleveland National Forest, then dropping and curving into El Cajon, massive Silverdome mountain dominant on our right, the cool sea air coming up to kiss away the memory of the desert, the city swallowing us up, the freeway packed with traffic, and the ocean straight ahead. San Diego: my adopted hometown. I would need to pack my emotions tight.
Among the benefits of being the former sheriff of one of America's most populous counties was cooperation from other law-enforcement agencies even after you were out of office. An added perk was that Al Kimbrough was a San Diego Police commander. I first met him when Peralta had hired me back at MCSO and Kimbrough had been a detective. He could have stayed and become chief of detectives, but San Diego made him a better offer, one that didn't include 108-degree May days. Peralta scheduled a meeting with him downtown and complained about the overcast.
"It's the onshore flow," I instructed. "The June Gloom."
"It's not June."
I rolled down the window and looked over the bay at one of the aircraft carriers moored at North Island. "It's cool. I'm happy."
He handed me a copy he had made of Grace Hunter's photo. "Why don't you take the truck and go talk to the boyfriend? Here's the address."
I didn't breathe for about five seconds and handed the address back to him.
"What, you're showing off your photographic memory to an old man?"
"No," I said. "I used to live there." I sat for a few minutes in silence as a big jet going into Lindbergh Field rattled the cab. Sometimes coincidences were serendipity. Not this one. This was creepy. But there was a job to do. "Drop me off at Old Town. I'll take the bus."
"That's nuts."
"I couldn't find a parking place, especially for your beast."
"You can always find a parking place if you're patient."
"Not in O.B."
As I gave directions to Old Town, he shook his head and shrugged. "You're one weird guy, Mapstone."
Fifteen minutes later, I was on a half-full 35 bus, rolling down Rosecrans Street, turning onto Midway Drive for the ride across the hump of Loma Portal and into Ocean Beach. Behind us was the bay, ahead was the ocean. I counted twenty Arizona license tags and quit counting. This time of year, San Diego was Phoenix West. Native San Diegans hated the invasion. When I lived here and rode this bus almost every day, I learned not to let on where I was from. The bus started downhill, with the sun beginning to burn off the clouds behind me, toward downtown. But ahead, it was still gray, the vast expanse of the Pacific a sheet of lead blending into the overcast. The Pacific played a trick of the eye, seeming to rise into the horizon, even though we were merely descending a long slope to the ocean.
If you stay on Interstate 8, you'd run right into O.B. But most tourists didn't. They went north of the San Diego River to Sea World, Mission Bay and the more popular neighborhoods of Mission Beach or Pacific Beach, or they went south on I-5 to downtown. San Diego had changed substantially since I had lived here, but Ocean Beach looked much the same: the narrow streets, quaint and pricey cottages, one-story businesses lining Newport Avenue and the long municipal pier jutting into the ocean. I had lived two lives in San Diego: pre-Patty in Ocean Beach and with Patty in La Jolla.
It reminded me of the old days, getting off at Cable and Newport, and then walking past the business district down to Santa Cruz Avenue. A couple of guys carrying surfboards walked past me, going west. Seagulls passed overhead making their distinctive calls. The old apartment building was still two stories, painted white, and shaped like a U surrounding an interior swimming pool. My unit had been on the second floor. The boyfriend's apartment was directly beside it. I felt an involuntary urge to check my mail, smiled at it, and walked up to No. 205. The windows were open, as was common here, and the drapes were drawn and partly hanging out.
The loud, angry voice coming from the apartment wasn't surprising, either. O.B. was an eclectic place, where CPAs lived alongside bikers. Once I had been kept up all night when one of the latter had engaged in an all-night screaming fight with his old lady. Now I would put a stop to it, but I was different then.
The voice was deep and menacing, the dispute involving a woman, money, and perhaps more. The dialogue was generally, "I want Scarlett, motherfucker, and your white ass is out of excuses. Where is she? I need her ass back out making money," on and on. "You think you can hide from me? Nobody gets away from me. I own her sweet little booty. Now where the fuck is she? Tell me now or I stomp your white ass to death and find her my own self."
My mind momentarily thought of search warrants and probable cause, but, as Peralta said, we weren't the law anymore. When I heard a fist connect with flesh, cartilage snap, and a man squeal, I opened the door.
"What the fuck?"
The voice belonged to a very large man with caramel-colored skin, mustard-yellow driving cap, delicately manicured beard, eyes way too small for his face. A Bluetooth device was attached to the left side of his head. He was my height and about a third wider. He wore a black T-shirt proclaiming RUN-D.M.C. Below his shorts were heavy stomp-your-white-ass boots.
"Who the fuck are you?"
"Life insurance." I smiled.
He raised his shirt so I could see the butt of a semiautomatic pistol in his waistband. Then he advanced toward me, one step, two...
I thrust my hand forward suddenly, open and straight-fingered into the middle of his windpipe. The small eyes burst wide, the cap and Bluetooth flew off, and he was gasping. Both his hands clutched his throat in what we had been taught in first-aid classes was the "universal choking symbol." Done properly, this was a useful move for incapacitating someone. Done wrong, it would kill him, which was why it had been discontinued by police agencies.
My next move, one second later, was to remove the Python from its shoulder holster and level it at his face.
"See, you never know when you might need life insurance."
He staggered back. From his open mouth came the sound of an ailing carburetor. His eyes showed the most primal emotions: surprise, pain, and the sense that he was suffocating. It was a testimony to his size and strength that he was still standing. That made me uneasy.
"Move back, asshole."
He did. When I was all the way inside, I kicked the door closed but made sure I was still facing him.
"Who are you?" This from a skinny, pale kid with bushy red hair, sitting on a sofa. He was probably the only person in O.B. without a tan. He was in pain, clutching his face. Seeing his hands occupied, I ignored him.
"Can you talk now?" I said this to the black man.
"Iiiihhhhhhhhhh."
I asked him whether he was right- or left-handed. He opened his mouth and showed a gold incisor. He finally managed, "Left."
"So use your left hand and pull out that gun very slowly and hand it to me." I knew he was lying about which hand he favored, or at least I took that chance. After I had possession of the Glock, I shoved him back onto the sofa next to the white kid. Gravity did most of the work. Large human objects are easier to push around when they can barely breathe.
"Should'a known you was a motherfucking cop." His voice was a shadow of its former booming self.
"I'm not a cop." I kept the .357 magnum leveled at his chest. The barrel was only four inches of thick ribbed steel, but the business end might as well have been the size of eternity.
"Now wait a motherfucking minute." He held out two big hands, palms facing me and tried to make himself smaller on the sofa, no easy task. His expression changed. He wasn't worrying about his throat any longer. "Motherfuck! I've heard about you. Big guy with a big motherfucking gun...."
I held up my hand. He stopped talking.
"Did you ever consider that repeating the same profanity over and over deprives it of any ability to shock? You might consider trying out a word such as 'mountebank' or 'scoundrel.'"
He lowered his hands and took a deep breath. "Look, man, I got no problem with Edward, man. I'm completely good with him. Why you think I'm here right now? This is between me and this skinny pale-ass mother..." He stopped. "Scoundrel."
I said, "Who is Tim Lewis?"
"He is." The black guy quickly pointed to the red-haired kid next to him.
"Then it's time for you to leave."
"What about my Glock?"
"Get another one."
He stood without protest, picked up his cap, and hurried out the door, quietly closing it. I locked it, expecting him to at least be muttering indignation and threats as he departed, but nothing. I heard heavy steps thudding along the concrete, down the stairs, and then they faded. The gate to the street clanged shut.
I waited a few seconds and holstered the Python. "Who is he?"
"I think my nose is broken!" His voice sounded like a teary fourteen-year-old.
"So who broke it?"
"You don't know? He knows you." His eyes were curious. "He calls himself AFP."
My mind did a sort: FDR, JFK, LBJ. I asked again.
Through his hands came a nasal response. "America's Finest Pimp."
Get it: San Diego called itself America's Finest City. I didn't smile. I leaned against the outer wall and stealthily looked out the drawn curtain. The courtyard was deserted. Nobody was at the pool that dominated the space. Beyond the fence, nobody was on the sidewalk.
From my pocket I produced the photo and held it out. "Do you know her?"
"That's Scarlett."
I worked hard to conceal my surprise. "Who?"
"Scarlett. My girlfriend."
"What's her last name?"
"Mason. Scarlett Mason. Do you know where she is?"
I nodded, put the picture away, and asked him what problem he had with America's Finest Pimp.
"I'm really hurting, dude!"
I checked him out in more detail. He might have had the kind of face teenage girls consider cute, at least before his nose had been broken, but to me it looked like a comic-book face, a cross between Archie and Jimmy Olsen. His face was so thin, a vein running up his forehead was prominent.
His body looked rangy and underweight beneath a gray T-shirt, droopy Lakers shorts, and teal flip-flops. A flaming tattoo wrapped itself up his left calf. His fingers, long and slender, were oozing bright red blood from where America's Finest Pimp had hit him, and now it was dripping onto his shirt. I walked to the kitchen, grabbed a dishtowel, and tossed it to him.
"Where is Scarlett? Please..." His tone was plaintive enough to be believable. I was about to tell him to get some ice on his nose so we could talk.
The next thing I heard sounded like a cat, until it didn't. My right hand was on the way back to my holster. "Who else is here?"
"The baby."
I grabbed him by the arm and pushed him ahead of me into the bedroom. Once I would hide behind books. Now I was using a human shield. Beside a box spring and mattress on the floor was a yellow hand-me-down crib. After ordering him to stand against the far wall so I could watch him, I approached it.
Sure enough, inside was a baby, incredibly tiny, with a tuft of brown hair and a very soiled diaper.
When I looked back at Lewis, he was kneeling, his head pointed down. "I don't feel good..."
"How long has this baby needed changing?"
"I don't know. AFP was here for couple of hours, waiting for Scarlett, telling me he'd kill me if I didn't give him the money she owed him..." He was sobbing. The vein up his forehead expanded. "I think I have a concussion. I'm dizzy. Can you change him please? I didn't mean to leave him back here alone."
I filed the money part away and let him alone. He was useless. I looked around for supplies. All I saw was a television, along with a video-game box and a cell phone sitting atop a plastic crate that doubled as a bedside table. Opening the closet, I found a shelf with a box of Pampers, wipes, and baby powder.
Back at crib-side, I felt pretty useless myself. As a young deputy, I had delivered a couple of babies in the backs of squad cars. Otherwise, I had spent a lifetime staying as far away from them as possible. At least until a year ago, I figured that would always be the case. But as I beheld this tiny, helpless creature, I was nearly overcome by a hurricane of feelings and instincts. The bracing stench coming from the diaper brought me back to reality. It wasn't as bad as a dead body left for a week inside a house during high summer in Phoenix.
I pulled out a clean diaper and slid it under the baby, who was squirming with more energy and squalling like a siren. Maybe I was painting myself into a very messy corner, but it was worth a try. Then I set the wipes on the mattress and gingerly undid one tab. The stench grew worse. Thankfully, the window was open and a faint sea breeze was coming in. So far, so good: I pulled the other tab, folded it in on itself, and lowered the front of the soiled diaper. Immediately a little fountain of urine shot all over my tie and shirt.
It was a boy.
Pleased with himself, he kicked and flung his arms. Back to it, I used wipes to clean off his front, between his legs, and under his scrotum, wadding them up and putting them on the soiled diaper. Feeling pretty good about myself now, I folded the diaper in on itself to provide a clean surface, lifted his legs, and cleaned off his backside. That took another four wipes. Then I slid out the bad diaper, rolled it up, and, voila, he was safe and sanitary on the new one. I hooked the tabs and lifted him into my arms, which did nothing to stop his wiggling and crying.
"Better?" I smiled. The big baby head stopped crying for a moment, then started squealing again as if I were torturing him with hot pokers.
Instantly, the silent-but-deadly cloud of odor hit me. The new diaper was heavy again and I felt something oozing out onto my hands.
"Well, hell."
I know a few things: the socio-economic issues of the Progressive Era, the revisionist arguments regarding the causes of World War I, how to prepare a class syllabus. I have some skills, including reloading the Python under pressure, properly tying a necktie with a dimple in the center, and effectively swinging a hammer. I know how to make a dry martini and make love to a woman. Here, I was over my head.
Muttering a lesson in profane oaths for the young master's linguistic instruction, I carried him into the bathroom and deposited him in the sink. The din of his crying was magnified by a power of ten.
So much for my clever first attempt, filled with hubris and baby-shit.
It took another fifteen minutes, a facecloth protectively placed over his dangerous little penis, much clumsiness on my part, and two diapers, but the baby was finally clean, powdered, and back in his crib. I put a rattle in his hand and shook it. He looked at me with a surprisingly grown-up expression, dropped the rattle, and conked out. After what we'd both been through, it seemed like a good idea to me, too.
I wished that Lindsey's face would stop flashing across my vision.
After I washed up and cleaned my tie, I retrieved Tim Lewis, who had slumped against the bedroom wall, silently watching my learning curve.
"Get up. We need to talk."
"Have you been crying, dude?"
"No."
"Thanks for the help."
I said nothing.
A few minutes later, he was back on the sofa and I was sitting across from him on a dining chair.
He stared at me over an icepack that I had improvised for his traumatized nose. A nasty black left eye was also materializing. He started shaking.
"Are you going to kill me?"
That's me: the diaper-changing, first-aid-giving hit man. I said, "I will kill you if you abuse that baby."
"I take good care of him! I love him! AFP wouldn't let me go back and change him. Since Grace left..."
He blinked and I knew he was hoping I hadn't noticed his slip.
I said, "So who was this Scarlett?"
He cursed at himself. "That was Grace's business name. Her brand."
I pulled out the photo again, turned it toward him, and tapped my finger on the pretty face.
"Her name is Grace Hunter," he said.
"Is that her baby?"
"It's our baby." Somewhere under the icepack, I heard a long sigh. "This has gone so wrong."
"What, that you're living with a prostitute?" I was careful to keep Grace in the present tense.
"She's not a prostitute." His face flushed with anger.
"Then what do you call it when a woman works for a pimp?"
I waited and he told it. It wasn't easy telling.
They had started dating as freshmen at San Diego State. He was studying theater and she was a business major. She had wanted to be in theater, too, but her father demanded that she declare a more practical major. Specifically, business. If she wanted money, he said, she could start her own business the same way he had done. Grace moved in with Tim. They were poor and not happy, working part-time at restaurants, already facing big student loans. They broke up. It was a big campus, so he didn't see her often. He dated some other girls but kept wishing he could get back with Grace.
Three years later, he saw her at Comic Con, the huge comic-book gathering at the convention center downtown. But she wasn't dressed like a nerd. She was in a tight but very expensive-looking mini-dress and on the arm of a guy in a suit who was old enough to be her father. He later learned that the man was a producer in Hollywood. She smiled and waved at Tim, and a week later she emailed him to get together.
Tim learned how much had changed in the time they had been apart.
Grace Hunter's entrepreneurial inspiration had come soon after their breakup. One night she went out and got drunk. An older man hit on her, she went back to his hotel room with him, and spent the night. When she woke up, he was gone but on the bedside table was a thousand dollars cash. Whatever weeks or hours of moral wrestling she did with herself, she realized that San Diego was full of male tourists and businessmen, almost all of them dreaming of a night with a California girl. And they would pay quite well.
She drew up a formal business plan on her laptop: her market was affluent, older married men, the startup costs consisted of the right clothes—bikini for the strand, nice dress or suit for a hotel—and her competitive advantage was that she didn't look like a call girl. The tax exposure was zero. Her brand was Scarlett.
For more than two years, she succeeded brilliantly. The men were usually nice, often terribly lonely, some wanted only to talk, and all were willing to use protection. Not one beat her up or even made her feel creepy. Once a month, she had herself tested for STDs and was always clean. That checkup report would ensure top dollar. She gathered regular clients and her discretion gained referrals. Thanks to her patrons, she stayed at the best hotels and resorts in the area. A few times, men paid her to be with them on more lavish adventures.
"Did she do kink?" I interrupted. "Bondage?"
"No," he said. "That doesn't sound like her at all."
I wondered how much he really knew her, but shut up and let him continue.
The money she earned was awesome. The Great Recession didn't hurt her profits. This sure beat taking on more student debt. She set up small accounts at banks around town, depositing cash as if it were her tips as a waitress. Over time, she consolidated them into a smaller number of bigger accounts. She took out loans from her father and paid them back, telling him that she had a job helping a woman stage condos and houses for sale. Her father's checks were clean to deposit. It was a crude way to launder money, but it was good enough.
The only thing Grace Hunter hadn't assessed for her business plan was the competition. And one night she was kidnapped, beaten, and raped by America's Finest Pimp. He told her that he ran the hotel girls in America's Finest City. He would control her liaisons and take seventy percent of her gross earnings. If she held out on him, he promised, he would beat her to death and take her body out on his boat, feeding her remains to the sharks. For the next three months, she lived in constant fear.
Then she saw Tim again.
He took off the icepack and shook his head. "We thought we'd be safe in O.B. She had money saved. Then she got pregnant and the baby came along. We were happy. She just got a job at Qualcomm and I was going to be a stay-at-home dad when I graduated. I guess she decided to leave me. But I can't understand how she could leave our baby."
Lindsey's face again, whose eyes were such a deep blue that in certain light and certain mood they appeared violet. I thought about the new life I had held in my hands, minutes after gripping the potential death of the Colt Python in the same hands. It was a corny thought, to be sure. But Lindsey's voice burned like acid on my face: You did this!
Focus, Mapstone. "Why didn't AFP get her addicted? That's the usual M.O. for a pimp."
"She convinced him she'd be worth more clean. She was good at convincing people. AFP sees himself as a businessman. She paid him straight, every week, until she disappeared and came to be with me."
"Did it bother you that she'd fucked all those men?"
I phrased it as crudely as I could and he stared at the carpet. He was a natural suspect. Jealousy was always a prime motive, wronged spouses and boyfriends always prime suspects.
"All those men, their dicks inside her." I spoke tawdry fluently. "It would sure bother me. It would bother me to find that my wife had been fucking even one man other than me."
Trust me. Only every second, splinters under my skin. But the splinters didn't want to make me kill her.
I said, "I know you're a nice guy, Tim. But didn't it get to you? Did you ever think about killing her when you thought about all those men..."
"No!" His face flushed apple-red.
I took my time, studying his expression and body language, and letting the silence work for me, having watched Peralta interrogate many suspects.
Finally, Tim drew up his wiry frame. "That was in the past. She regretted it. I loved her. I'd rather die than hurt her."
I believed him. He didn't have murder in him.
"Did she ever talk about a man named Larry Zisman? He used to be a pro football player. Owned a condo downtown."
"Was that one of her clients?"
I didn't answer.
"The name doesn't sound familiar," he said. "And she didn't talk about those men. I didn't want to know and she didn't tell me."
"So you guys lived alone here. What about friends?"
"We'd say hi to neighbors. It's that kind of place. Grace stayed in touch with Addison..."
"Who the hell is that? A man or a woman?"
"A woman. She was her best friend."
"Did she visit?"
Tim said that Addison had visited several times, but they never left O.B.
"Addison didn't know anything about Grace's, you know, business."
"I need her contact information." Then I asked when he had seen Grace last.
"The morning of April twenty-second. I had classes. When I came home, she was gone. I never even got a text goodbye. All her stuff is still here. It doesn't make any sense."
"Are you afraid she's gone back to the life?"
He shook his head. "She said she was done and I believed her. She got rid of her old phone, even. We were good together." He sighed. "I wanted to save her from the past."
Tim Lewis looked like a weak reed of a white knight, but his sincerity was obvious. I had gone through my white-knight phase. Now I was covered with tarnish. I made him go through the day she disappeared in detail. He had gone to classes at eight-thirty that morning. Grace was with the baby at home. When he returned around three that afternoon, she was gone. All she took was her purse and cell phone. She always carried pepper spray and a knife in that purse. Nothing had seemed unusual in their apartment.
"Why didn't you go to the police?"
"I filed a missing person's report the next day. The cops made me wait twenty-four hours and even then they didn't take me very seriously. I could tell. They thought she'd left me. They said she was an adult and there wasn't much they could do unless I had evidence of foul play. Of course, I couldn't tell them she used to be a call girl." He shook his head. "Anyway, AFP pays the cops off. Grace warned me. I was sure I'd eventually hear from her. I called hospitals for a week. Nothing."
Grace would have been dead by the time he went to the police. But things fell through the cracks in every police department.
"Where's her family?"
"They lived in Arizona."
I asked him to get me their address and he did.
"What about a brother? Big guy? My size with close-cropped hair and a prosthesis on his lower leg?"
"She was an only child."
I looked at the skinny kid with the cat crawling up his leg: I thought, dear old dad. I said, "Who is this Edward that the pimp was talking about?"
"I have no idea. I swear!"
So I told him she was dead and waited as he cried. It was a long wait. He said over and over that Grace would never kill herself, especially after the baby came.
Finally, I asked if he had any place he could go.
"My parents live up in Riverside. It's a boring hellhole."
"My advice is to go there. Right now. And stay awhile."
He nodded, but it was obvious he was descending into a fog of grief, in addition to being beaten up. I made him repeat what he would do.
Go.
Now.
I handed him my business card.
"Private investigator," he said quietly. "Are you trying to find out who killed Grace?"
"Yes."
"I want to hire you."
"We already have a client."
He repeated his request. "I've got to know what happened to Grace. And I want the bastard who killed her to burn." Misery shone in his watery, pale eyes.
"Okay."
He reached under the cushions of the sofa and I tensed.
"Here's five hundred." He handed me a wad of cash. "Is that enough for a start?"
"Sure. But I'll do this pro-bono."
"No," he said. "I don't want your charity. I want you to work for me, and cash talks. Grace taught me that."
I realized it might be good to have a living client, especially because the man who had hired us yesterday was dead and Peralta had lied to the Phoenix Police, saying he had never even come into our office. I took the cash and wrote out a receipt for it on a blank sheet of paper.
He rooted around in the kitchen and returned with a flash drive. "This has her client list. The regulars."
"Have you seen it?"
He shook his head. I could understand why he wouldn't want to look.
I took it and told him we'd be in touch, but that he should call me when he got to Riverside.
His voice stopped me as I was halfway out the door.
"Thank you again for changing the baby. Do you have kids?"
I didn't answer.
"They totally change the way you look at life."
Personal history: the day I arrived in San Diego to take an Assistant Professor of History position at the same university that Grace Hunter would later attend, I drove all the way to the end of Interstate 8. It put me in Ocean Beach. I had never been there before. Unlike today, when I was growing up Phoenicians didn't go to San Diego every summer by the tens of thousands. I had visited the city a total of one time before, staying at Hotel Circle in Mission Valley. I had no idea of this magical enclave called Ocean Beach.
But that day I had taken the freeway as far as it would go. After growing up in the desert and then spending several years completing my Ph.D. and teaching in the Midwest, it was as if I had landed in my own little paradise. Ocean Beach immediately felt like home. That evening I walked the 1,971 feet to the end of the municipal pier, turned around, and looked at the neighborhood as it rose up to the spine of the Point Loma Peninsula. The lights in the houses looked like Japanese lanterns and I made a vow out loud:
"I'll never leave."
A few hours before, I had rented my apartment a block-and-a-half from the ocean. I was neither a surfer nor much of a beach person. As a native Phoenician, the idea of tanning went along with the promise of ruined skin soon and melanoma later. But I loved O.B. The only thing that could pry me out was that I loved Patty more.
Patty.
I met her at the ugly main San Diego State University library. We both reached for the same book at the same time, Paul Fussell's The Great War and Modern Memory. She was an English professor and, with the Sharon Stone jaw line, classic Wayfarers, and lush wheat-yellow hair, you might mistake her for another shallow Southern California beauty. With the millionaire developer father and house in La Jolla, you might assume she was spoiled, too.
I never made that mistake. I judge a woman by the books she reaches for.
My life was so unfurnished when we met. I had a fairly new doctorate in history, boxes of books, and the old house in Phoenix that had belonged to my grandparents, now rented out. I happily let her help make me the man I became, in all good ways. She taught me how to open a Champagne bottle like a man of the world. Opened my ears to jazz.
Patty appreciated my love of history, ability to dress well, being "debonair," as she put it, for turning out well-balanced and kind, despite having lost my parents before I could even remember them. She called me a mensch, one of the best compliments I ever received.
It pleased her that I loved ethnic food and had a very dry sense of humor and possessed an eclectic past that included working for five years as a deputy sheriff trained by a tough older cop named Peralta. I had published my first book, Rocky Hard Times: The Great Depression in the Intermountain West, and it had been favorably reviewed. This also pleased her. We made a sophisticated, good-looking couple. But I knew I was marrying up.
She spoke French well. Not well enough to satisfy the most obnoxious waiter in Paris, but her French was better than my Spanish. Thanks to Patty, I learned fun and useful phrases: cherchez la femme, which proved to be true in cracking one cold case. Dragueur, a skirt chaser. Terribles simplificateurs: the world was full of those, Arizona especially. Billets-doux: love letters, the writing of which she excelled. La petit mort: orgasm. The vocabulary she had taught me was coming back now with the sea breeze.
It was things like this that made me cluelessly happy being with her.
I was one of the few who were allowed to call her Patty. To the rest of the world, she was Patricia. She teased me about spray-painting her name on a wall of I-5. For a long time, I wondered if we would have stayed together if I had committed that simple act of vandalism, decorating the concrete spaghetti with eight letters, leaving drivers to wonder what passion had stirred a man to do such a thing?
A man who would have done that could have kept up when she got on tenure track at the University of California at San Diego, an infinitely more prestigious appointment. He would not have been content being a good teacher, nor would he have bridled at the intentionally dull and social-science-y conventions of academic historiography.
He would have realized that even if I didn't feel in competition with her, she expected me to overachieve, as her father had demanded of her. The impetuous one with the spray paint would have done more than appreciate, support, and learn from her seemingly infinite avocations, from cooking to film history and painting. He would have tried harder to match her imaginative gift giving even though it couldn't be done.
That man sure as hell would have focused on publishing more so as to ensure tenure at second-rate San Diego State.
Who knows? I can argue this history one way and then the other. Participants don't usually make good historians. Even Churchill had his flaws. As for Patty, she was needy and broken, too. She was as insecure as I was. Our insecurities together acted as an accelerant to burn up our marriage. I taught her things of the world, too, made her happy for a time. The collapse of our marriage wasn't all my fault. Just mostly my fault.
I am too close to the events to recount them dispassionately.
I do know two things. One is that we married too soon. We weren't the people we would become. And I know a simple, transcendent fact...
She was the Glory Fuck of My Young Life.
Now I stood at the end of the same pier, the longest on the West Coast if I remembered correctly. A man fished off the south side and pairs of lovers strolled out toward me. My chest was tight and I could feel my heart trying to make its escape, my throat tightening. It was merely a panic attack. I knew that now. They never came in situations where a normal person would panic, only when I was quiet and alone. If I couldn't stop them, at least I could get away from other people so the attacks wouldn't cause me to do something inappropriate. Like tell the truth. Whatever.
I thought again about Patty. Contrary to Peralta's baiting, I wasn't afraid of seeing her. It would be nice, actually, to know she was happy.
As for my native prudence, that had gone away in the preceding months. Now I had barged into a stranger's apartment and assaulted a man with a move that could kill, and I wasn't even a cop anymore. Get me a can of spray paint.
I wondered if she remarried and had children.
Now it was hard to imagine that lost love as even real, especially after Lindsey.
I remembered the Fussell book Patty and I had both been reaching for. Writing about World War I, he meditated about how our age couldn't understand why hundreds of thousands of British soldiers had gone "over the top" to certain death from German machine gunners for something as abstract as honor. But for them, that sense of honor and obligation was as real as our age, drowning in illegitimacy and irony, is for us.
What a pity. Quel dommage.
I had brought Lindsey to O.B. exactly once, when we had first become a couple and I worried that I was falling for her too fast, this magical younger woman with the fair skin and nearly black hair. She had browsed the postcards and made fun of the tourists. The memories caused me to pull out my iPhone and text her:
"I'm in San Diego with Peralta, on a case."
It was a fool's errand. She wouldn't respond. I didn't say I loved her, even though I did. Why set myself up for the disappointment of her silence? She wasn't wearing her wedding ring now. I still wore mine, even though I operated heavy equipment: large-caliber firearms. I studied my ring and my hands that had changed the baby. I didn't even know the baby's name, but I remembered his tiny hands and arms struggling against me, struggling against a world of trouble.
This little soul who hadn't asked to be brought into that world. I didn't even know his name.
That tattooed kid who was his father had better be on his way to Riverside.
Lindsey had worried whether she would make a good mother.
Now this child's mother was dead. After meeting America's Finest Pimp and learning about Grace's venture as Scarlett, I wondered if the man in our office yesterday had been right to question the circumstances of her death. He hadn't said a word about Grace being a call girl. Had he not known? Hell, I didn't even know who he really was.
The pimp had mentioned a big man, an enforcer, someone he was afraid of enough to clear out and leave us alone. Was that the big man from yesterday, assassinated on Grand Avenue? And who was Edward, someone else the pimp feared?
Too damned many questions and barely twenty-four hours into our first case. I felt only my lack of ability. This was not what I had done as the Sheriff's Office Historian. It was no cold case but was uncomfortably warm. Maybe I should have chucked Robin's fancy that I be Peralta's partner and found some community college where I could teach.
The idea of coming to San Diego wasn't unpleasant because of Patty. It was bitter because San Diego represented my spectacular failures.
Looking up the hill at O.B., I remembered that I had broken my vow. I had left my little paradise.
The phone buzzed in my hand. The screen read: "Peralta."
I gave him an abbreviated report over the comforting noise of the surf. The beach wasn't crowded and the onshore flow was still keeping things soothingly cool.
"I went to Balboa Park," he said. "Really beautiful."
I agreed. It was a very un-Peralta like thing to do.
"It was where they held the 1915 Panama-California Exposition," he went on.
Yes, I knew that, but quietly noticed his uncharacteristic interest in something that didn't involve law enforcement.
"We're checked in to the Marriott on K Street. Know it?"
It was in the Gaslamp Quarter which had been built long after I had left, but I knew how to get there.
"Your key is at the front desk."
My own room. I wouldn't have to listen to him snore. He hung up before I could ask how his end of the investigation had gone.
"Mister?"
The small voice behind me went with a small, slender girl with long brown hair that looked as if it hadn't been washed in a week.
"Do you want a date?"
I told her I didn't.
"I'll suck your cock for twenty bucks."
She was jonesing from whatever she was addicted to, visibly shaking, looking like a drowned kitten. I asked her how old she was.
"Eighteen," she said. "I'll suck your cock for twenty bucks. I need to get something to eat. I know a place we can go."
She looked sixteen at the most, probably younger. I asked her if I could call a shelter for her, told her she didn't have to live on the streets. She asked if I was a cop.
"Not anymore."
"I'll suck you for fifteen."
I left her there and walked off the pier and up Newport Avenue to catch the bus back downtown. My heart decided to stay inside me, at least for a while.
The phone buzzed again. Lindsey had actually answered me.
Her text read, "Be careful, Dave."
San Diego had changed extensively since I had lived there, and, unlike Phoenix, mostly for the good. It was a major high-tech center now, not merely a tourist-and-Navy town. It had less population than Phoenix but surpassed it in almost any measure of quality. About the only thing that seemed the same was the mediocrity of the newspaper, formerly the San Diego Union-Tribune, now under new ownership with its name contracted to U-T. It sounded like a far campus of the University of Texas, but I'm sure a consultant charged big bucks for a new "brand."
Downtown, thrown away in the 1960s and 1970s, had made a stunning comeback, including the Gaslamp Quarter with its lovingly restored historic buildings and Horton Plaza urban mall. Nobody would know it used to be skid row. Walking to the Marriott, I was struck for the gazillionth time how Anglo the city seemed, even though it sat right on the Mexican border. The barrios south and east of downtown had been carefully tucked away and so it remained.
I showed my driver's license at the front desk and got my key card to a room on the eighth floor. Before going up, I went into the business center and booted up the computer. I am a lifelong Mac user and couldn't understand why anyone would use Windows. So I waited, and waited.
Then I plugged in the flash drive and clicked on the icon.
A window popped up and the screen went blank. Then Grace Hunter was talking to me.
"Hi, babe. I bet you'd like to know what's on this drive. But if you don't have the code, too bad."
A white box appeared and I had nothing to enter. The screen went dark again. But for a few seconds she had been alive. I could see her allure with her wide smile, the elegant movement to push her hair out of her face, the sexy taunt in her voice. I popped out the drive and stuck it in my pocket.
When I stepped out of the elevator, a woman was walking toward me: black, shoulder-length hair, attractive if older, elegantly dressed. As she came closer, I was sure I was wrong. I saw plenty of ghosts in my dreams.
But, no...
"Sharon?"
"David!"
She ran to me and gave me a long hug.
Her face was flushed and, up close, her usually perfect hair was mussed.
All I could do was sputter words. "What? Why?"
She grinned at my discomfort.
"What's wrong?"
Where to begin? She was Peralta's ex-wife. She had moved away to San Francisco in as final a breakup as I could imagine. I had known both of them for most of my adult life. And here she was, having obviously been in his room. But it was none of those things. I felt the embarrassment of nearly coming across my parents having sex.
"It's all right, David." She laughed that full-out laugh that always put me at ease. She studied me. "You've lost weight."
Her eyes held concern rather than a compliment. I knew the suit was now almost hanging on me.
I said, "So you're why he went to Balboa Park. I thought something was odd."
"Maybe he can grow a little after all," she said. "I was down here for a conference, so..."
So, indeed.
She hugged me again, made me promise we would get together for drinks or coffee before we left, and disappeared into the elevator.
After a minute to collect myself, I knocked on his door. He greeted me in a bathrobe.
"Why are you blushing?" he demanded.
"I got too much sun at the beach."
"Why is your shirt and tie a mess?"
"A baby peed on me, okay? You change and I'll come back."
"I'm fine," he said and walked inside, leaving the door open. I reluctantly followed him.
He plopped down on the unmade bed. I sat on a sofa and filled him in on Tim Lewis, the baby, and Grace Hunter's small business. He closed his eyes and grunted after every few sentences, taking it in as he always did. He offered no more reaction when I showed him the flash drive. We would have to find someone to break the code.
The room was too warm for my suit.
I wrapped it up. "Tim Lewis has parents in Riverside. I told him to take the baby and go there today."
"Did you get their address?"
"Yes." I said it a bit too testily.
"What's wrong?" His Mister Innocent voice. Then, "Look next to you, on the desk. It's the entire case file on the girl's suicide."
I swiveled to see several thick folders bound with a large red rubber band.
"Man, you have the pull," I said. "How is Kimbrough doing?"
"He's happy." He slurped on a Diet Coke. "I'd like to say it was my pull, but remember that suicide in Coronado? The girlfriend of the millionaire from north Scottsdale who allegedly hanged herself?"
I remembered. It had happened at the Spreckles Mansion in the rich, idyllic town that sat on a spit across from San Diego. The rich guy had purchased the iconic house. As I recalled, he made his money from acne products and cosmetics. The girlfriend, young enough to be his daughter of course, had been alone when his young son had tripped and fallen over a balustrade in the mansion. The child had died.
The next day the girlfriend had been found hanging from a second-story balcony, naked, a cloth in her mouth, and her hands bound with rope. As with Grace, the authorities had pronounced it a suicide.
Peralta shook his head. "I can see your mind making connections, Mapstone. They're not there. It has nothing to do with our case. Bill Gross is a good friend of mine." That would be the San Diego County Sheriff. "His department was called in because Coronado PD doesn't have the expertise for a complex death investigation. The media put Bill through hell on this one. News choppers overhead got pictures of the body and pretty soon it was on the Internet. Everybody became an amateur sleuth. They even got Dr. Phil involved."
He shook his head. "But the woman in Coronado really did kill herself based on the evidence. Hell, the sheriff's department even put up a special page with the information on their Web site. Kimbrough said his chief didn't want Grace Hunter to turn into another media circus. So we lucked out and have copies of everything."
"So what about our young woman?" I asked. "Suicide?"
"You'll have time for light reading." He pointed at the stack of case files, in case I had forgotten. "The short answer is they believe it was a suicide."
"What do you believe?"
He shrugged the big shoulders. "I'll wait for your report. Kimbrough brought along the night detective who was the first to respond to the call."
"Night detective?"
A quarter of one side of his mouth attempted a smile. "I'm showing my age, Mapstone."
I looked at the rumpled sheets and doubted that.
He continued, "Departments used to have night detective bureaus to cover the late shift, so the investigation into a major crime could begin immediately. Now it's almost all in-house with each unit, so, for instance, homicide has its own people on call. That's the case here. I was using old-time cop talk. Did I ever tell you about the night detective I met when Miranda bought it?"
He was being so uncharacteristically loquacious, and actually talking about himself, that I stifled my impatience.
"It was 1976, and Miranda was out of prison. He actually went around signing Miranda warning cards. Somewhere I have one he signed for me. Anyway, I was a green deputy and was serving a warrant down in the Deuce. The old La Amapola bar. Means 'little poppy.' I must have gotten there the second after Miranda got in a fight and was stabbed. People were scattering. The first PPD unit was a night detective. This tall guy named Cal. They called him the Red Dude on account of his hair. He marched my ass out in a hurry. We became friends later. Never did find the suspect I was trying to arrest."
If I had my geography right, the bar where Ernesto Miranda died was located where the Phoenix Suns arena now stood. Mike Peralta, historian. It made me wish he would talk more about his past, but we had business and he moved right along. I tried to imagine a time when he had ever been a rookie and uncertain of himself.
Night detective. It had a nice ring.
"Anyway, I talked to the detective. You would have liked her. First name Isabel. Cute little chica. Make you forget about Patty."
"Will you stop that shit!" I pulled off my suit jacket and threw it on the floor. It would have to go to the cleaners anyway.
His eyes followed the garment's flight, then fixed his gaze on me again. "Grace's body was found on the concrete by the pool. It was a straight fall and she landed on her head. Massive trauma, loads of blood. She was handcuffed from the back, nude, and no real note was left, like our guy said in the office yesterday."
"What do you mean 'real note.' "
"I want you to read the reports. Hang with me and I'll give you the overall run-down. So the uniforms that initially respond go upstairs and the door to the condo was locked. The manager lets them inside."
He folded one brawny brown calf over the other and told me the cops found no sign of a break-in. The lock was a deadbolt, so nobody could simply close the door behind them and cause it to automatically lock. It had to be secured from the inside, as if Grace had done it, or from the outside with a key. The only ones with keys were Zisman and his wife. She wasn't in San Diego on the twenty-second. There were no signs of struggle. Grace's purse was there with a hundred dollars in it, her keys, and a brand-new cell phone.
I said, "The handcuffs didn't arouse suspicion?"
"Sure. But sometimes people who want to kill themselves bind their hands so they can't change their minds. I've seen those calls in Phoenix. That was the case with that girl in Coronado, although she used rope and not cuffs. SDPD thinks the same was true here. Kimbrough had Isabel demonstrate how a person could do it. Then walk to a balcony and go over."
"Where'd Grace get the cuffs?"
"Apparently the former quarterback likes bondage. They used them during their playtime."
I tried to ignore his bulk in a bathrobe lying in a bed where he had had some "playtime" of his own. This was something I did not want to visualize or even contemplate.
"Does he own this condo?" I asked.
"He did. It's for sale now. He was away at his boat when Grace killed herself and the alibi's good. The owner at the slip next door saw him there during the time of the suicide. Zisman told the cops she was his girlfriend and she'd been feeling depressed, but he had no idea she might do something like this, yada-yada-yada."
"And they believed him?"
"Zisman is a reserve police officer in Phoenix," Peralta continued. "He showed his badge and identification. That might have bought him a little professional courtesy the night Grace died. He cooperated fully. I'm sure he was scared shitless this would make the papers or television and the missus back in Arizona would find out."
I told him newspapers usually didn't report suicides out of concern that there might be copycats. Grace had died at night, with no television news choppers in the air.
"So Zisman walked?"
He nodded. "There was no evidence of his involvement. No probable cause to hold him, much less get an indictment. If they arrested every Arizonan who had a mistress stashed in San Diego, they'd have to build a new jail."
It was nearly five but Peralta wanted to go out again. He had scheduled a meeting with a real-estate agent to see the condo.
I changed into casual clothes, a light-blue shirt and cargo shorts. The Python was too big to carry, which was why I had invested in a Smith & Wesson 340PD Airlite, an eleven-ounce, snubnosed .357 magnum that slipped easily into the right-side pocket of the shorts. I stashed the Glock that I had confiscated from America's Finest Pimp in a drawer. Who knew how many unsolved shootings or homicides it was connected to? I would deal with that later. Peralta was out of the robe, thank goodness, and in tan slacks, dress shirt, and blue blazer.
We walked ten blocks down Broadway toward the waterfront. The condo was hard to miss: more than forty stories, right across from the beautifully restored railroad station, with its blue Santa Fe railroad sign on the roof. In the lobby was a watchful concierge and, sitting on the edge of a chair with perfect posture, an auburn-haired, middle-aged woman who exuded perkiness. The Realtor. We made introductions and she took us up the elevator to the nineteenth floor.
We must have looked like the oddest gay couple she had ever dealt with.
"I have so many clients from Phoenix," she chirped. "This is the place to be."
The deadbolt turned with decisive effort and opened onto an empty living room. The condo hadn't been staged for the sale. What most stood out was the handsome hardwood floor. And then the view, of course. Asking price: $599,000.
I let her walk Peralta through the rooms and wandered off by myself to the balcony. It was amusing to hear her calling him "Mike" in nearly every sentence. Nobody but Sharon called him Mike. But he was as convivial as could be, a skill he had learned over the years while wooing voters. Not that he had needed to put on a front. His record as sheriff was spotless, with crime down, jail conditions excellent, response times across the county top-notch, and his history professor solving high-profile old cases. All that didn't matter when his opponent ran against him claiming he was soft on illegal aliens. I pushed that out of my mind, opened the glass door, and stepped outside.
The view of downtown and the harbor was not as stunning as you could get for one or two million bucks on the upper floors, but it would do. If you had the money to escape the summer hell and dust storms of Phoenix, San Diego would be about as close to heaven as you could get.
The sun had burned off all the clouds and was now angled to throw the city into enchanting relief. The water was flawlessly blue and full of pleasure boats, which were dwarfed by the carrier at its mooring on North Island. The Navy kept the Nimitz-class carriers there because they wouldn't fit under the bridge that connected San Diego to Coronado, even though it soared 1,880 feet, a blue arch, across the channel that led to the Pacific Fleet's base.
I drove that bridge many times but was glad not to be going over it this trip. I was glad not to make connections between Grace Hunter and the suicide at the Spreckles Mansion. As I got older, I didn't like heights, didn't like bridges. I didn't like being on this balcony with the restless wind, distorted and accelerated by the other skyscrapers, flapping against my shirt. San Diego didn't really get earthquakes. A small fault line ran through Rose Canyon east of La Jolla, but otherwise it was pretty safe. That made me happy, nineteen stories over downtown.
At the edge, I looked down on the pool. A party was going on and the people looked very small. As I recalled, a body fell at thirty-two-feet-per-second, accelerating as it went down. It was a long damned time to contemplate death, to wonder if you'd made a big mistake.
What desolation must this young woman have felt to want to kill herself, sure that the terror of the fall and the pain of impact would be brief, and then nothing, comforting oblivion. If that was what really awaited us. Who really knew? I reached under my shirt and ran my finger along my totem, Robin's cross.
"Hey, babe..." The video of Grace on the flash drive was vivid in my mind. The confident, teasing voice and smile. It fit perfectly with Tim's description of a young woman who started her own business, however illicit, and was the consort of men who would pay thousands for her company. Would that same woman commit suicide?
I stared down at the concrete for a good five minutes.
The railing was at my belly button, but I was about ten inches taller than Grace Hunter. If I were suicidal and athletic, I was tall enough to hike one foot to the top of the railing and launch myself off. No fuss, no time for second thoughts. Grace couldn't do that. Based on the description of a five-foot-four woman, her legs weren't long enough. Handcuffed, she would have had to do a bit of a gymnastics move to go head first. Or maybe she hopped up on the railing backwards and pushed out into the sea-kissed air.
I was gripping the railing so hard my hands started to hurt. Making myself stop, I ran them along the smooth metal. The balcony was secluded, so nobody from an adjoining unit could see what was happening there. Other condos, offices, and hotels were too distant to give a detailed view, so witnesses were unlikely, especially after dark. I'm sure the cops had checked that out.
Such a lovely place to stand. How could you look out on this city and see anything but pleasure and hope? I knew better.
"Et in Arcadia ego..."
The Latin phrase came into my mind. "Even in Arcadia, I, death, hold sway."
If she didn't kill herself, who did? Not America's Finest Pimp: he was searching for her, didn't know she was dead. Zisman? It still couldn't be ruled out. Alibis can fall apart with a little push. I wondered about this Edward that AFP had mentioned with dread. I wondered more why Grace, safe with Tim in Ocean Beach, with a new baby and seemingly much to live for, had gone to see Zisman.
"David, I see you like the view!" the agent chirped behind me. Her voice gave me a start. "Oh, I'm sorry!"
"David is a little jumpy," Peralta said behind her. Two beats later, he asked, "Was this the condo where that girl fell from?"
She quickly herded us back into the living room. "Yes, it was a terrible thing. A suicide. Young people have such a hard time..."
After a few minutes more, she loaded us with marketing materials and we left, walking in silence. Peralta wanted to eat at the Grant Grill in the restored U.S. Grant Hotel, so we waited in the bar, me with a martini, him with a Budweiser, surrounded by tourists. Only three people came up to say hello to him and say how much they wished he were still sheriff. They meant well. It made me angry.
After two more martinis and a fabulous supper, I felt better. Peralta and I went back over what we knew as we ate. He wanted to visit Grace's parents. I wanted to check out the list of regular clients. It had only taken me a day to go from not wanting to take this case to full buy-in. I had even landed another client. Was this Peralta's usual ability to rope me in, or had I done it myself? Better to follow this case than to sit at home alone with only my thoughts, memories, and regrets. My mind was a bad neighborhood. I didn't want to wander around there alone.
"If we visit her parents, maybe they'll agree to become our clients," he said, polishing off the king salmon. I wondered if he was joking. It sounded a bit like a used-car salesman on the make.
I told him about Tim Lewis hiring us and said we didn't have to worry about having a real, live client. His expression was unreadable, but I didn't think he was happy about my effort at business development. He was not worried about spending our dead client Felix's money at this posh restaurant, however.
I used the silence to fold and refold my napkin. I reached in my pocket and slid out my iPhone, slid it back. Then: "So you don't think it's a suicide?"
"I want you to read the reports and give me your opinion, Mapstone. But, based on what you've told me, what she was into, and the cops didn't know about it..." His voice trailed off, his meaning obvious. He ate and chewed, thinking.
He said, "I don't know why SDPD wouldn't have had Grace in its computer when her boyfriend filed the missing person's report. Maybe a lag. Maybe a system glitch."
"Maybe somebody paid off."
He poked his fork at me. "Why do you keep checking your phone? If you want to call Lindsey, call her."
"Like you called Sharon?"
He smiled slyly. A rare, actual smile.
But my phone-checking wasn't about Lindsey, to the extent that anything I did wasn't about Lindsey.
"It's past nine now," I said. "Grace's boyfriend ought to be in Riverside. He ought to have been there hours ago, even with the worst traffic jam in California. I told him to call me, and I've heard nothing."
He stared past me in thought.
"Maybe a careless kid. He's there and safe."
"At first he was afraid I was going to kill him," I said. "I don't think he would space this."
I told him I wanted to go back to O.B. and check.
"Want me to go with you?"
I told him no. "Sauve qui peut." Every man for himself.
"Why are you speaking French, Mapstone?"
I smiled. "Memories." To be a show-off, I added: "Pourquoi pas?" Why not?
"Bonne chance," said the simple boy from the barrio.
With that, I walked out front where I gave the U.S. Grant Hotel doorman five bucks to hail me a cab.
The cab let me out in front of the apartment building at a quarter of ten. All the street parking was taken, probably all the way down to the business district, if not beyond. Your own parking space was a precious thing in O.B. I stood there as a black Dodge Ram truck slid by on Santa Cruz. The truck had a tag frame that read, "I (heart) Rancho Bernardo."
I shook my head. "Good luck finding a parking spot this time of night, suburban boy."
Then I was alone. When I lived here, O.B. had been dimly lit by yellow streetlights, a program the city had begun to cut the light pollution and protect the Palomar Observatory. Now the streetlights looked new and were definitely brighter, reflecting off the gray ceiling of the returning clouds. It was probably bad for the astronomers but good for me. I could see that the sidewalks were deserted, a good thing because I felt itchy with anxiety.
With all the windows open, I could hear televisions, a couple making love, and the subtle resonance of the surf a block and a half west. It brought back memories of the rare nights when there was fog and I would hear the foghorn coming from down by the pier. Tonight, it was so still I could hear my steps on the concrete.
It was ten degrees cooler than downtown. For a few minutes, I let the temperature help me feel normal again instead of breathless from the Phoenix heat. Then I walked to the gate and stared up at the apartment. The windows were closed, curtains open, and lights off. The tension that had been swelling for hours in my middle relaxed. The kid was gone and had forgotten to call me. He was mourning. He had a baby to take care of.
I thought about walking down to Newport and taking the bus downtown, but it was better to be sure. The vocal passion coming from the southeast apartment had subsided, so the gate loudly protested against me pulling 1950s metal hinges against each other. It put me on guard, but no curtains parted to see who was coming in. The pool was deserted and the water sat perfectly still and inviting.
When I looked up this time, I could see Tim's door was partly ajar.
The dread wouldn't let me go. Sure, there was a chance he was sitting inside, enjoying the breeze through the cracked door, playing a video game on headphones while the baby slept.
But only a fool would believe that.
I took the stairs two at a time, careful to keep my footfall quiet.
By the time I reached my old unit, I had the lightweight Smith & Wesson in my right hand. The windows to Tim's apartment were on the far side of the door so I couldn't see what was inside the apartment. I tapped lightly on the hollow door and called Tim's name. The door was open three inches. Beyond was darkness. Now it didn't seem like such a good idea to have come alone. Second-nature almost got the better of me: I almost called, "Deputy sheriff!" but I pushed the door all the way open and stepped silently into the room.
I moved to the side, to avoid providing a backlit target.
The outside light streamed in through the windows. Tim was sitting upright in the dining chair I had used earlier that day. It was directly facing the door. His face was tombstone white and the blood from his slit throat had flooded his T-shirt. There was no point in checking for a pulse. His dead eyes stared at nothing and his hands were in his lap, bound with handcuffs that glinted from the ambient light. Something like a big, curved bar of soap was in his lap. It had probably come from the kitchen.
Tim was gagged with a dish towel wrapped with duct tape. My eyes were drawn to his hands. Every one of his fingers had been broken. They had tortured him before they slit his throat.
The killer had also tossed the place. Clothing, food, video games, books, cushions and the flotsam of daily life were strewn around. Every drawer had been pulled all the way out and turned over, in case something had been taped beneath it. The pillows had been slit open and their stuffing pulled out.
Why didn't you leave when I told you? I forced back that thought. Right now, I had to secure the scene and observe, even if I wasn't the law anymore.
There was enough blood to do finger painting on the south wall of the living room. The red characters were uneven and drippy, but the words were familiar.
PERALTA AND MAPSTONE, P.C.
PRIVATE INVESTIGATORS
The moron had left his fingerprints in Tim's blood.
A closer look would have to wait.
I hurried into the bedroom and swept it with the barrel of the revolver, fearing what I would find. No bad guys. And no baby in the crib. I quickly checked the bathroom and the closet. No baby. I felt my own pulse slamming against my temples. The bedroom had also been thoroughly gone through. Whoever had done it, and taken the baby, hadn't bothered with the baby supplies.
I pulled out my phone to call Peralta and then the police.
Then something clicked in my brain. I dropped the phone back in my pocket, barely feeling my hand.
The object in Tim's lap was not a bar of soap. And it had lettering that suddenly opened a file in my vast memory of trivia.
The lettering carved into the object said,
FRONT
TOWARD ENEMY.
"Oh, God."
I heard a voice say those words. It was my voice, but my mind was desperately processing my options. I don't know if I made a conscious decision because my next memory is reaching the walkway outside the door and letting adrenaline heft my right foot to the top of the railing, balancing myself with my left hand. Then I was midair headed down for the pool.
Hoping that I remembered which side was the deep end.
The smooth surface came up suddenly and next I was underwater, surprised by the liquid cold, my terror-filled muscles acting in concert with only one goal: dive deep. I touched the bottom and started counting but only got to three before feeling a sharp concussion overhead. It popped my ears and pushed me violently against the far wall of the pool. I swear my brain felt about to burst. Something large and heavy missed my head by no more than six inches. It was half of a cinderblock.
When I came up, gasping for air, Tim's apartment was gone and the smoke made it difficult to assess the damage to other units. The surface of the water was coated with glass fragments, burning drapes, a can of Pringles, papers, the debris of daily life—and little metal balls. Those had been ejected from the Claymore anti-personnel mine that had detonated. Robin's cross floated on the surface, glinting under the light, still attached to the chain around my neck. The revolver was still in my hand.
Something soft bobbed against my arm.
It was Tim's head, face up, hair like seaweed, staring at the overcast.
We started back to Phoenix at dusk the next day, driving through the desert at night the way people used to do, before advanced automobile cooling systems. Back in the days when only a fool would cross the wilderness without an adequate supply of water.
Before we left, Peralta found a deserted space where he could park and get into the steel storage compartment that sat in the extended cab behind our seats.
It was a gun case.
"Time for heavy metal," he said, and I didn't think he was about to break out some Black Sabbath CDs.
Ten minutes later we were speeding east on I-8. I had received a tutorial on a Kel-Tec RFB assault rifle, "a bull pup," he called it. Barely more than two feet long, it was black and homely. But with the fire-selector capable of semiautomatic and a twenty-round box magazine, it didn't need to win a beauty contest. I slid it beside me, barrel down, safety on. Peralta slid an assault rifle into the well between his seat and the door. It looked a little like an M-16, but it was matte black with a retracting stock and a rough-edged thing on the barrel that might have been a flash-suppressor or a hand-guard—or not. He didn't bother to explain besides telling me it was a Colt AR15 Magpul Special.
"A good truck gun," he said.
My world was still a little blurry from the blast. My stupid question: "Why?"
"I want to have an edge," he said. "Are you steady enough for this?"
"Yes."
The question irritated me, but I had no time for that. I had no time for sentimental thoughts about departing from my second hometown as we climbed out of Mission Valley into El Cajon and began the long uphill grind—away from Ocean Beach, away from my other life in this beautiful city and its balm of cooler weather. I opened the glove box, pulled out the gun-cleaning kit, unloaded my Airlite, and began cleaning and oiling it to avoid any trouble from its contact with the pool. My hands shook.
"Sorry you didn't have any time for fun here," Peralta said, trying his best to sound sympathetic. "I should have at least set up drinks for you and Isabel, the night detective. To talk over what she found. Anyway, she was cute."
"You're trying to set me up? You're the one who keeps saying Lindsey will come back."
"You need to get laid, Mapstone. It'd do you a world of good."
"Like it did you." I heard my voice, joyless and raw.
Grace, Isabel the detective, Grace's friend Addison. Oh, I felt old and in a foreign country. The young women's names sounded either like they belonged to old ladies or unfeminine and strange. I shouldn't have been so judgmental.
But I was particular in my female names. I liked boomer names like Susan, Amy, and Karen. Pamela: three syllables of sexy. Lisa and Linda were nice. And Patty. I had preferences for Generation X names, too. Heather and Melissa. And Lindsey. And Robin. Addison? No. Leave it to me to start categorizing and analyzing even small things. Maybe it was a good sign. Or maybe I was leaking blood inside my brain from effects of the explosion.
I wanted to take a nap. But then the dreams would come.
This was the first time we had spent alone together since the blast and I briefed Peralta as much as I could. My head hurt despite nearly overdosing on Advil, everything felt slowed down, and concentration was difficult. My shoes, the only casual pair I brought, were still soggy. The one constant thought I could hold was the missing baby.
I did my best to brief him.
He immediately interrupted. "You're one lucky bastard. The kill zone of a Claymore can be fifty meters. It's a shaped charge, meant to explode in the direction that it's pointed. You might have been better off running to the bedroom and getting under the bed. That way you wouldn't have been directly in front of it."
"Trust me, there wasn't anything left of the bedroom, and there was no bed frame." I started to zone out a little. "Hell, I don't know. I reacted with instinct. How did they detonate it?"
"Could have been anything nowadays: timer, laser, plus the good old fashioned wires." One big hand was enough to handle the steering wheel. "We used to set up Claymores to ambush NVA columns. They'd come down a jungle trail and we'd let the gooks get well inside the kill zone. Then we'd set off one at the front of the column and they'd naturally run backwards. That's when we'd set off the Claymores from the back, going forward." He laughed malignly.
"Sounds like fun."
"You don't know. You weren't there." He said this without irony.
"Thank you for your service to the country, sir. Now, may I fucking continue?"
"Sure," he said. "But how did you realize it was a Claymore?"
"I read about it in a book."
When my eyes were closed, I started to get dizzy. When I opened them, the car lights from the freeway hurt. Looking off to the shoulder, I was overcome by the fear someone would suddenly step in front of us. So I stared into my lap.
After the explosion, I pulled myself from the pool. My cell phone was ruined, of course. But my gun was fine. It wasn't needed. No bad guys were there to finish the job. Instead, people were shouting and screaming. I went from apartment to apartment, getting people out, sending them to the street until the fire department could arrive. That seemed to take forever. One man living in my old unit looked badly injured. I found him last, under the remains of a heavy desk that probably saved him, and I stayed with him until the first cop came in the door with a flashlight and a gun.
It was a miracle that the damage wasn't worse. One person in critical condition, two more suffered less-serious injuries. It helped that the people directly below Tim's apartment were gone; the same with the residents of the unit directly to the south. No fire followed the explosion and the emergency crews quickly shut off the gas.
I remembered choppers overhead and a bright beam from the sky.
Then, after a cursory checkup by the paramedics, it was all cops, all the time. I never got a chance to have coffee with Sharon. Nor did I have time to order a new cell phone. Instead, I spent the hours telling my story to seven different San Diego cops, including Kimbrough, who was not at all happy to see me. Then ATF showed up and took me downtown to talk more. What sleep I got came from leaning my head against a wall while waiting for the next round of questions.
I was fortunate for a law passed after 9/11, giving retired police officers in good standing the power to carry a concealed firearm in any state. Otherwise, things could have gotten very disagreeable. Somehow Peralta had pulled some levers before he left office and I was able to "retire" with a combined fifteen years service to the Sheriff's Office. The pension was shit, so don't judge me as a greedy public employee. But the conceal-carry benefit probably kept me out of jail.
The cops and feds didn't think I did it—"it" being called a "possible act of domestic terrorism" on the television crawler I saw while waiting in one of the fed's offices. But they didn't like that I was in San Diego as a private investigator and that my client was dead. I wondered if they'd force us to stay in town for further questions. Instead, it was a wonder that we weren't escorted to the city limits. I thought momentarily of my unread George Kennan biography and how he had been declared persona non grata by Stalin, his ambassadorship to Moscow cut short. I was persona non grata in San Diego at the moment and for better reason.
An Amber Alert was issued for the missing baby. Detectives had called Tim's parents in Riverside and assembled more information: a photograph of the now-orphaned infant and his name.
His name was David.
"I should have gone with you," Peralta said.
"You couldn't have moved as fast as I did."
"I wouldn't have mistaken a Claymore for a big bar of soap."
He had me there. I went on and tried to tell him everything, step by step.
"Did you tell them about the pimp?"
I said yes.
"Did they believe you?"
"They did when I gave them the Glock I took off him."
I had no doubt that America's Finest Pimp was now sitting in one of America's Finest Interrogation Rooms, but I didn't make him for the killer. He had been too unnerved by my arrival and my assumed connection to the unnerving Edward to return to the apartment. Anyway, the pimp didn't strike me as the throat-slashing kind and certainly not as a bomb maker. But I didn't even know his name. The cops told me nothing. There was no professional courtesy to give to a private investigator.
When the de-brief had exhausted me, I asked Peralta a question. Did it pass the smell test? The rich guy leaving a thousand dollars on the nightstand for Grace, and then her setting up a business based on that kind of sum? Not a twenty-five-dollar blowjob from a hooker on Van Buren, but hundreds, even thousands of dollars.
"Sex is big business," he said. "Don't forget Eliot Spitzer. Didn't he pay four or five grand every time? I've seen plenty of investigations into high-end prostitution. We took down a county supervisor while you were away teaching, for putting hookers on his county credit card. The single-girl-on-her-own part of it is unusual, but she eventually got caught by a pimp. That sounds real."
I put away the gun-cleaning kit, reloaded my revolver, and slid it back into my pocket.
"If you'd gotten gun oil on the carpet, I would have killed you," he said.
I ignored him. "Why would a man pay for sex, especially when there's so much free stuff around? Especially why would a rich man do it?"
"Tiger Woods spent something like four million bucks a year on prostitutes."
"Your mind is an amazing thing," I said, repeating a phrase he usually applied to me. Having my brain rocked like a Jell-O salad had addled my mind at the moment.
His big shoulders shrugged. "What can I say? I'm a golfer."
"Do you spend four million...? Never mind." I really did not want to know.
Even in my driest spell, in my twenties when young women weren't drawn to a guy who read books and talked about history, I didn't contemplate going to a prostitute.
"Sharon could tell you the psychology," he said. "With a young woman and older man, it's called the Lolita Complex, I think. Some men are drawn specifically to prostitutes. Rich men want the privacy that the right prostitute can provide. Most of these guys are married, remember, and they don't want their wives to divorce them and take half of their wealth in a community property state. Politicians are willing to take the risk. A prostitute never says no, never has a headache, and she'll do kinky stuff the missus might not do."
"And it's a huge human trafficking problem."
"That, too."
Back in El Centro and the heat, we went through the Wendy's drive-thru and pulled to an empty part of the parking lot to eat.
"So," Peralta said, "what didn't you tell the police?"
He had parked the truck so we could see anybody coming into the lot and escape through two different driveways. His caution was good.
"Fuck!"
My concussed brain coughed up something essential.
"I forgot the flash drive. I forgot to give them the flash drive."
Peralta was silent.
"I've got to get it to them."
"Anything else?"
Yes, there was. I unpacked another chamber of my addled brain and told him about the writing on the wall: our names written in blood. Of course this critical piece of evidence didn't survive the blast.
He paused mid-bite. "How would the suspect know about us?"
"I gave Tim our card."
Peralta was silent and it was a long time, for him at least, before he resumed eating. About fifty seconds.
Many things about this case were unknown, but one was becoming clearer. The killers weren't only after our clients. They might be after us. I stroked the ugly little rifle beside me, glad that Peralta was into this kind of heavy metal.
"What should I do about the flash drive?"
"Keep it," he said. "Let's see who's on it."
We finished our meal and stopped at a truck stop, where I bought a cheap cell phone to get me by until I could order an iPhone. Then we returned to the Interstate, one of America's great accomplishments of the past century. Today the nation refused to do great things but that didn't keep people from crowing about our "exceptionalism." I had bigger problems than the fate of nations, but I let Peralta mind the rearview mirror.
Who knew how many killers roamed the anonymous Interstates of America tonight? How many truck-stop prostitutes would disappear tonight, meeting terrifying deaths, mourned by none? Except for the infrequent tractor-trailer rig, I-8 was mostly empty and carbon dark, as though the moonless night was trying to steal the beams our headlights threw ahead. Above was a vault of stars that most urban humans rarely saw in person. In my grandparents' generation, it had merely been the night sky. Against it, my problems seemed very small. We were only here for nanoseconds of cosmic time. Inside the cab of Peralta's super-truck, there was no song of the wind or moan of the engine, no sense that our onrushing feet rested only a few inches above the pitiless land.
Back in Phoenix, our office was in what passed for perfect shape. Every tube on the neon sign out front was operating flawlessly. The house on Cypress appeared safe, too. Even the air was better, the smoke from the forest fires clearing out while we were gone. Nobody had left a message on the answering machine. A neighbor had neatly stacked the newspapers beside the front step. Only the New York Times was on my daily routine now.
I couldn't stand to read the Arizona Republic any more, the stories about the antics of the new sheriff and the other buffoons that had taken over state politics. I didn't like the way the writers referred to the place as "the Valley," using the touristy Valley of the Sun, not even the geographic Salt River Valley. Here we had one of the most magical city names in the world: Phoenix. And yet the suburbanites insisted on "the Valley." Silicon Valley? The Red River Valley? Shenandoah Valley? And these were the same people who moved from suburban Chicago but said they were "from Chicago." It drove me nuts. The local papers went straight to recycling.
Then I unpacked the flash drive and plugged it into my Mac laptop to see Grace tease me again. The ghost in the machine.
Lindsey could get into the drive but Lindsey was gone.
In the living room, I laid it behind a volume of Will and Ariel Durant's The Story of Civilization on the top bookshelf by the staircase. It wouldn't survive an extensive search of the house, but this dusty spot would do for now.
In a few months, I had gone from a deputy sheriff with a clean record to a civilian, a "private dick," as Robin teased me with her delightful lascivious smile, concealing evidence. The top of the book held a sheen of dust. I didn't blow it off. This had been part of my grandparents' library passed on to me. When I was gone, it would be broken up in an estate sale or tossed in the dump.
After lying awake a long time, I slept badly with two guns to keep me company. Many dreams interrupted my sleep but the details were gone after I opened my eyes. If Tim and Grace had shown up as new dramatis personae, I couldn't recall. Robin was there. I couldn't remember what she said. I got up in the night to check the Amber alert and the San Diego media Web sites several times. Nothing was new.
By half past seven Sunday morning, a hitherto ungodly hour for me, I opened the automatic gate and pulled into the office, then shut it behind me. The high temperature was only supposed to be in the nineties today, the old normal for May when the dry heat was bearable and even pleasant in the shade. At this hour, the air was cool. No bad guys were waiting inside, merely a stale odor and the same old furniture. I dropped my briefcase on the floor and my Panama hat on my desk, crown down, and flopped onto the sofa to drink my mocha and eat a bagel. Remembering Sharon's reaction to my gaunt appearance, I tried to make a commitment to eating more regularly.
Peralta arrived fifteen minutes later wearing a Stetson and jeans. He peered at me over his sunglasses, surprised that I had beaten him into work.
"How ya feeling?" He tossed the cowboy hat on his desk, letting it fall where it landed.
I told him San Diego had been a blast. He didn't smile, disappearing into the Danger Room to either bring out more weapons or admire his prizes or whatever he did in there. How was I? I hurt like hell and the tension inside me was thrumming like a tuning fork. Otherwise, I was great.
When he returned, he leaned against the doorjamb, all six-feet-five of him. Maybe half of a supermodel could have squeezed through the remaining space.
"I'd like to bring Sharon into our practice. Is that all right with you? What the hell are you smiling at?"
That last part was more like it. I wasn't accustomed to Peralta being solicitous of my opinion. In the old days, he barked orders and made demands, alternating between the "good" Peralta who was a natural leader and inspiring peace officer, and the "bad" Peralta, who could be manipulative, micromanaging, and Vesuvius when he didn't get what he wanted.
In my office on the fourth floor of the old courthouse, I had been somewhat insulated from the worst of his personality. Getting laid had obviously done him a world of good. And his term "our practice" sounded both professional and ironically on target. We were definitely practicing. I told him none of this. Why was I smiling?
"You," I said. "Of course, great if Sharon joins us. I love Sharon. Why would she want to work with us?"
"We need her expertise. She's been consulting for San Francisco PD, you know."
I didn't. I knew she had moved there to be closer to their grown daughters. She had stopped her popular radio show and quit writing the best-selling self-help books that had made her a wealthy woman.
"So you don't mind?"
"Of course not."
"We can put Lindsey on the payroll when she comes back, too."
That should have made me smile. We had no payroll besides the ten grand from Client No. 1 and Tim Lewis' five hundred. Outside of business cards, our practice was only getting started. But I didn't smile or answer directly. Lindsey wasn't coming back, except to get her things and move away permanently to be with her lover or lovers to come.
"Are you and Sharon getting back together?"
He evaded.
"Now I want you to think about this, Mapstone. Every police agency in Southern California is looking for that baby. It's a big deal and we're going to get in the way. The feds are investigating the explosion, who got his hands on a Claymore, and if we get in their way, we could compromise an undercover operation."
"We have other strands we can follow," I said. "Grace's friend and parents. Her list of johns. Tim's parents. Larry Zisman."
He nodded. "But we're going to make enemies if we get on the wrong side of law enforcement. We might get prosecuted. Are you sure you want to stay on this case?"
I was momentarily confused, recalling his insistence that we couldn't allow our clients to be killed. But it didn't last long. "I do."
"Why?"
I repeated his rationale back to him. Then, "I remember our names painted in blood on the apartment wall. Whoever set that Claymore was counting on me coming back. They watched me go into the apartment and get well inside it before they set it off. So we've made enemies whether we want them or not. Then there's the little matter of withholding evidence. You didn't tell the Phoenix cops about our client. I didn't tell the San Diego cops about Grace's business, or about the flash drive."
"You gave them the pimp."
"Sure, but only that he was a guy threatening Tim when I showed up. I told them that's all I knew. Seems to me, if we're not pro-active, the bad guys will come to us, and if we don't solve the case, the good guys could come to us, too, and not in a good way."
He sighed. "I guess my point is, that I can take this one, if you want to bow out."
Now he hurt my feelings. It was that petty and selfish on my part.
I said, "No way."
"Are you sure you want to do this?"
I told him that I was sure.
He strode over to his desk and picked up his hat.
"Then bring your breakfast and saddle up." He pointed to my desk. "You might want to leave your fancy headgear here."
Up Grand Avenue, we had a fast ride cutting northwest through the checkerboard street grid of Phoenix and Glendale.
"So where are we going?"
"To see a guy I know," Peralta said.
"A guy you know?"
He nodded. It was going to be that kind of day.
"I want to talk to Larry Zip," I said.
"Not yet. Read the report. Then I want us to strategize before we interview him."
With that, he fell into his customary silence. What he was feeling from the contradictory events of the past few days, I wouldn't hazard a guess. Peralta's emotions were a deep ocean trench where leviathans stirred.
I distracted myself with the ritual obligation of memory.
I remembered when produce sheds and the remains of icing platforms for refrigerator railcars lined the Santa Fe railroad that ran parallel to the highway. I remembered passenger trains. Farm fields separated Phoenix from what was then the little town of Glendale. In grade school, we rode the train to the Glendale station. I even recalled one or two dilapidated farmhouses sitting right across the tracks.
Now it had all been filled in. Although the railroad was still there, the area around it mostly consisted of tilt-up warehouses, along with anonymous low-slung buildings, most with for-lease signs, and a gigantic Home Depot. Passenger trains were long gone. So, too, was the agricultural bounty that the Salt River Valley growers sent back east by rail. The children and grandchildren of the farmers who owned this land were living in places like San Diego thanks to the profits made selling it for development.
The road soon clogged up and stayed that way for miles. Much of Grand Avenue in the city of Phoenix had been turned into flyovers, back when the planners, such as were allowed here, thought about turning it into a freeway to Las Vegas. Like so many Phoenix dreams, this one didn't work out.
As a result, when we reached the "boomburbs" of Peoria, Sun City, Sun City West, and Surprise—yes, that's the town's name—Grand hit a six-point intersection at least every mile and other stoplights in between. And nearly every light was red. Traffic was miserable. The built landscape was new, cheap, and monotonous—made to speed by in an automobile. Smog smudged the views of the mountains.
Most of these had once been little hamlets on the railroad, but now they were home to hundreds of thousands populating the subdivisions that had been smeared across the broad basin that spread out from the actual Salt River Valley toward the White Tank Mountains and was labeled, incorrectly, "the West Valley." They came from the suburban Midwest or inland California and most thought life couldn't be better.
The metropolitan blob was slowly working its way northwest to Wickenburg, a combination quaint former mining town and home to celebrity rehab centers. I loved Wickenburg. It was authentic and charming, everything suburban Phoenix wasn't. As a young deputy, when I was working my way through my bachelor's and master's degrees, I had worked a patrol beat out here. The state had about four-and-a-half million fewer people and the land was empty, majestic, and mysterious. Wickenburg and the other little desert towns huddled to themselves. A lone deputy had many square miles to cover, usually alone, and traffic stops were always risky. So were family fights, where a husband and wife that had been trying to kill each other a few moments before were suddenly united in trying to kill you.
But we weren't going as far as Wickenburg today. Peralta turned left into the shabby little desert village of Wittman and drove west. After five miles or so and several turns, the last remnants of settlement were gone, the roads turned to dirt, and we were surrounded by desert. The smog hadn't reached this far north today, so the Vulture Mountains stood out starkly ahead. Go far enough and you'd find the fabled and long-ago played-out Vulture gold mine and who knows what else hiding in the desert. We bounced over the bed of the meandering Hassayampa River, dry this time of year. As a Boy Scout, I had learned the legend that if a person took a drink from the Hassayampa, he would never tell the truth again.
Immediately ahead, the country turned hilly and rugged, good terrain for saguaros. I was glad I brought two frozen bottles of water. But even in the air-conditioned truck cab, they were already half melted. It was only ninety-eight degrees outside. Inside my body, I was sore everywhere from my dive out of the apartment. Even my face hurt.
The bare impersonation of a trail appeared on the right and Peralta took it. Another mile and we reached a rusted metal gate. Peralta honked six times: three short, three long.
"Get down in the seat," he commanded.
"What?"
"You heard me."
I did as I was told as he shut off the engine, opened the door, and stepped out. He raised his hands high and his voice boomed. "Don't you shoot me, you paranoid son of a bitch. We need to talk."
This didn't seem promising.
The longest pause came to an end with a shout from the distance, "Go away!"
"I'm coming in if you don't come out!"
"Is that you, Peralta? Go back to your lettuce field, beaner! I'm done with the law. Got nothing to say."
Peralta shouted back: "Why aren't you on your reservation and cleaning toilets at a fucking casino, bow-twanger? Get your redskin butt down here!"
"If I do, it's only gonna be to kick your wet-back ass!"
"Good luck trying, wagon-burner!"
"Watch me do it, spic!"
"Bring it on, breed!"
It was, needless to say, not faculty-lounge language. And although Peralta was my least politically correct acquaintance, the outburst seemed out of character. Suddenly the yelling stopped. After too long a silence, I reached for the Colt Python and prepared for the worst. But when I rose up, the gate was open and Peralta and another man were shaking hands and embracing.
"Who's the white eyes?"
"David Mapstone, meet Ed Cartwright."
The shorter man beside Peralta was stocky in jeans and a Western shirt, with a long mane of lead-colored hair pulled back in a ponytail. His face looked like the Indian in the environmental ad way back, with a tear running down his face from the damage we had done to the land. He was tearless in appraising me. When we shook hands, I noticed the pistol on his belt. He handed me a business card with only his name and a phone number. I gave him one of my new private detective cards. The ones I once carried, with the gold badge, were only for my scrapbook.
I followed the two of them as they walked through the gate along a rutted, dusty trail to an adobe house that sat on a rise maybe a quarter of a mile away. Beside it, in a carport, was a restored Chevy El Dorado with a bumper sticker that read, AMERICAN INDIAN AND PROUD OF IT. The sun was frying my skin and I wished we could have driven the distance.
"Still waiting for the apocalypse, Ed?" Peralta asked.
"Yup."
"Show Mapstone your bunker."
That didn't sound like a good idea but pretty soon we were trekking off into the desert while Peralta stayed behind. The land was lush with sage, prickly pear, thick stands of cholla—jumping cactus—and ancient, towering saguaros with four and five arms. Those saguaros had watched the procession of humanity through this land for hundreds of years. Unlike their brothers in places such as Fountain Hills, they had avoided the bulldozers, at least for now. The silence was surreal and healing, except for the temperature and the fact that I was on high rattlesnake alert, walking heavily so the vibrations of my tread would give the poisonous snakes plenty of time to get out of our way. Cartwright was spry and walked fast. I worked to keep up and the muscles in my legs and back burned with pain.
Before we reached another hill, he led me around a lush palo verde tree. Beyond was a well-concealed cut that was obviously man-made. It led down until it was below ground level and zigzagged. It reminded me of the way trenches had been constructed on the Western Front in World War I. They zigzagged so an enemy soldier couldn't stand above the trench and take out an entire company with his rifle. We were on the verge of the hundred-year mark of that cataclysm that changed the world, but few Americans paid any attention to the past.
Cartwright broke my reverie. "Peralta and I were in 'Nam together. The sheriff's a good man in a shitty situation."
I agreed that he was.
We zagged to a stop. Cartwright hefted away a tumbleweed and unlocked a door that blended perfectly with the tan soil.
"This was an old mine," he said. "There's probably hundreds of them out here."
Now I was really worried about rattlers. But beyond the door, I could see only bright lights and a clean concrete floor.
Getting inside required another sharp turn beyond the entrance. Nobody could open the door and start shooting at the occupant of Cartwright's keep. We walked down a long flight of concrete stairs and made an abrupt turn into a short hall. He unlocked another door, metal and heavy, and closed it behind us.
We entered a space that looked about twenty feet long and wide enough for two men to stand comfortably. The ceiling was a foot above my head. On both sides, shelves rose six feet high holding meals-ready-to-eat, canned food, water, first-aid supplies, and ammunition. Boxes and boxes of ammunition for several calibers of firearms.
Beyond this supply area, the shelter opened up and held a bed, two chairs, and a desk with exotic radio equipment and other electronics. A well-stocked gun cabinet took up one wall. An American flag was posted to another. It was a forty-eight-star flag, the way it would have looked after Arizona was admitted to the union in 1912. Beside it were highly detailed U.S. Geological Survey maps of the area. The map fiend in me wanted to study them, but I felt mildly claustrophobic and unsure of my host.
"Ventilates to the outside," he said. "But I've got filters against fallout and biological attacks. I can air-condition it, if I need to. Got two generators and plenty of fuel farther back into the mine. Redundancy on everything. There's an emergency exit that comes out half a mile on the other side. I built it all myself."
He was plainly proud of it and I suppose there were worse retirement hobbies as long as he didn't wander down Tegner Street in Wickenburg and start mowing people down with one of the M-16s in that gun cabinet. The place was surprisingly free of dust and noticeably cooler than the outside, but I could feel myself only a few internal degrees from heat exhaustion.
I tried to be convivial, in an end-of-the world way, complementing his bunker. He seemed amiable enough, for an armed survivalist vet who might suddenly snap and kill me, stuffing my remains somewhere back in the old mine as varmint treats.
"Were you military, son?"
"No."
I could have let him judge me in silence, but I made an effort to keep the conversation going. Get people to talk about themselves, as Grandmother always advised.
"So this is where you ride out doomsday?"
"You bet your life. We came within an ass-hair of blowing up the world in 1983. The Soviets picked up a launch signal from the continental U.S. Their computer system said it was an incoming American missile. It was a glitch, but they didn't know this. They always expected an American first strike, and their strategy was launch on warning, so our missiles would hit empty silos."
He jabbed a finger my way. "If it hadn't been for a Russian colonel who suspected it was a false alarm and refused to send on the alert, they would have fired every ICBM they had over the pole at us and adios, baby. Hardly anybody knows how close we came."
"Stanislav Petrov," I said. That was the Soviet lieutenant colonel who perhaps saved us all.
"Very good. Don't think it won't happen again. All those missiles are still sitting there, waiting to be used. Damned Russians are building an underground city that's as big as Washington. You look on the Internet. Israel and Iran. North Korea. China's got miles of tunnels to hold their nuclear forces. Hell, we're even allies with Hanoi now against China. Makes you wonder why anybody even wants to live."
His agitation grew as he talked and he paced over to the gun cabinet and I placed a hand on the butt of my Python. My spinal cord was filling with ice.
"We got seven billion people on the planet, climate change, ebola and diseases we don't even know about that can't be killed by antibiotics. Your people did this." His expression was accusing, his voice angry. "Couldn't leave well enough alone. Had to conquer nature, but she won't be conquered, kid."
He sighed. "Anyway, it might not even go down that way. You take away the power and gasoline from five million people in Phoenix in high summer, and watch what happens. I'll be fine."
I had no doubt.
After fifteen minutes of this cheery conversation, we arrived back at the adobe, where Peralta was standing under the shade of the porch, smoking a cigar, and surveying the jagged treeless mountains on the horizon.
"You got another Cuban, Sheriff?"
Peralta produced a cigar and Cartwright ran it under his nose, inhaling like a connoisseur. "You people wouldn't even have tobacco if it wasn't for us."
"Apaches didn't have tobacco," Peralta said.
"Well, then we would have killed the Indians that did and taken it. Thanks for the cigar. Now I don't have to kill you." He carefully slipped the stogie into his front pocket. "I see the kid here is a revolver man." He pointed to the Colt Python in the Galco high-ride holster on my belt.
"He doesn't trust semi-autos, thinks they might jam." Peralta raised an eyebrow, an act of raucous comedy coming from that face.
"It can happen," Cartwright said. "May I?"
Every instinct told me to decline, yet I handed the heavy revolver over, butt-first. He opened the cylinder, dropped out the six rounds in his left palm, and dry-fired it against the wall: click, click, click.
"The Combat Magnum. Listen how clean that action is." His tone was that of a wine connoisseur. "It was the first gun to be bore-sighted with a laser, you know. Finest mechanism you'll ever find in a revolver. Tight cylinder. Highly accurate." He handed the gun and ammo back. My pulse pulled off the fast lane. I was fortunate that the house was air conditioned and dark inside, to cool me down and conceal my apprehension.
The living room was furnished with handsome leather chairs and sofas. Books were everywhere: in floor-to-ceiling shelves, on tabletops, and sitting in stacks on the hardwood floor. They were not of the Anarchist's Cookbook genre. Instead, literature, philosophy, poetry, political science, and, of course, history filled the room. Classics and new, important works. I'll admit it: I took stock of a person by the presence of books and their titles, and I almost started to let down my guard. I could see no television or newspapers. He might not even know that Peralta was no longer sheriff.
Cartwright returned from the kitchen with bottles of Modelo Especial and we sat.
"What brings you out to my humble outpost, Sheriff?"
"One guy shot and killed earlier in the week with an AK-47." Peralta took a swig and a puff. "Then my partner here almost bought it with a Claymore."
Cartwright made a tisk-tisk-tisk kind of sound. "Walk down memory lane, eh? Did you tell him about the way we used Claymores to ambush the slants back in the day?"
Peralta nodded. "Whoever did the shooting with the Kalashnikov was damned good. Pumped ten rounds into the victim sitting in a car. The shooter was in another car. Only one shot failed to hit the target. And this was daylight, right on Grand Avenue down in town."
"Sounds interesting."
Peralta waited.
Cartwright sighed. "I'm retired, Sheriff."
"Bullshit. You know things. You know more than me when it comes to assholes seeking illegal weapons."
"Is there such a thing as an illegal weapon in Arizona anymore?"
"If there is, you're selling it," Peralta said.
So he was an arms dealer.
"Not true," Cartwright said. "Drive back to Wittman or Circle City or Mesa for that matter and you'll find guys who can fix you up with anything you want."
Peralta sat back, wreathed in cigar smoke, his expression losing its amiability.
He said quietly, "They can't fix you up with a Claymore."
Cartwright spoke softly, too. "I'm not a rat. Never have been."
Peralta had handled the tribulations of the past several months better than me. Of course, some of them hadn't affected him quite so personally. Still, I was the one who seemed angrier about his loss of the election and the ugly, racist campaign that preceded it. He had turned philosophical and, if such a word could be applied to him, mellow.
But watching his face now, I could see the flickering of the old anger and impatience. Cartwright spotted the launch signal, too, and knew it wasn't a glitch. Still, he tried to escape.
"You know I'm not in the game anymore. Give an old man a break. I'm tired now. I need to rest."
"You were up to your ears in Fast and Furious," Peralta said, referring to the federal operation meant to disrupt the flow of guns to Mexico that had gone horribly wrong. It had cost the U.S. Attorney his job, brought hearings in Congress, and even become an issue in the presidential campaign.
"My part worked." Cartwright glared back at him.
The two dark stone faces faced off. Cartwright's was cut with gullies in geometric precision, while Peralta's aging congregated around the crow's feet beside his eyes. His hair was still naturally jet black. He was actually better looking than he'd been at thirty-five. He wore distinguished well.
Neither seemed willing to give. I tried to imagine them as young infantrymen, fighting for a country with a poor record of treatment for Apaches or Mexican-Americans and yet there they were, brothers in arms, in Southeast Asia. That bond showed in their expressions, too.
Finally, Cartwright stood and walked slowly at first, as if his hip hurt. Then he strode out of the room. In five minutes, I heard his tread and something landed in my lap. It wasn't as heavy as I imagined.
"Your boy's pretty cool," Cartwright said.
Peralta watched me. I can't tell you why I didn't make the jump of the startled or run screaming from the house once I saw he had dropped a Claymore on me. Instead, I carefully studied it: "FRONT TOWARD ENEMY" the same as the one in Tim and Grace's apartment, two sets of extendable legs, and a small housing on top where wires, or another kind of detonation mechanism, could go.
Cartwright eased himself into a chair across from me. "You're lucky to be alive, son."
He hefted an AK-47 in his hands. "Mikhail Kalashnikov's baby. Cheap to make, easy to use. One of the first true, mass-produced assault rifles. Seventy-five million of 'em all around the world." He quickly field stripped it and put it back together, his pudgy fingers working expertly. Anybody who watched television had seen AKs in the hands of freedom fighters or terrorists, take your pick.
"How do you know your guy was killed with an AK? Was the weapon recovered?"
"No," Peralta said. "I heard it."
Cartwright nodded. He understood.
"Anybody can buy an AK. You know that. Using it with such precision is another matter. And why would you want to? There's too many good, modern weapons available. Maybe your suspect has a thing for the gun? Maybe it's his bad-ass signature. You should run that through ViCAP." The FBI's violent criminal database. "It's probably not some disgruntled 'Nam vet. We're getting too damned old. But the older we get, the tougher we were."
He chuckled. Peralta didn't.
I was half-listening to the ordnance talk. The Claymore sat a few millimeters from my genitals. I kept looking at the instructions stamped on the front. Such a funny thing. So you don't forget and aim it wrong. I shouldn't even be here right now. Why did I get over that apartment railing and into the pool with only seconds to spare, when Robin hadn't been safe in our back yard? Contingency was the god damndest thing. Robin would have made the better mark on the world if she had lived and I had died.
Peralta tapped an inch of ash into an amber glass ashtray. "I've thought about all that, Ed. Quit stalling."
"The Claymore is a different matter entirely." He cocked his head. "Is this connected to the explosion in San Diego on Friday night?"
So much for being cut off from the world.
Peralta said, "You know it is, so quit playing games."
To me, he said, "How far did you get into that apartment before you realized you were in the danger zone?"
I told him.
He let out a long whistle.
"So you see," Peralta said, "This is personal and it might get a hell of a lot more personal."
Cartwright set the rifle in his lap.
"Do you know how far my ass is already in a sling even by talking to you?" he said. "Even by you being here?"
"I don't care." Peralta swiveled his head.
"So give me something to work with?" Cartwright folded his hands over the assault rifle. "Who was killed with the AK?"
"Anglo, thirty-five or so," Peralta said and went on to describe our first client including the expensive prosthetic leg and the multiple names and identifications.
"Nobody I know," Cartwright said.
I said, "He had yellow eyes. Very well dressed. And he had a silver Desert Eagle on his passenger seat when he was killed."
Cartwright shook his head slowly, but I caught the involuntary tic of his left eye.
"Didn't do him much good," he said. "You're probably lucky he got killed when you weren't in the line of fire. One less dirtbag in the world and the kid here survived. What's not to like? Now I need to take a nap."
Suddenly, a fury rose in me. Tim Lewis' face hovered in my mind. And the baby I had held in my arms.
Cartwright asked me what I was doing.
"How do you set this thing off?" I was fiddling with the Claymore.
"You can't." He smiled at me like I was an idiot. "It's disarmed."
That did it. I threw the Claymore straight at his face. When he reached to catch it, I was up, crossed the eight feet separating us, and picked up the AK-47 from his lap.
"What the..." He let the dummy Claymore fall. It clattered on the wood floor. Next he reached for the pistol on his belt.
I chambered a round in the AK-47, although I didn't aim it at him. Yet.
Peralta said, "I wouldn't move, Ed. Mapstone here had a run-in with Los Zetas where they tried to put a hand grenade in his mouth, so he's PTSD'd to the moon."
Through his teeth, Cartwright said, "Why is he alive then?"
Peralta spoke softly. "That's why I wouldn't move."
He spoke quietly, "How do you even know how to work that thing, kid?"
"A million child soldiers in Africa can work it. Want to take a chance that I can't?"
He studied me through angry but uncertain eyes, his hand still on the butt of his sidearm.
If Cartwright had even started to pull the weapon, I would have pumped several shots into him before anything like judgment could have caught up with the rage I felt. A savage stranger's voice started speaking. It was coming from my mouth.
"You listen to me, old man." I spat out the last two words. "I've got two young people murdered and a missing baby. Now I've got an armed whacko survivalist sitting in front of me who thinks he can get off a shot before I send him to hell. Who knows how many weeks before they find your body? What I don't have is time to waste finding that baby, and that means you don't have time."
"All right, son. Please calm down."
I swung the barrel to his chest.
"Now you have ten seconds less time."
He saw my finger was on the trigger and a sheen of sweat appeared across his forehead.
"A dozen Claymores went missing from Fort Huachuca last month," Cartwright said.
Peralta shook his head. "That's an intelligence installation. What are anti-personnel mines doing there?"
"The military has this stuff everywhere. Makes it hard as hell to track. Who knows how much walks away from bases and nobody ever knows?"
I wanted to know who took it.
"Word is, soldiers."
"Active-duty soldiers?"
He nodded. I didn't lower the weapon.
He swallowed. "White supremacists are in the military. That's not new. You remember a guy named McVeigh in Oklahoma City. Now there's more of them. We've spent more than a decade at war, and we're sending home killing machines." He sighed. "Anyway, the word is, that's who took the Claymores. I don't know if it was to sell or to use."
"What about prostitutes? Are they involved in running high-end whores?
"That's all I've heard, son," he said. "Do what you please."
He closed his eyes and in the terrible silence that followed he put his hand in his lap. I lowered the assault rifle.
Peralta said, "Give me that and wait for me at the truck."
My blood was still up but I did as he asked.
Before I walked out, I heard Cartwright's voice.
"You have an unusual name, kid. I read a book by somebody with that name once, about the Great Depression."
"He wrote it," Peralta said.
"It wasn't bad," he said. "But you should have written more about the effect on the tribes."
He was right. I closed the door behind me.
Half an hour later, we hit solid pavement and Peralta spoke for the first time since he had returned to the pickup truck.
"There was a day when he would have killed you."
I let my breathing return all the way to normal before speaking.
"Ed? As in Edward? America's Finest Pimp thought I was the enforcer of some guy named Edward. He was afraid of Edward, and he didn't strike me as the kind who was afraid of many people. The man he described as Edward's muscle sounded a hell of a lot like Felix."
"That's not this Ed," Peralta said.
"How do you know? Did you see the 'tell' when I told him about Felix? He was lying."
"He had a loaded AK-47 being held by a crazy man, Mapstone. That's not a 'tell' you can trust."
"Maybe. His name is still Edward."
"Ed was a decorated FBI agent before his end-of-the-world fetish got him in trouble and he was fired. Only that's not the whole story. He's quietly enjoying his FBI pension and an honorable retirement."
"So tell me the whole story."
"Being known as a disgraced, bitter former special agent gives him cred. He deals guns to skinheads and bikers, cartels, Mexican Mafia, whoever pays. Gives 'em training, if they need it. And any takedowns happen so far down the line that nobody suspects crazy old Ed Cartwright."
"I never heard of him."
"You wouldn't," Peralta said. "He doesn't work for the FBI, doesn't work for Alcohol, Tobacco and Firearms. He reports to higher authorities. Maybe where your wife works, Mapstone. Nobody else in Phoenix law enforcement even knows about him, except as another reclusive old coot living out in Wittman with his guns. That's the way it's supposed to work."
"Why would white supremacists deal with somebody who has brown skin?"
"They must dig the whole Apache noble savage thing."
My breathing return to normal. It would have helped if Peralta had given me the whole story before we went visiting, to know what his play was. That kind of non-disclosure was like the old Peralta. It would have helped if Cartwright could have done a better job of connecting the Claymore to an apartment in San Diego, a young woman's fall out of a condo tower, and her boyfriend's violent death. Was he her boyfriend or husband? I didn't even know. How nuts was that?
"We've got white supremacists in the armed forces," I said. "I thought that was the least racist institution in America."
"Not after you break the force in two long wars," he said. "And drop recruiting standards. And have a black guy as commander in chief, which has brought out all the whackos. You remember the group they arrested in Georgia? Five soldiers were stockpiling weapons. They wanted to poison the apple crop, set off bombs, and overthrow the government? Thank God for stupid criminals."
"Until some smart criminals show up," I said. "So I assume Cartwright's bosses will be on this."
Peralta shrugged and we rode the rest of the way in silence back through the antic monstrosity of the suburban asteroid belt and into Phoenix. It might have been quicker to take the Loop 101 down to the Papago Freeway but Peralta stuck to surface streets. What were all these people doing out? Driving was the local sport.
Outside the office, he shifted the truck into park and turned to face me.
"Mapstone, I know you're not yourself, but..."
Peralta's voice trailed off. He set his jaw and turned forward.
I got out. He drove off.
A few minutes later I returned home, eating a Popsicle to cool down and brooding. For the first time in my life, I had come close to being the worst thing possible in law enforcement: a hot dog. Cartwright was Peralta's play and I should have stayed in the shadows, let him handle it, and listened. I had allowed Cartwright and my anxiety over the baby to strip away my professionalism, send me into a tantrum with a loaded gun. He was going to get the idea that he couldn't count on me, never a good place to be with Peralta.
But the reality was that Peralta wasn't getting anywhere and Cartwright was holding back. Now I knew about the stolen Claymores but the connection from there was tenuous. The idea of white supremacist cells in the military was frightening. Some future Gibbon would write about that. For much of America's history, the nation had frowned on large standing armies. Now it was part of our economic policies and if you didn't "support the troops," you were a commie. And how all this tied in to Grace, Tim, and baby David, I did not know.
I was still running out of time. Sure, every law enforcement agency in Southern California would be working this case. But it was on me. That, at least, was how my concussed brain parsed it.
I checked the Amber Alert for the tenth time that day. The baby was still missing. He hadn't been in the apartment when I went through it. That much I was sure of. I was also certain that this was no child-custody issue but a kidnapping. Soon the FBI would come knocking. I was, after all, not merely a washed-up historian of cold cases. I was an expert on lost children.
But there would be nothing helpful I could tell them. My lost child was the one conceived with Lindsey. The child that never made it past four months in her womb before the miscarriage. Lindsey fled to the job in Washington, insisting that Robin continue to live here. Her only demand of me was that I keep Robin safe, a task at which I had failed. Now they were all gone and a house I had expected to be filled with unaccustomed baby sounds was silent. There was no Amber Alert for our loss.
These were thoughts I had become a master at stuffing in a closet of my brain and locking away. Patty had complained that I lacked the discipline to write serious history. She didn't know how disciplined I could become. I hardly cried now. It helped that I had hidden away all the photos of Lindsey and Robin. I didn't speak of it. No pity party for me. I didn't want Sharon Peralta to shrink my head, get me in touch with my feelings. Those would kill me. I was not special. The world was a planetary engine fueled by loss. Some things could never be made right.
At the Sheriff's Office, one of the cardinal rules of "incident command," say a hostage situation, was "stay in your own lane." One of Peralta's commandments was to avoid what he called a jurisdictional goat fuck. We were gone from the Sheriff's Office, and yet I knew he still thought that way. So I tried to choose a lane to make my own, take one of the many strands that had dropped on us in the past few days and follow it.
That would be the suspicious death of Grace Hunter.
Putting the Mac to sleep, I moved over to the leather chair, the police files of Grace Hunter's death on my lap. My body gradually cooled off from the hike through the desert to Cartwright's bunker. From the study, I could see the big picture window of the living room and I dawdled, staring out at the lush shady landscape of the street. Phoenix had done so many things wrong as a city. One of the things it had done right was preserving some of the old neighborhoods such as Willo.
I meditated on all the things I had done wrong in San Diego. Whores always lied. This was something I had learned as a young deputy. Most had drug habits. Most had children. So much could go wrong in their lives. Telling the truth was an occupational hazard. I should have interrogated Tim in much more depth. Who knew if Grace was even telling him the whole story? On the other hand, I had never dealt with high-priced call girls. Yet something told me the same operating procedure, the lying, applied. Many of them were victims. Some were not. None had a heart of gold, whatever Tim's conviction that she wanted to leave the life and be a mommy in Ocean Beach.
I should have gotten much more information about both Tim's and Grace's pasts, should have asked to see her computer and especially her emails. When she didn't come home that day, did he try to call or text her? I failed to ask. Did they have any friends in town with whom they socialized, besides her girlfriend Addison? Had any new neighbors moved in, or were there folks they had recently started to hang with, and had any taken an interest in Grace? I didn't know.
I had assumed we would have more time for a follow-up, a foolish supposition considering that our first client, the one who had launched us on this quest, had been gunned down outside our office. Most of all, I should have waited until Tim was safely in his car with the baby and driving out the alley and toward the freeway to Riverside and his parents' house. I was not at the top of my game.
The police files had a familiar feel. Every department used standardized reporting now. But the information from my first spin through the paperwork showed me I wasn't the only one who had made mistakes. I pulled over a legal pad from the desk and started making notes.
On April twenty-second, the first emergency call came in at 11:54 p.m., a woman had fallen or jumped from a condo balcony. It was placed by another female who had been with her boyfriend in the Jacuzzi by the pool. Grace landed no more than twenty feet away and the caller was sure she had come off the nineteenth floor. The woman lived on twenty and her balcony was distinctive because of its hanging plants.
The first units to arrive, three minutes later, were a fire department engine company, then, five minutes after the first call, a paramedic unit. Grace was obviously DOA. All the witnesses agreed she was nude and handcuffed from behind.
Unfortunately, the building's night concierge was there, too, and opened the condo door for the firefighters. And they walked right in, finding the place empty, and the door leading to the balcony open. Uniformed police got there six minutes after the call, which had originally been routed to the fire department as a medical emergency. That's not surprising. The woman in the Jacuzzi was probably hysterical when she made the call and was perhaps assuming, with the hope of the traumatized, that Grace was only hurt. But it was several minutes before the officers could secure the condo.
That meant the first investigator to arrive, Isabel Sanchez, the "night detective" to use Peralta's lingo, was already facing a contaminated scene. I thought about Peralta telling me she was a looker, how it would do me a world of good to get laid. What would my opening line be? "Hi, I'm David and I'm concealing evidence. What's your sign?"
Back to work, Mapstone. At least two firefighters, the night concierge, and perhaps some curious civilians had been in the unit before the uniformed officers ran them out. The "fact" that the condo had been locked with a deadbolt also came second-hand, from the night concierge.
The inventory of Grace's belongings was minimalist: one pair of jeans, a cotton blouse, bra and panties, and black sneakers. But the clothes had not been neatly folded. The blouse was draped over a chair in the living room and the bra was two feet away on the floor. The jeans and panties were in the bathroom, again on the floor. There was a small amount of standing water on the bathroom floor, but the towels were dry and the shower hadn't been recently used. One sneaker was in the hallway leading to the bedroom. The other was in the kitchen.
Her purse was on the living room sofa, a small black satchel, tipped on its side. Inside were a wallet, cell phone, tissues, keys, sunglasses, and hand lotion. I studied the photos of each. No pepper spray or knife, which Tim had said she always carried.
A cynical street cop would say Grace was wandering the condo, stripping down as she cried or raved or hallucinated. Maybe she splashed water from the sink onto her face—and onto the floor—trying to make herself snap out of it. But then she found the nerve, walked to the door, opened it, and stepped out on the balcony. Next, to make sure she couldn't change her mind again, she handcuffed herself.
It might have gone down that way.
Wide-angle shots of the rooms showed nothing to indicate violence. Lamps were in their place. A big, fragile piece of pueblo pottery was undisturbed on a table nearest to the glass door. The bed was made. The kitchen looked as if no one ever cooked in it. The floors were clean except for the discarded clothing.
Another photo showed the inside of the front door. Writing in red said, I AM SO SORRY. It jarred me momentarily, bringing back memories of the wall in Tim's apartment. But it wasn't blood. A lab report said it was lipstick.
My cynical street cop, who had spent so many hours filling out paperwork for successful and attempted suicides, said to me: "See, she offed herself. Leave it to a Southern California airhead to have no writing implements other than her lipstick."
The only problem was that the inventory showed no lipstick, no cosmetics of any kind. I did a grad-school speed skim of the rest of the incident report, supplemental reports, and Detective Sanchez's case notes. No lipstick tube.
Sanchez tried Grace's keys on the condo door. None fit. Her driver's license had an out-of-date address. But her parents' names and phone number were in her wallet, and they were the ones Sanchez called to notify. They flew over and identified her remains. If Grace's wallet also had baby photos, they weren't listed in the inventory. There was no mention of a baby anywhere.
The medical examiner rushed through the autopsy and toxicology reports, the cops no doubt being mindful of the Coronado case. Its findings were what you would expect from a fall of at least one-hundred-ninety feet onto concrete. Massive head trauma, burst organs, collapsed lungs, and dozens of fractured bones. "The cause of death was the fall," one comment noted. Her bloodstream contained neither drugs nor alcohol. Her wrists showed abrasions where the handcuffs had been placed, but this could be consistent with a falling person's panic. She had semen in her vagina.
I got up and returned to the laptop, using Google to search for keywords about what happened that night. Not one story appeared. I remembered what my friend Lorie Pope, once a reporter at the Arizona Republic, had told me: that newspapers had reassigned or pushed out the old-time police reporters, the ones who went on calls and formed close relationships with the cops. That was how she and I had first met, me a deputy, Lorie working the police beat on the city desk. Now the newspapers only wrote up what the police public information officers told them. Crime upset readers, or so the editors thought. Television was a different matter, but only if they had "visuals," and no station seemed to pick up on the action that night. It was nearly midnight when she fell, after the end of the eleven o'clock news. If SDPD feared another suspicious death circus, they didn't need to worry.
The death and autopsy photos were what I expected. About all that was recognizably left of Grace Hunter was the butterscotch hair with blond streaks.
Larry Zisman returned at one, a little more than an hour after Grace's long dive. Sanchez initially interviewed him downstairs in the owners' lounge. Then she took him to the station to sign statements.
Zisman and his wife had owned the condo for six years. They often spent the summer months there. His wife worked for Intel and sometimes traveled, so Zisman was frequently there alone. Zisman described Hunter as his girlfriend and begged the police not to tell his wife. They had five children. He had never been unfaithful before. The usual routine. He also referred to Grace as "Scarlett," a discrepancy Sanchez noted and questioned him on. "Subject states that he only knew deceased as Scarlett, a student at SDSU."
He had met her three years before at a convention and they became occasional lovers. He denied giving her money, only some presents. As for the handcuffs, they belonged to Zisman, the reserve Phoenix police officer. He said they used them for light bondage during sex.
Zisman said they had a fight the night of her death. She wanted to see him more regularly and demanded that he leave his wife. She became hysterical and was generally "emotional" and "high strung." It did not become physical, according to Zisman's statement, only shouting and tears. "Subject stated that he left the condo at approximately 2230 hours with Hunter still there, to let the situation cool down."
According to Zisman, they did not have sex that night.
Zisman said he went to his boat, which was moored at a marina less than a mile away, and stayed there reading until half past midnight. Coming and going, he talked with a couple that owned a craft at the next slip. This checked out.
So if Zisman left the condo at half past ten that night, Grace was alone for little more than an hour before her death. If she was, in fact, alone.
I thought about what Tim had told me. He had gone to classes that morning and not returned until after three. Grace was gone. So even if she went out right before three, it left many unaccounted hours before her death. I found nothing that gave an indication of how they might have been spent. Grace had a new cell phone and it had no information on it, no calls, no texts, either incoming or outgoing. Would the mother described by Tim simply walk out of the Ocean Beach apartment, leaving the baby, and not even check in?
If Sanchez asked Zisman the last time prior to that night he had seen Grace, or Scarlett, it wasn't in the report. If he hadn't seen her for, say, eighteen months, then Tim's story might be accurate. If he had been with her four times that month, Tim had been betrayed. Unfortunately, after my encounter with the Claymore mine, I didn't think Isabel Sanchez would tell me diddly.
If Tim was telling the truth about them hiding out for more than a year in O.B. as Grace gave birth to their baby, one of the most important questions remained unanswered: why Grace would have gone downtown to meet up with a former john?
I had to hand it to Zisman. From the case notes, he came across with exactly the right mixture of distraught, surprise, and forthcoming. And maybe all that was genuine. But he was also in law enforcement. The only ones better at lying than whores were cops.
Then there was the missing lipstick tube.
I walked around the house and returned to the chair, spending two hours reading in more detail. This didn't change my initial impression. The man with the prosthetic leg had been right.
This was not a suicide.
Back at the desk in the study, I wrote a report for Peralta. This was my element, as it was back in the days at the Sheriff's Office, where I was his in-house egghead clearing cold cases. In thirty minutes, I had shot all the holes I could find in the shoddy and rushed work of the San Diego cops, and then I printed out the report.
A few seconds later, the house phone rang. We had cordless phones in other rooms, but the phone on the desk had belonged to Grandfather, with a real dial, now a retro amenity, and the old-fashioned ring that could put a nail of dread into your brain. As in the old days at the Sheriff's Office, I would answer and he would announce one word, "Progress?"
Only it wasn't Peralta.
"Doctor Mapstone." It was an average voice, neither young nor old, lacking any accent, be it from the Midwest or from the heart of Mexico. An unfamiliar voice. And he was using Doctor Mapstone, exactly as Felix did—the inflection even reminded me of Bobby Hamid, the gangster that Peralta sent to prison for life. I was easy to find. I was in the phone book of whatever company currently owned the former Mountain Bell. I switched on the recording device that Lindsey had added.
"Yes."
"How could anyone fail to get tenure at San Diego State University?"
"I always surprise. Are you calling to offer me a job?"
"Perhaps. In a way."
"Go on. I need options."
A fine laugh followed, not villainous at all. The kind of laugh you could have a beer with. "I bet you do. You have something that I want."
"What would that be?"
"Don't fuck with me." No emphasis or emotion entered his voice. He might as well have said, "Excuse me."
He went on, "You have something I want, and I am willing to give you something in return."
"Did you make that same offer to Tim Lewis when you were breaking his fingers one by one?"
"Yes, I did."
At least that was settled. I wasn't talking to the tenure committee, calling to say they had made a mistake and wanted me back as a professor. I was talking to a stone-cold killer. Caution flooded my nervous system: he had the baby. I had to be careful not to bait him into further death. I wasn't sure one could negotiate with a man who would slit Tim Lewis' throat and paint the wall with blood, but I had to try.
"He claimed that he didn't have it. So I can only assume that you do. But you present a bigger challenge because of your law-enforcement connections. I have to take a different approach."
"I bet. You're a murderer and I'm coming for you."
So much for David the Negotiator.
The voice remained steady and calm. I hadn't gotten under his skin. He sure as hell was under mine. Smooth and calm and very sure that I wasn't tracing him. He seemed in no hurry to hang up.
"I know you want to come for me, Doctor Mapstone. I learned how you settled your Mexican trouble. Impressive. I don't underestimate you."
"I didn't have Mexican trouble. I had criminal trouble. You're no different."
"You're wrong there. We might even be on the same side, using different methods."
"I don't think so."
He cleared his throat. "If I told you we were in a battle for this country, for whether it can remain a white nation, you'll dismiss me as a racist nut. You work for a Mexican. And you're an academic. You've been brainwashed. Samuel Huntington has it right about the clash of civilizations, but it's happening in America. In fact, it's killing this country. You don't want to face the facts."
He was mighty chatty and had done some reading. I said, "I'm not sure Professor Huntington would agree with you." God, I wish I were tracing the call.
"David," he switched to the familiar, "I don't want to kill you. I had a chance already. I didn't take it."
I wondered if he even had the ability to change baby David's diaper and feed him. I asked him about that chance to kill me that he didn't take.
"The apartment. You're very physically impressive for a man your age, getting out the way you did. But I helped you by waiting to detonate the Claymore. Do you have a new cell number since your old phone went with you into the pool?"
A man my age? Fuck you.
I said, "So you were watching."
"How do you think I gave you time to get out of the apartment? Now answer my question because I won't give you time ever again."
I gave him the number of the temporary phone I had bought at the truck stop in El Centro.
"Drive to the Park Central parking lot. Be on the south side of the lot in five minutes. Come unarmed. Wait and I'll call. Stay in your car. If any police are with you or near you or I even suspect you're fucking with me, you won't get what you want."
Even the profanity was said with a businesslike calm.
Then the line went dead.
I switched off the recorder, checked my watch, and called Peralta. After I gave him a quick update, he told me to get moving, follow the caller's instructions.
"Where are you?"
"Camelback Mountain."
"It'd be nice to have some backup."
"Get moving, Mapstone. I'll get there when I can."
Wondering what he was up to, I pulled the heavy Python and its holster out of my belt and set them on the desk. The freshly cleaned Airlite .357 magnum slid into my pants pocket and was barely noticeable. I added a small Ka-Bar "last option" knife, with a serrated blade, inside my waistband. As the name implies, it is for use if you get in a fight for your weapon and are afraid you'll lose. I further armed myself with a bottle of frozen water.
Halfway to the car, I stopped, turned around, and went back in the house. In the office, I rooted around in Lindsey's tech drawer and found a blank flash drive. If I had to trade, at least this might buy me some time to assess the situation. If the mystery caller actually made contact with me, I could show it. Then he would have to show the baby. Then things would get interesting.
Less than three minutes later, I pulled into the bland assortment of low-slung buildings that was Park Central.
When I was a child, it had been the first shopping mall in Phoenix, a sweet, modest, open-air affair on the site of an old dairy. For the past two decades, since the Midtown skyscraper boom collapsed with the 1990 real-estate crash and retail fled toward Scottsdale, it had been mostly offices. Most were medical-related, tied to gigantic St. Joseph's Hospital and Medical Center to the west.
Local wags were calling St. Joe's "Mr. Joe's" now, after the bishop had withdrawn its Catholic affiliation because an abortion was performed to save a woman's life. It was a mere kerfuffle in the many local rows of the moment. Even without reading the newspaper, I knew Peralta's successor was doing "sweeps" to pick up illegal immigrants and holding news conferences to trumpet this as a huge triumph of justice. The Legislature was an insane asylum. Nobody in power was talking about anything real or important.
Like almost everything in Phoenix, Park Central was surrounded by a large surface parking lot. In Phoenix, the word "park" often doesn't mean a recreational space but an asphalt place to leave your vehicle. Those were among the reasons why the weather kept changing for the worse. Today the lot was nearly empty, except for cars near the popular Starbucks on the southeast corner of the former mall. All of Midtown Phoenix was virtually deserted on Sunday. I eased the Prelude in that direction, hoping to find some shade from the skyscrapers lining Central Avenue.
The best I could do was to catch a little relief from the sun in the shadow of the Bank of America building. Nothing looked unusual: nobody following me, nobody sitting in a car waiting. A light-rail train slid by, its electronic bell penetrating the sealed passenger compartment of the Prelude. Across the street were two mid-century towers, one tall, one short. The tall one once had an outside glassed-in elevator, which, I am told, was a popular spot for a quickie before it reached its only stop at the top floor. The short tower had been the site of the Phoenix Playboy Club. Back in the day, Midtown Phoenix could swing.
Ten minutes later, the phone rang. It made an annoying xylophone sound. I listened on my headset.
"You've done well so far. Go through the Jack in the Box and get something to eat from the drive through. There are only two cars in line."
The incoming call readout said, UNKNOWN.
I sped a hundred yards over to the Earll Drive exit, drummed my hands on the steering wheel for the light to change, and turned north on Central. It was so nice that he was concerned about my eating. Sure enough, only two cars were ahead of me. I checked the mirrors, craned my neck around. Nothing.
My destination put me more on edge. The drive-thru was hemmed in on one side by the restaurant and on the north and east by the fancy One Lexington condo tower and its swimming pool. I had heard stories about those pool parties. Maybe the caller had a different party in mind for me. If he wanted to trap me from behind, then walk up and spray the Prelude with machine-gun fire, this would be the ideal place.
"Welcome to Jack in the Box, would you like to try two Jack Tacos for a dollar?"
The disembodied voice startled me. I said two Jack Tacos were fine and waited for the line to move, my hand on the Airlite. Did I want to add an order of curly fries? Hell no. So far, no car pulled behind me. An ancient Toyota Camry was belching fumes ahead, two young children yelling and beating on their mother's shoulder with shopping bags.
He might have been sitting under one of the umbrellas outside the Starbucks across the street, but no—he couldn't have seen the Jack in the Box line with such precision. The caller was mobile. He implied that he had something I wanted. That could only be the baby. And what I had was the flash drive. How did he know I had it? Torturing Tim Lewis would easily have given up that answer.
I got my order and pulled to the edge of Central. The phone rang again.
"What?"
"Drive north to Indian School. Go the speed limit. I'm watching."
"I want to hear..." I wanted to hear the baby's voice before continuing, but he didn't give me a chance to finish. Now would be a really good time for Peralta to show up. But he didn't call, and I didn't dare tie up the phone calling him again.
So I did as told. I let a few cars pass. Nobody goes the speed limit in Phoenix. Aside from people waiting at the light-rail stations in the middle of the avenue, I could count the pedestrians on one hand. One was a man in a red cap running south with six dogs on leashes in front of him.
Once upon a time, the city's leaders had intended Central Avenue to be Phoenix's version of Wilshire Boulevard in Los Angeles. It didn't work out. The old corporate headquarters, banks, and shops that made Central the most important business location in town were bought or closed or lost in the savings-and-loan crash. Much of what remained moved out to Scottsdale or other suburbs as Phoenix became a back-office town, heavily dependent on population growth and real-estate hustles, and then the Great Recession had cleaned our clock.
As a result, this part of Central held an assortment of bland skyscrapers with massive parking garages behind them. A few quirks of the older city remained, such as the curved "punch card" building that was once the headquarters of Western Savings. When the thrift went bust in the savings-and-loan scandal, the tower went mostly empty. A pair of domed, two-story circular buildings complemented it. They had been retail bank branches and their mid-century appeal endured. I recalled that inside, each building had a sunburst skylight. In a healthy city, one such as San Diego, they might have been remade into a jazz club or a restaurant. Instead, both sat empty, the blinds hanging as if the savings and loan had closed the day before.
Other Central landmarks passed by as I drove north: an inverted pyramid and Macayo's restaurant, whose façade was meant to resemble an Aztec temple. And much empty land.
Somebody had bought it long ago, gotten the City Council to zone it for high-rise, and cleared the old single-story buildings or bungalows. But development never happened. Speculation and flipping did. Because it was zoned for high-rises, no developer could afford to do less than build towers—for which there was no demand. So the land stayed bare, aside from an occasional plan that went nowhere.
Farther north, back in the 1980s, somebody had promised to build the tallest building in the country. That site was still bare. Absentee owners were banking it on their asset sheets for someday, even as decades went by. This was the heart of the nation's sixth-largest city, but these parcels might as well have been World War II Dresden after the debris from carpet-bombing had been carted away. I had read that forty-three percent of the city of Phoenix was empty land. What were a few art projects on vacant lots against that?
Hardly anyone knew what had been lost, first by the skyscraper rush, then by the mass abandonment. But these blocks had once held hundreds of bungalows and adobes of the same kind that were now protected and coveted in districts such as Willo. These had been neighborhoods people cared about.
Back in the 1940s, most of Central had been lined with queen palms and handsome haciendas, irrigation ditches, and citrus groves. When I was young, most of these had been replaced with shops and businesses, but it was a vibrant street. "Cruising Central" was an essential tradition on Friday and Saturday nights until the police banned it.
Now the most influential people rarely if ever saw the torn heart of the city, much less gave a damn. It was the only place I felt at home. I drove the streets and the history of the place formed the effortless backbeat in my mind, memory stitching together memory. I was helpless against it.
Half a mile later, I reached Indian School Road and he called again.
"Go into the park and wait."
Steele Indian School Park stood on part of the grounds of the old Phoenix Indian School. The choice acres facing Central had been given to a big developer and, naturally, sat empty. The rest, meant to be a grand central park for the city, wasn't much. The imposing brick entrance signs were the most impressive features. Otherwise, it lacked the shade trees of old Encanto Park and the city never seemed to have the money or vision to make it into anything beyond sun-blasted grass with a couple of historic buildings saved from the Indian school and plenty of hot concrete sidewalks.
Today, it was nearly empty. A Hispanic family was having a picnic under an awning. I put the car into park and munched on a Jack Taco, waiting, watching. No other cars came or went. He could see me from the parking lot of the VA hospital to the east. Note to self: keep some binoculars in the car. Lindsey's Prelude smelled of fast food. My watch ticked around to five minutes, then ten, fifteen, twenty.
At twenty-one minutes, he called again and ran me around. My efforts to start a conversation were immediately cut off. I followed the instructions and drove east on Indian School to Sixteenth Street, south a mile to Thomas, and then back west into the core. Traffic remained light. If someone was following me, he was doing a very good job staying hidden. I thought about pulling off into a sidestreet, backing into an alley, and seeing if I could catch him behind me. But it was too big a chance. I stayed with his itinerary.
Another call: "Go to the McDonald's on Central. Pull into the parking lot facing east. I'll be there in a few minutes."
This next leg took me about five more minutes, back the way I had come before, a few blocks north of the punch card building. I drove to the east end of the long lot and waited. Ironically, the FBI offices were in my view to the northeast. Otherwise, it was acres of empty blight, adorned here and there with a dead palm tree looking like a giant burned matchstick. During the 2000s boom, before the biggest-ever collapse of Phoenix's only real industry, real-estate speculation, these lots facing Central were supposed to become twin, sixty-story condo towers. I don't think anybody ever believed it would really happen. It didn't.
As I drank cold water, the airplane came in low from the east, a small, single-engine private craft. It was flying very low. Dangerously low. Immediately before it passed to my right, it jettisoned something. That something fell straight down and landed in a plume of dust on one of the empty lots. I didn't need a phone call to make me speed a block to the landing zone. If I had that pair of binoculars, I might have gotten a tail number, but probably not. The airplane pulled up and disappeared into the sun.
I slammed the gearshift into park and sprinted into the empty lot. It was a stupid thing to do, but I was powered by a panicky instinct, adrenaline, and dread. Dust was still in the air as I approached a parcel little more than a foot long wrapped in brown paper and tied with twine. Something red was leaking through. My chest felt as if all its bones had suddenly collapsed.
That stopped me enough to return to the car for the evidence gloves that I had always kept there from my time as a deputy. I got the gloves and scanned the side streets: nothing. Then I saw him: a man was walking toward the parcel. He was tall and thin with stringy hair the color of urine.
"Stop!"
He ignored me. He looked like a homeless man, but I had my hand on the revolver as I walked back across the empty ground.
We faced off.
"My stuff!"
Was he really a street person or a watcher? I decided on the former.
"This is police business."
He looked at my PI credentials, not too closely thank goodness, and shuffled quickly toward Central.
I watched him go, then pocketed the wallet and pulled on the gloves. Call the police, call the FBI—this was what my interior voice was saying. I ignored it, dropped to my haunches, pulled out a small knife, and cut the twine. It might be another bomb, came the interior governor that had saved me so often in the past. I ignored that too, and carefully unwrapped the parcel.
It held a baby doll, covered in blood. It looked as if a plastic bag of stage blood had been inserted into the package so it would burst on impact.
"Hey!"
I looked up and the homeless man was fifty yards away, a maniacal look on his face. "You see me comin' on the street, it's lights out!"
I waved. My stomach felt as if it was going to climb out of my throat. Yeats was running through my brain, I have walked and prayed for this young child...
The doll's plastic smile mocked me as I pulled it out of the blood, seeing if the package contained a note. But there was none. The sun beat down on me as I realized the real baby was dead. An entire family wiped out on my watch. It was always going to turn out this way. The baby was going to die. Why did I think it could turn out otherwise? You can't bargain with kidnappers.
There had always been a chance to save the baby? Hadn't there? That we could rescue this child while the bad guys kept it alive and either bargained with us or prepared to sell it on the adoption black market? Hadn't there been a chance?
No.
I banged my fist into the dirt, catching a bunch of burrs that punctured through the latex into my flesh.
I had not reacted as a professional, but as a hysterical civilian. Now I needed that professional core to return and save me.
Scooping up the fresh evidence I was about to conceal, I carried it to the car and laid it in the trunk. I peeled off the sweat-filled, blood-and-burr covered evidence gloves and tossed them in, too. Then I waited in the car, blasting the air-conditioning on my face, for another half hour. The cell phone didn't ring again, no matter how loudly I shouted at it. Not one car appeared on the sidestreets. Finally, I pushed the anger inside and felt very cold. I called Peralta but went to his voice mail. Two things seemed clear: more than one individual was involved, the man who called me and at least one more piloting the airplane. And I needed to get into Grace's flash drive, find out what was so valuable.
There was nothing more I could do but drive back home, Yeats still in my ears, his great gloom in my mind.
An intellectual hatred is the worst.
I would find who did this.
And then kill them all.
The chilled numbness I felt deepened on the doorstep. The door was unlocked. I had gone off in such a hurry that I hadn't even set the alarm. If they were looking for the flash drive, they had come to the right place. I could walk back to the car and get help, but didn't. If they were careful, they had already seen me through the picture window. I had five bullets on my side against their automatic weapons or Claymore mines. Maybe they weren't careful and I would catch them searching. We would settle accounts.
I stepped inside. Heather Nova was on the stereo. I thought about Frank Sinatra's quip about committing suicide listening to Sarah Vaughn. Heather wasn't bad background music to die by.
"Is that you, Dave?"
Lindsey Faith Adams Mapstone was in the kitchen, on her knees scrubbing the floor in front of the refrigerator, her brown-black hair in her face. She rose and hugged me, and, after a long time, I put my arms around her, too.
Her voice was a whisper in my ear. "I have messed up so bad."
"Me, too."
I heard the engine of Peralta's truck roaring up Cypress before he walked in the door without knocking. He hugged Lindsey and I took him outside to the Prelude.
"I told you she'd come back," he said.
I ignored that and told him about the call, the runaround, the airplane, and its bombing run. The caller had referred to me as "Doctor Mapstone," exactly as Felix had done. He knew about me, the failed historian and the failed lawman. The newspapers had written up some of the big cases I had broken, but this wasn't an innocent informed reader. He had done some homework. On top of that, he admitted that he had set off the Claymore. All I needed to tie it up in a bow was for him to confess to killing Felix. Unfortunately, the only bow I had was the twine from the package with the bloody doll.
I opened the trunk. Sheltered by the shade, the heat left us alone.
Kicking at the driveway, I said, "The baby's dead."
He slipped on a fresh pair of evidence gloves and carefully examined the bloody doll.
"Let's not get ahead of ourselves." He leaned in, reading off the brand of the doll, noting the quality of the wrapping paper and twine. "Where's the flash drive?"
I told him about the hiding spot in the library.
"It must have something important if he's willing to put on this show," Peralta said.
I told him about my inability to get past Grace's pre-recorded greeting.
Half of his upper lip tilted up, a wide smile for him. "Lindsey can take care of that."
"Tell me we'll find these guys."
He raised up and studied me. "We will."
I followed him back to his truck, where he produced a garbage bag, then we returned to the Prelude, where he slid the evidence inside. I didn't want Lindsey to know about this bloody baby doll and what it implied. Anyway, was she visiting? Was she back for good? I didn't know. This case could only deepen her grief.
"Should I call the FBI?"
"No," he said. "In an hour, you'll have twenty agents setting up shop in your living room. Any chance we have will be lost."
"We don't have a chance. The baby is dead."
"No," Peralta said. I took no comfort from his tone of certainty. He went on: "If the baby was dead, he'd have nothing to bargain with. He did this as a warning. He wanted to throw you a scare in the most dramatic way possible. He'll call again. You ought to check out historic cases and see about bodies being dropped from airplanes."
Now he was trying to distract me.
As I leaned against the fender, which had cooled off enough that it didn't burn me, I thought about being eight years old, coming home from Kenilworth School full of joy to be free, catching the limb of a small tree outside, and getting stung by a bee. As Grandmother removed the stinger, I thought it was the worst thing that could possibly happen, it hurt so much.
Now that same tree was grown tall but my branch was sawed off. It was a long time coming. I was a surprise baby and then my parents died before I even knew them. No brothers or sisters, no aunts or uncles. So many times at the Sheriff's Office, I had eluded violent death. It was something our unborn child couldn't do and saving Lindsey's life meant we couldn't have another.
"Mapstone." Peralta knew I was too deep in my head. "I'll take the package to a private lab and get a workup. Let's go inside."
"What about Cartwright?" The thought had just come to me.
"Stop obsessing about the Edward thing."
"That's not what I mean," I said, unaware of any irony involving a man I was prepared to shoot the day before. "He might be dead or in danger. They're playing us, jerking us around. They kill people we talk to, but they won't come straight for us."
His heavy hand guided my shoulder toward the front door. "Ed will be fine, whether it's nuclear war or some dudes coming onto his property. If anybody was stupid enough to make a move against Ed, he'd have them buried in the desert within the hour."
Inside, he slipped on the pair of Bose earphones that Lindsey had gotten me for Christmas five years ago and listened to the voice on the recorder. He replayed it several times. Then he settled into the leather chair, put on his reading glasses, and studied my report on Grace Hunter. Lindsey was still cleaning the kitchen.
He folded his glasses. "I agree."
"Why would she leave her baby and go see Zisman?"
He ticked off scenarios on his fingers. She received a call like I had, perhaps threatening to kill Tim if she didn't go. Maybe not go to Zisman's condo. Maybe instructions to go to the corner to meet someone, and she had been taken.
Or Zisman himself had coerced her with some form of blackmail or reward.
Or an abductor had gained entry to the apartment and made her leave with him. A gun at one's back is a good persuader—could even make a mother leave her newborn baby.
Or she had gone willingly because she and Zisman were still lovers and Tim Lewis didn't know it, but something had gone wrong, and he had arranged to have her killed while he was on his boat.
"The last one seems improbable," I said. "She wouldn't leave the baby."
"We used to see that all the time," Peralta said. "Mother abandoning a baby so she can go party. Leaving them inside their cars in the summer while they go shop. Remember the mother who drove off with the baby still on the car roof?"
"It doesn't fit Grace, and not only the woman described by Tim Lewis, but the woman's actions. She started her own illegal business, where it required discretion and care. She did it for some time without attracting a pimp, and then she eventually gave him the slip. She wasn't an airhead."
"That's true."
When I waited in silence, he continued. "I was talking to her father."
"I'm surprised he would talk to you."
"I didn't give him much of a choice. When I went to his place, the housekeeper told me he'd gone hiking. He's one of these idiots who climbs Camelback every day, even in the summer. So I got a description and waited at the Echo Canyon trailhead for him. Someday Phoenix Fire will have to airlift him off if he keeps this up. He was so heat exhausted that I didn't have trouble getting him into the cab of the truck."
"You shouldn't have gone alone," I said.
"Why?" He snapped it out in a harsh tone. "I can take care of myself."
"That's not my point. We've had two clients killed and the murderer is at large."
"I'm not an old man, Mapstone. You handled the kidnapper's call on your own. I did this. We don't have an entire department backing us up any more. Sometimes we have to work separately to get results in a hurry. I sure as hell can take care of myself."
I shut up. It was the first time I had sensed that he was not as philosophical about losing the election as he appeared. My concern about him going alone, and my frustration that he hadn't been around to back me up, came off as questioning his abilities.
When he cooled down, Peralta described Grace's father: a self-made man, owning a successful company in Chandler that sold garage-door mechanisms. In a metropolitan area where big garages were almost as sacred as unlimited gun rights and red-light running, it was a very good business. The daughter he described to Peralta was smart, a National Merit Scholar finalist, but a young woman with a rebellious side. Her father had wanted her to attend Stanford, so of course she had chosen San Diego State. In retaliation, he had made her pay her own way.
"They didn't get along?" I asked.
"Didn't sound like it," he said. "The guy struck me as a prick. Chip on his shoulder. Sense of entitlement. And he's got a wife half his age, so he's desperately trying to stay in shape and be the extreme athlete, totally focused on trying to be her age. He's had work done, I could tell."
"She's not Grace's mother..."
"No. They divorced when Grace was a freshman in college and her mother found out dad had a girlfriend on the side that was his daughter's age. He said Grace blamed him for the divorce, but the parents had been fighting for years. Grace couldn't wait to get out of that house."
"The dad told you all this?"
"No," he said. "The housekeeper did. I don't know whether she's legal or not, but let's say she was a fan. 'My Sheriff,' she called me. She was happy to help."
"Where was the new wife?"
"Where else? The spa at the Sanctuary."
After the divorce, Grace had come home to the Phoenix less often, and had visited her dad less still. She hated the young woman who, in her eyes, had broken up her parents' marriage, and refused even to see her.
So her father was surprised and proud when Grace asked him for a loan to start her own business in San Diego. He was even happier when she paid him back.
"How did he seem to be taking her death?"
"Like a tough guy," Peralta said, "but I could tell it's eating at him."
"Did he bring up Zisman?"
"No, but I did. He claimed he didn't know Zisman. Grace never mentioned the guy to either parent."
"Or what her real business was."
"Right. But Grace had no known enemies and she was emotionally stable, even the housekeeper backed that up. She said Grace was the only nice person in the family. No history of suicide attempts. Later, I talked to her mother on the phone and it all jibed. The mother moved back to Iowa and hadn't seen Grace for a year."
"Do they think it was suicide?"
"They don't know what to think. The dad wanted to know who hired us, and of course he had never heard of Felix Smith. They didn't know about her boyfriend, either."
"And they didn't know they were grandparents?"
He shook his head.
Her mother last spoke to her on the phone the day before she died and Grace said she wanted to tell her some good news. She said it was a complicated story. But her mom was at work, so they decided to talk about it the next day. But the call never came."
That made me even more suspicious: additional witnesses that Grace was not depressive, not suicidal. And a phone call promising good news: I assumed that meant telling her about her new baby. This was not a woman who killed herself.
"We have missing time to fill in," I said. "On April twenty-second, Grace was gone when Tim returned at three that afternoon. She didn't die until nearly midnight. None of that time was spent calling her mom in Iowa. So what was she doing?"
I also didn't like the cell-phone situation. Someone Grace's age couldn't live without constant texting. And yet she had a new cell with nothing on it. I looked once more at my phone, willing Mister UNKNOWN to call again. He didn't.
"San Diego PD will re-open this as a homicide based on your report," Peralta said. "It will take time, but they can find her other phone records."
"We don't have time." My temples were starting to ache from stress.
"Maybe I can help." Lindsey was behind me.
I didn't know how much she had heard. But I didn't want her anywhere near a case that involved what would no doubt be a dead baby. Yet before I could speak, Peralta said, "That would be great, Lindsey." To me, "Give her the flash drive with Grace's clients. It's encrypted."
"I need to go to the Apple store at the Biltmore," she said. "And a Radio Shack. Then I can get started."
I handed her the keys to the Prelude.
By the time Lindsey returned, Peralta was gone. She set a large bag down on the desk where I was working.
"I want to see the garage apartment."
I wasn't sure that was a good idea for either of us. That had been Robin's space, where she had lived with us after coming back into Lindsey's life following a long absence, lived for two years rent-free after she lost her job as curator of a man's art collection. He lost it all in the real-estate collapse and she was looking for her next adventure. I hadn't been up there since her death.
"I want to see it," Lindsey insisted.
I tried very hard not to sigh. We walked up the staircase, bookshelves on one side and a wrought-iron railing on the other, to the landing that overlooked the living room, then across the walkway above the interior courtyard where Lindsey's garden had sat neglected. I fumbled with the keys and opened the door.
Heat greeted us so I turned on the window air-conditioner. It was a simple space, one large room with a bed and a couple of chairs, an alcove for a little kitchen, and a bathroom. A back door led to an outside staircase on the north end of the building. Grandmother had kept her sewing room up here when I was a child. Robin had added several social realism posters—her specialty in art history—and two of her own oil paintings, abstracts with geometric lines and vivid colors, illuminated by the afternoon sun. Her easel stood in one corner, an empty canvas on it.
Lindsey walked around, lightly touching the edges of the paintings. Opening the closet, she examined Robin's clothes, holding a blouse up to her face.
Her dark hair grew fast and it was now down to her shoulders with bangs added. It fell thick and pin-straight. Women would kill for Lindsey's hair. The edge of it brushed around the nape of her neck as she ran a hand against Robin's clothes. They would kill for her fair skin and the lovely contrasts between dark hair, fair skin, and blue eyes. She looked familiar and yet a stranger. In so many ways, I did not know my wife.
So many times, I had imagined what was next for us, wondered whether I even wanted her to come back. It was a terrible thought, but she had left me once before, when we were first dating, and it had lacerated my heart. She had come back on Christmas Eve and that lyrical return had become a part of our story.
This time when she left, after losing the baby and taking the job offered to her by the governor of Arizona who was becoming Secretary of Homeland Security, the story turned darker. I tried to understand her need to grieve. She had to get out of this house, she had said at one point she didn't know if she could ever stand to be here again, and yet here she was. I tried to understand and yet I had been hurting, too. It was our baby that was dead, not only hers. In the blurred months she had been gone, we had talked nothing out and I had given up trying. I didn't know if I wanted her to come back because I didn't know if I could open up to the pain of another abandonment. The fortifications I had built against her were not strong.
She turned and studied me, a big smile playing on her sensual lips. In her eyes was a nothing look that was densely underlain with meaning. It lasted a few seconds.
"Did you fuck her there?"
She nodded toward the bed.
Before I could answer—the answer was no—she strode quickly over and slapped me. The blow was so hard it brought little lighted planets and asteroids to the edge of my vision.
"Did you fuck my sister in that bed?" The smile was gone and her eyes were burning violet with emotion.
She was about to deliver another blow but I caught it. She was strong as hell. With her other hand, she shoved me against the wall and mashed her mouth roughly against mine. You could call it a kiss if you called knuckle-breaking a handshake. I twirled her around and slammed her back into the plaster and our tongues fought. She was making sounds that were half whimpers, half snarls. We were both sweating. Buttons were popping off my shirt. I was jerking her jeans and panties down despite the fact that not a millimeter of space separated our half-wrestling, half-embracing bodies.
We both fell on the floor and the rest of the clothes came off. What happened next was the angriest lovemaking I could ever imagine. The hardness of the floor was apt. A bed would have been out of place. Her hair fell into my mouth.
Usually there was a part of me standing outside every interaction observing. That me fluttered on the perimeter for only a few seconds, managing to remember Chrissie Hynde lyrics about love and hate and the thin line, wondering if I was taking my woman back or she was taking her man back, noticing that my formerly inhibited Lindsey had been practicing moves and not with me, or if any of this even mattered beyond these minutes and that rough floor, and then the observer was sucked back inside our primal bout and lost. She moaned "fuck me" among many untranslatable sounds.
Her orgasm was more intense and longer lasting than any I ever remembered, and finally she collapsed on top of me. The only familiar gesture was her tucking her feet under my legs.
In the silence, I could only hear the fronds of a palm tree brushing against the window. Spasms ripped through my back as my San Diego dive caught up with me, and my face was still stinging from her slap. All other thoughts had been torn away like our clothes.
My shoulder was suddenly wet. And then she started sobbing, in heaving, loud convulsions that seemed too big to be coming out of her slender body. All I could do was hold her and stroke her hair. She initially resisted even as she mashed herself against me, until she finally gave in and held me too. Her arm wrapped so tightly around my neck that I almost passed out. It was a long time before she was simply crying.
Afterwards, we lay side by side, one of her long legs over mine. The room was finally cooling down. Somebody could have come in and killed us right then and it would have been okay.
She retrieved a pack of Gauloises out of her wadded up jeans and lit one, then exhaled a long, blue vapor trail. Tobacco mingled with the pervasive scent of sex.
I followed the smoke out into the room, down her lovely body, past our reddened knees, and noticed the tattoo on the top of her right foot. The word "Emma" was surrounded by brambles.
It made me smoke one of her French cigarettes, too, even though it would probably make me slightly ill. I am an old guy, so in my mental world body art is confined to Melville's whalers, real sailors and enlisted Marines, and trailer trash. It is elitism out of step with the age, but I find tattoos barbarous. And here was one on the perfect fair foot of my wife. True, she had worn a small stud in her nose when we had first dated, but that was years ago and Lindsey was no longer twenty-eight. This made her feel more alien and distant from me.
The tattoo's provenance was no mystery: Emma, at least for Lindsey, was our lost daughter. Emma wasn't a name I would have chosen. We didn't even know the gender of the baby.
It was better to make light conversation as she lay against me, both of us staring at the ceiling.
"How was the Apple Store?"
"I got a new laptop," she said. "And other stuff."
I had never seen Lindsey travel without a computer. "What happened to yours?"
She blew a smoke ring, then a second.
"They confiscated it after they took away my security clearance and fired me."
Lindsey dropped me at the office the next morning. Even though it was ten, I had beaten Peralta there again. I was so sore from the various explosions in my life that my first few steps were like an old man's. I wasn't complaining about the ones that involved Lindsey but I was out of practice. The night before, I went to bed while Lindsey worked on her new computer. She had claimed a space on the landing above the living room and sat cross-legged with her back against the wall. When I was a child, the stairs and landing had seemed exceptionally high. Now, having grown to six-feet-two, I could touch the landing with my hand. Such was perspective and context.
Sleep hadn't come easily, so I was still awake when Lindsey had slipped in bed and curled up against me. It was so much like what Robin had done that first night that it kept me awake even longer. At first I thought my dreams had turned into a hallucination. But, no, it was Lindsey. Robin was taller and bustier. We fit together beautifully. Robin was dead.
Lindsey woke me from two nightmares, but when she wanted to know what I was dreaming, I said I couldn't remember. Hearing about other people's dreams was as tedious as watching their vacation videos and Lindsey sure didn't want to know about my dreams lately.
Around five, we had sex again, this time without the anger, but she was as loud as her half-sister, something new about my wife.
We used to play a game over cocktails. Lindsey had been endlessly entertained about my adventures before we got together, but she had drawn the line at knowing about my former girlfriends. It was better for her mental health not to know, or so she had said. As we had enjoyed martinis, I would tease her: "I'll tell you anything, all you have to do is ask."
"No thanks," she would say.
When I had asked about her life, she would say, "I lived a boring life before you, Dave. There's nothing to tell." I had never believed that, even though I was older than she and had lived perhaps more adventures, but she didn't talk easily about herself. I knew she had grown up in chaos, run away to join the Air Force where she had learned computers, and had claimed one boyfriend before me. Perhaps this was even the truth.
Now I wondered how much I wanted to know about the past months of her life. I imagined her boyfriend in D.C. as wealthy, handsome, and definitely better endowed than me. Maybe he was a black guy. Maybe her lover was a woman. And now I knew this person had mined a deeper lode of sexual passion from her than I had ever been able to reach. For that to happen, a woman had to be willing to really let her lover in, really open herself. She had not done that for me. I didn't realize it at the time, but the past twenty-four hours had shown me different. Did I really want to know about those past months?
After we lost the baby, Lindsey could barely endure being touched. That changed yesterday as we bounced the historic floorboards in the garage apartment. My wife, who had never even used the word "ass" before, was now talking dirty during sex.
I supposed I should thank the son of a bitch.
Now I was slipping my report on Grace Hunter into a file folder when the office phone rang and the readout was a San Diego area code.
"This is Detective Sanchez with the San Diego Police," came a pleasant voice on the other end. So Isabel Sanchez was going to talk to me after all.
"How may I help you?"
"How about opening your gate so I can come in."
This was not good. I wished Peralta were here but pressed the button to open the gate.
The night detective was about five-four with a size two figure, dark eyes with long lashes, and long, black hair that looked as if it had caught a gust off the Pacific at that exact second. Her pregnancy was also beginning to show. The man with her was a few inches shorter than me but very buff with yellow surfer-boy hair. San Diego had the best-looking cops in the country.
"This is my partner, Detective Jones," she said. I invited them to sit down, thinking: sure, Jones—he probably had multiple IDs and aliases, too.
"Deputy Chief Kimbrough speaks highly of you," she said.
"That's nice. He's a great cop."
"That's why we're not filing a charge to ask our friends in Phoenix to arrest you on," said the pleasant voice.
So it was going to be like this.
Several charges came to mind, but she wouldn't know about those.
I said quietly, "I was a victim of a crime in your city."
"I can understand how you might still feel badge-heavy, Mapstone," said Jones, who, with his mean little eyes, looked exactly like a badge-heavy cop. "But you're not a deputy sheriff anymore."
We went through this small talk, all designed to get a rise out of me, for about ten minutes. None of it worked. Jones gave me the cop stare. I returned it with the amiable look of a concerned civilian. I didn't even feel the need to bring up their rushed and shoddy investigation into Grace's death. From their attitude, it seemed clear that Kimbrough had already done that, based on my report that Peralta had emailed to him yesterday.
Sanchez said, "Grace Hunter phoned your office the day she died."
I looked at her evenly, which probably made her more suspicious. But this was the way I always reacted to shocking news. It took me a moment to deny it, but then she produced a copy of the LUDs—local usage details—from Grace's phone.
She handed me the sheet. Sure enough, our 602 area code number stood out, call placed at four-ten p.m. on the day she died. The call lasted two minutes. I memorized Grace's phone number to write down once they had left.
"Care to explain?" Sanchez looked at me sweetly.
I cared a great deal and had no explanation. I turned on my laptop and opened up the office calendar. It showed that Peralta had given a speech that afternoon at a law-enforcement conference. We hadn't been in the office when the call came in. No one had left a message. Sanchez walked around, looked over my shoulder, and examined the listing.
"How long did you know Grace Hunter?" Jones asked unsweetly.
"I didn't know her when she was alive. There was no message left here. You can see from the LUD that it was a quick call. It probably rang to the answering machine and the caller hung up."
Jones leaned forward in his chair. "Want to try again?"
"No."
We sat for a good five minutes with only the sound of the air conditioner to keep us company. I struggled to maintain my agreeable, relaxed look, but the reality was that it sucked being on the other side of an interrogation. I wasn't used to it. This would be a good time for Peralta to arrive.
"It seems too coincidental," Sanchez said, walking in a circle around the office, studying the large, framed maps of Arizona and Phoenix that I had bought at Wide World of Maps to decorate the place. "You go to San Diego and find her husband, Tim Lewis, murdered. His apartment blows up. Now we know that Grace Hunter called you before she died."
"If she did, we didn't know that," I said. So they were married. "And Tim was a client. He asked us to look into her suspicious death..."
"We know all that." Detective Jones dismissed me with a chop of his hand. "We found your receipt in the blast debris. Hand written on blank paper and signed by you. Real professional operation you have going here, Mapstone. No answering service. Hand-written records."
It was my turn to lean toward him. "We all have our shortcomings, Jones. Like when Tim filed a missing person's report on Grace with your department and nobody made the connection that she was already dead and misclassified as a suicide."
Jones' ears started turning red.
"Wait for me in the car, Brent," Sanchez said. He noisily pushed back the chair and slammed the door behind him.
She leaned against Peralta's desk and watched her partner leave, then turned her head toward me.
"Are you the good cop?" I asked.
"Dream on. So no call from Grace Hunter?"
"We never talked to her."
"But she called you."
"Somebody called here with her phone. She was found dead with a new phone that didn't have any called numbers on it. That was in your report."
Sanchez persisted. "Why would this somebody call here?"
I told her the truth: I didn't know. Maybe it was Tim, using her phone. Considering he didn't know she was dead when I first met him, that seemed unlikely, but no need to tell her that.
I didn't say how this call to our office indicated that whoever killed Grace, set off the Claymore mine, and took the baby had made that call to frame us, or at least slow us down, knowing the police would track the LUDs. This had been planned well ahead of the moment Felix walked in that door.
The only alternative was that Grace herself had actually tried to call us. But why? She didn't even know us.
"I can make your life miserable." Sanchez sat in the chair in front of my desk, crossed her legs, and placed long fingers protectively across her belly. "Losing your license will only be the start of the hurt I can put on you."
"I don't doubt it," I said. "But Kimbrough and Peralta go back a long way, and you've got a bungled investigation on your hands. Let me ask you a question, if you don't mind: you pulled Grace's LUDs. Do they match with the phone found in her purse that night?"
Sanchez deflated by degrees. Even her hair deflated.
"No. They don't match. The phone she was carrying that night was scrubbed clean of recent calls. We traced it to a seventy-year-old woman who lives on Clairemont Mesa. It was stolen from her in a purse snatching at Fashion Valley mall."
"So whoever pushed her off that balcony took her real phone."
She nodded.
"How is the hunt for the baby progressing?"
She forced her expression to harden. "That's confidential law-enforcement information and you're only a private dick."
Robin's words again. I stifled a smile.
"Come on, Isabel. You don't have to mimic your jerk colleague."
Two beats, three.
Then: "We don't have anything. Not a damned thing. If I had known she was married or had a kid..." She shook her head. "The vic didn't have any of that information in her purse. Her parents didn't tell us, either."
"I understand." I thought about the wall with our names painted in blood, information I had held back for our protection, and asked about fingerprints.
"The apartment was destroyed. It could take ATF weeks to sort through things and see if there are any usable prints." She cleared her throat. "What do you make of Larry Zisman?"
I laid out the backgrounding I had done. Among a certain group, people who had lived here a long time, Zisman was still beloved for his college-football days. He was a razzle-dazzle quarterback in the glory years of Sun Devil football. He left less of a mark in the NFL, playing for five teams before being forced to retire early.
Zisman was a native Arizonan, attended the old East High School, and came back here to live after he retired from the NFL. Not only that, but to live year-round, not only keep a casita at one of the resorts for the winter months. He had started a non-profit to fund athletics for inner-city schools. He was in demand to give speeches at Kiwanis and Rotary, but removed enough from celebrity to be under the radar in a city with so many comings and goings.
"Did it surprise you that he had a lover on the side?"
I held out empty hands. "Who ever knows? But, yes, a little. From what I picked up, Larry Zip was so full of clean living that he might have been mistaken for a Mormon."
"Do you think he killed Grace Hunter?"
"He's physically capable of it. Former athlete. As a reserve officer, he would have gone through police academy training."
She made a few notes.
I said, "It would be pretty stupid, though, to push her off his own condo balcony. He'd know that he would be the prime suspect. Better to strangle her and dump her body in the East County."
"Unless," she said, "it was an act of passion and he did it in the moment."
"Right. But then you have the problem of the alibi, of him being on his boat."
I was only trying to be convivial enough to get Detective Sanchez out of the office. This couldn't be a mutually beneficial relationship because Peralta and I were concealing critical information. We had dug this hole a little scoop at a time, for good reasons at the moment, and now we were in deep. Too deep.
She thought about what I had said regarding Zisman, twirling a strand of her hair.
"I think he could have done it."
"You interviewed him that night and cleared him," I said.
"I read your report," she said. "After our ass-chewing from Kimbrough and before we got on the plane, I dug a little more. The man at the next boat is a good friend with Zisman, you know. He's from Arizona, too. You people really need to find another summer escape. The man is a developer who used Zisman as a spokesman for some of his properties. He might be lying for him."
Zisman hadn't figured in any of my theories about the case—not that I had formed many yet. I had been focused on getting out of that apartment before my body was turned into an aerosol state, and then on examining whether Grace had actually committed suicide.
"What about Tim?"
I cocked my head.
She went on. "Maybe he followed her to Zisman's condo and found out she was cheating on him. Oldest motive in the world."
To me, he barely had the guts to change a baby's diaper, much less kill his wife or have the strength to do it in such a physical manner. Sure, people would surprise you, especially if money or sex were concerned. If so, he would have had to do a good job feigning surprise and sorrow when I told him Grace was dead. And been tough enough to slit his own throat and wire his apartment to explode.
I remembered a case in Scottsdale years ago, where a man cut the throats of his family, shot them, set the house on fire, and blew it up. They never caught him.
Detective Sanchez also didn't know that our names had been written in blood on the apartment wall. Tim Lewis didn't do that in the seconds before his carotid arteries bled out. Then there was yesterday's phone call, Mister UNKNOWN saying he had detonated the Claymore and with his aerial theater implying he either had the baby or had murdered it.
"Tim was genuinely torn apart when I told him Grace was dead," I said. "And remember, the pimp was beating him up when I got there. And if Tim was Grace's killer, who took the baby?"
She sighed. "I wish I could keep things simple. Occam's Razor, right? My ass is on the line for this now, and there's a hundred local, state, and federal investigators living in my shit because of that explosion and kidnapping."
I appreciated a woman who could quote the classics, but this was one instance where the least complex hypothesis wouldn't do.
"The pimp is Keavon William Briscoe," she said, spelling the first name. "He's middling, not a big player. This is a guy who provides prostitutes for sailors and Marines on leave and runs streetwalkers, not escorts for big-time executives and legislators."
"He claimed Grace worked for him."
"Maybe she did. It wouldn't be the first time a coed made some money on the side. The reason I don't like Briscoe for this is that he was in jail on the night of April twenty-second, a parole violation. He had a baggie of pot in the car. He'll probably go back to prison but it gives him an alibi for the one-eighty-seven." The homicide.
"How did he find where she lived?"
"That's the thing," she said. "He was cruising O.B. on April twenty-first and said he saw her, followed her home, and was driving around the block for a parking space when a marked unit stopped him and arrested him. His sister didn't bail him out for several days."
"Did you execute a search warrant?"
"Don't piss me off, Mapstone." The dark eyes deepened. "I usually don't fuck up cases. Yes, we gave his place a total colonoscopy and didn't even find a cheap gun, much less explosives. That brings me back to Zisman. If Zisman found out that Grace was tricking on the side, he would have even more motive to kill her. Maybe it's his baby. Maybe he has access to military explosives."
I nodded, but I had seen this so many times: a detective latches onto a theory and does whatever it takes to make it stick and clear the case. Back when I untangled cold cases for the Sheriff's Office, this was often the original sin in what turned out to be an unsolved case, or worse, one that sent an innocent person to prison.
I also appreciated the heat she was feeling from the brass.
Sanchez didn't know the full extent of Grace's entrepreneurship. It sounded as if she was unsure if she had even been a real prostitute or only a wild child.
"What about her friend, Addison?"
"Addison Conway," Sanchez said. "Jones talked to her. She went back home to Oklahoma at the end of the semester. Grace hadn't made a call to her since March."
"So did Zisman and Grace have contact the day of her death?"
She sighed. "It's not in the LUDs. I went back through two years of records and didn't find his number. Grace called her mother on the twenty-first. She received a call from the human resources department at Qualcomm that same day. She called your office on the twenty-second. That's the only call she made on the day she died. The other thing is, the semen inside her doesn't match Tim's DNA. In fact, it shows evidence that she had sex with three different men, but none of them her husband."
The information exchange was definitely working in my favor. I was processing it, thinking out loud. "Grace had gone to a lot of trouble to drop out and get away from guys like Larry Zisman..."
A big smile played across her face. "Until she needed him. Come on, Mapstone, don't be naïve. Babies are expensive and there's college coming right up on a parent. You probably have kids, so you understand. She hadn't even started her job at Qualcomm. Her bank account was drawn way down, only six hundred dollars."
I wondered if they had checked all her bank accounts, but said nothing.
Sanchez continued: "What if she showed up at Zisman's condo unannounced and wanted money? Former pro football player—she's got to figure he's loaded. Pay up or I'll tell your wife. Better than that, pay up or I'll tell your wife I had your baby. Zisman loses it and tosses her off the balcony, goes to his boat, and has his friends cover for him."
"Wouldn't Grace have been seen coming into the lobby? Or him going?"
"The night concierge didn't come on duty until eleven," she said. "Nobody was at the front desk for eight hours that day. They've been having staffing problems. In San Diego, 'sunshine dollars' only go so far."
I thought back to our visit to the condo. "But the building has a card-key entrance. Nobody could get in without using the card."
"Unless somebody coming in held the door for them. Anyway, after the body hit the concrete, the concierge runs out to the pool area. So if Zisman left, nobody would see him."
"Cameras?"
She shook her head. "The lobby cam was broken all week."
It didn't seem so neat to me. But the former football hero was in her radar lock.
"Have you interviewed Zisman again?"
The luminous black hair shook. "He's not answering his phone. But I've got a lot of questions when he resurfaces."
I still wondered about the missing hours in Grace's day. I said, "Why would she leave the apartment without telling Tim?"
She shrugged. "Men aren't the only ones who lie about sex."
Peralta still wasn't at the office when I had finished writing up the notes from my meeting with the San Diego detectives, and I was starting to worry, which was silly given Peralta's ability to protect himself and others.
My concern was forgotten when I buzzed open the gate for Lindsey. It closed and locked automatically after she pulled in. Lindsey in a miniskirt would chase away every concern, to be shamelessly shallow about it. She also carried lunch and my new iPhone, which FedEx had delivered that morning. After putting down the bag, she gave me a kiss and a hug that seemed almost normal. Her hand went up inside my shirt across my belly and onto my chest.
"Was this Robin's?" She touched the cross of Navajo silver.
I hesitated, then nodded.
"May I have it?"
"Of course." I removed it and slipped the chain over her head. She bent toward me as if receiving some kind of decoration.
The Order of the Lost Sister.
I pulled the cross around to fall above her breasts and fluffed out her hair.
"Thank you." She was trying not to cry, so she made herself laugh. "This way it won't tickle me when you're on top."
I tried to hold her, but almost immediately she dropped to her knees and started unzipping my slacks.
"Lindsey." I pulled her up and hugged her. "Just be with me."
"Yeah." Her voice was one notch above a whisper but I heard the sardonic tremolo. She was barely with me. Lindsey's body was in my arms but Lindsey was somewhere else. This appearance was conditional. She wasn't wearing her wedding band. It wasn't her fault. All she had of her child was a tattoo.
A few days ago I had nearly died, despite the claim by UNKNOWN that he waited before detonating the mine. I remembered the chunk of wall torpedoing into the pool inches from my head. I was living on bonus time but did she care? She had said that she had messed up, but maybe that meant getting fired, not leaving me. Lindsey, just be with me. What a damned fool I was.
I pushed her over to the desk, kissing her, caressing the soft skin beneath the hem of her shirt. After enough kissing to feel her body relax and even wilt, I lifted her onto the desktop, removed her sandals, and slid off her panties. Sitting in the chair, I started sucking her toes and licking her perfect ankles, slowly working my way north with my mouth and tongue. The fabric of her miniskirt tickled the top of my nose. She didn't resist. I held my arms behind her so she could lean back against my hands. She clutched my head with her hands, bent her knees, and rested her warm feet atop my shoulders.
Circles and slides and figure eights. Cheerleader legs. I played her, made it go on a long time, loving being so connected to everything she was feeling, loving giving her pleasure. I even knew when she was ready to intertwine her hands in mine, gripping me for the grand last movement.
Afterward, she slid into my lap and this time didn't resist being held.
"I love you." I couldn't help myself. It came out involuntarily.
She didn't say anything, but nestled closer.
I was a fool. The Bettye LaVette song played in my head: Everything Is Broken.
Sex would keep anxiety and time and death at bay. I never have panic attacks if I am getting laid. I had to be satisfied with this eternal truth for the moment. But sex with Lindsey made me lose focus, made me forget, made me fall in love with her again, ensured that I might withdraw my emotional siege machines.
Steps on broken pavement.
The sound was so soft I wasn't sure I had even heard it over the periodic whoosh of cars on Grand. Lindsey noticed my expression and I held up a hand. Someone was walking across the lot, very slowly. It couldn't be Peralta, whose entrance was announced with the alert of the gate opening, followed by roaring engine and bumping suspension. My blood stopped pumping for a couple of seconds. Someone had jumped the fence, no easy maneuver. It could be anybody. The office door was unlocked.
Mail, she mouthed?
I shook my head. The mail lady came later in the afternoon and the gate was locked.
"Get under the desk."
She didn't question me and scrambled into the cave where my legs would normally go. I pulled out the Python, dropped to my knees, and stayed close.
"Are you armed?" I whispered.
She shook her head.
I slipped the Airlite from my pocket and handed it to her.
The only fancy furniture in our office was our chairs and the leather sofa. Otherwise, most of the rest was second-hand, including the two heavy Steelcase desks that looked as if they had once been part of a 1960 secretarial pool. You could fire a rocket-propelled grenade at them and barely make a dent.
I waited for the door to open. Maybe the gate had somehow jammed open, an innocuous malfunction, and the footsteps belonged to a new client, a traveling salesman, or a Jehovah's Witness who would knock and say, "Hello, is anyone here?"
The room was silent.
I didn't dare move to catch a glimpse. The desk sat so close to the ground, I was confident that if someone did come in he couldn't see us. That would change if he walked behind Peralta's desk, or toward the Danger Room. By then, I would have him in my gun sights, unless he was prepared.
If I get hit, come out blazing, I telepathed to the frightened blue eyes watching me.
The floor was old and creaked when you walked on it. The hinges squeaked when the door opened. But nobody tried to enter. The sound of footsteps came again, this time from the carport. Whoever had come into the lot was still out there. The palm of my hand was sweating into the custom combat grips of the Python.
Then, nothing.
I had to let a good five minutes pass before I dared slither out on the far side of the desk, ready for action. But no one was there. Waiting was the safe way. But it also ensured that I couldn't see if our visitor had a vehicle. For that matter, I also couldn't get a license tag number. We waited. Finally, I stood and locked the door. Peering out the blinds, I could see the gate was indeed shut.
Not long afterwards, Peralta arrived, sweeping into the room like a parade.
"Lindsey."
"Sheriff."
She was sitting on my desk. I stopped stroking her knees, said nothing, and resolved to avoid his glance.
"Lindsey!" Sharon's voice. I looked up, and she walked in carrying a bag of hot dogs from Johnnie's on Thomas. This was fun food.
As Lindsey and Sharon embraced, Peralta's eyes found mine, and he knew what we had been doing, and his eyes actually twinkled like a tough Saint Nick of nooners. I felt my face flushing.
"We're all here together, like it should be," Peralta announced like the paterfamilias. As if anything were settled. "So let's eat and get to work!"
Lindsey had fixed us healthy salads, to which I added a Chicago dog from Johnnie's.
"He's too gaunt," Sharon whispered to Lindsey.
I told Peralta about the visit from the San Diego cops and the mystery guest who had been in the parking lot but never came in. His forehead tightened as he listened, but he only dived into lunch.
Peralta, with his mouth full: "Sharon talked to Tim Lewis' parents," which I translated from shawob awked a wimoois barents. It had taken many years of listening to Peralta over breakfasts at Susan's Diner and lunches at Durant's to master this particular dialect.
I said, "They talked to you?"
"I used my winning people skills," she said, pulling a chair closer to his desk as she ate her salad like a lady. "Empathy, trust, respect..."
"She flashed her credentials," Peralta said, amazingly pausing in his eating. "Show them."
She held a wallet identifying her as a police psychologist for the San Francisco Police Department.
"After being married to him for thirty years, who could be more qualified?" She winked at him.
"Plus, Tim's mother had all of Sharon's books," he said.
"As I was saying..." Sharon reclaimed the floor, and Peralta, uncharacteristically, shut up. "The mother's name is Vicki, father named Mike. They were both there, a nice couple, and were very generous with their time considering all they've been through. They're devastated by Tim's death and sick about their grandson. The police have tapped their phones, but they haven't heard anything, much less a ransom demand. They don't understand why anyone would have killed Tim or Grace."
I actually swallowed my food before speaking. "So they knew Grace?"
Sharon nodded. "They met her when she and Tim first started dating. After they got together again, they saw her more than a dozen times, including at their wedding, which was held in Riverside, and when she gave birth. They loved her. That was the word each one used."
I listened to Sharon and was so glad to see her. She was a couple of inches shorter than Lindsey's five-seven, but was still in great shape with the black hair and angelic face off a tapestry in a Mexican church. In a way words couldn't describe, she centered our world. I had known her when she was a young, uncertain mother, then as she put herself through college and graduate school, not always with Peralta's emotional support. This had been one of the old battlegrounds between Peralta and me. Then she had hit it big and finally she had divorced him. But apparently "finally" had a second act.
She said what we had heard before: Grace was stable, not suicidal, and had no enemies. Tim's childhood sounded suburban normal, the kind that produced golf pros or lone mass shooters. And Grace had done a very good job of keeping people from knowing how she had made money working through college.
"They didn't have a clue," Sharon said. "But the world of high-end call girls can be very different from the sexual exploitation you find with streetwalkers or immigrants from Eastern Europe who thought they were getting a trip to America for a job in a factory and it turns out to be a very different kind of assembly line. What Grace was doing was even more specialized, working on her own. Most work for agencies. But powerful men will pay very well for the services."
"I bet." Peralta licked his fingers. Sharon shot him a civilizing glance and he stopped, using his napkins instead.
"These men pay for the sexual skills, no question. The more versatile, the better. They think they have a woman in her sexual prime who really wants to have sex and enjoys it. Many of them are narcissists who want a beautiful young woman on their arms. It's a prestige thing. If he's an executive, it's gotten too risky to hit on subordinates. So a discreet hooker is the thing."
Lindsey said, "Is it only about the prestige and the sex?"
Sharon shook her head. "Many of the johns also want an emotional connection that they feel they aren't getting from their wives. If Grace was all these things, plus polished, cheerful, intelligent, sophisticated, and romantic, then she could get top dollar. In San Francisco, I met call girls who were getting more than five thousand an hour."
"An hour?" Peralta raised an eyebrow.
"Yes."
"Considering she was their daughter-in-law," I said, "it's better that they didn't know her past."
I certainly didn't know Lindsey's recent past.
Sharon said. "They liked being a family to her. It sounds as if Grace's mother was totally self-absorbed and her father was even worse. Tim's parents went to her graduation last year. Neither of Grace's parents did. Her father was at a golf tournament with his buddy, some washed up pro football player."
I stopped in mid-bite and pushed the hot dog away.
"Larry Zisman?" I asked.
"That the name," she said. "He was a star for the Sun Devils back in the seventies. I remember."
Occam's Razor, indeed.
Peralta attacked his second chili dog with more aggression than usual. A Scottsdale McMansion of possibilities had opened up. One room contained the obvious, that Zisman was a client. Another held the possibility that Zisman had hit on his buddy's daughter and gotten it for free. The rest of the floor plan was too twisted to think about over lunch.
I said, "Maybe Zisman wasn't her client."
"He wasn't," Lindsey said.
Everybody turned to her.
"It took me about two minutes to break into that flash drive," she said. "It contained an Excel spread sheet with sixty two clients: names, Social Security numbers, driver's license numbers, dates, and amounts. No Larry Zisman."
Nobody took a bite.
I said, "The johns gave her that information?"
"They would have to do that for an escort agency," Sharon said. "It helps ensure safety."
"But," I said, "Zisman knew Grace's father and covered it up."
I could see the slow burn on Peralta's features over Hunter lying to him. Maybe Grace's father didn't know his daughter had been intimate with Larry Zip. That was the most charitable explanation. But he sure knew that Grace had fallen out of Zip's condo, and yet he hadn't admitted their friendship either to Peralta or Isabel Sanchez.
"There's something else." Lindsey nodded toward our front parking lot. "The Prelude has a GPS tracking device tucked inside the front fender. You can buy one in any spy shop."
My legs and feet felt very heavy on the floor.
"What did you do with it?" Peralta asked.
"I left it there." She ate a bite of salad and dabbed her lovely, orgasm-flushed face with a paper napkin. "If they don't know we found it, we have an edge. From what Dave says, San Diego PD has a hard-on for Zisman now that Dave's shown that Grace didn't kill herself. Maybe we can work with them."
"They're not going to work with private detectives," Peralta said.
After a long silence, I looked at him. "These scumbags have had the upper hand from before Felix walked in that door. They placed a call to our number using Grace's cell phone so the cops would be suspicious of us. I'm tired of playing defense. What's our next move?"
He inhaled and rose up in his chair. "I've heard a person's cell phone can be tracked. Not only the calls they make and receive, but the locations of the user at any point. Is that true?"
"Absolutely," Lindsey said. "Wherever you go, your cell phone sends data and it's mapped. And the cell providers keep those records. So somebody could find out Grace's moves on any given date." She paused and looked into her lap, and then she pushed her hair out of her face. "These companies have very sophisticated security and firewalls."
"Can you hack it?"
My appetite fled. I stood and stalked the six feet to his desk. "I can't believe what you asked her to do. That's a federal crime."
He shot up out of his chair and stabbed a finger at me. "What's your plan, Mapstone? Get blown up again? You might not be so fast next time. We've been played for chumps and our clients are dead. Do you know why? I don't. What I do know is it's only a matter of time before we're dead, especially if they get that flash drive."
"Then we'll take them on. Why bring Lindsey and Sharon into it?"
"Because they're already in it with us." He spat the words. "These assholes are cleaning up loose ends. Tim and the baby were loose ends. Why do they have a tracker on your car? Because they're afraid of you? No. So they can find you and kill you when the time is right. Who's going to help you? Your new buddy, Isabel? Not when she finds out you've been withholding evidence."
He wasn't the only one running hot. I went from zero to asshole in three seconds. I barked, "Lindsey could go to prison! Put your own ass on the line. Put mine. But leave her out of it! Let San Diego PD track Grace's movements. Somebody cased our office. My god, are you nuts? We're not safe here. We're not safe at home. You said it yourself. We're loose ends."
So much for our convivial reunited family.
And then Vesuvius went dormant. He sat back in his executive chair and pushed his hair back with both hands. In a conversational voice: "We are safe as long as they are willing to bargain for the flash drive. That's our hole card. They want it badly. If they hurt us or kill us, no flash drive."
"Did the guy in the parking lot know that?" I told him about our visitor.
"Yes. He was probably some vagrant. If not, he was only on a recon mission."
He looked so damned sure of himself.
"Now," he said, "As for San Diego PD, I would leave this to them, David, but I don't know how sophisticated they are or how big their caseload is. They might figure this out tomorrow or next month or never. The more I meddle, the more suspicious Kimbrough is going to be that we're holding back evidence. I would hack those phone records myself, but I don't know how. Lindsey does. She spent eight years in the Sheriff's Office Cybercrimes Unit. She can reverse-engineer that knowledge."
"I know how to be a hacker." Lindsey's voice was small but sounded weightier than our explosions.
It wasn't as easy for me to dial back my anger, but I tried to match her soft voice. "Don't do this, Lindsey, please."
I had just, maybe, gotten her back. Now I would lose her again.
She took in my imploring glance, studied Sharon's practiced calm, and then looked back at Peralta.
"Can you cover your tracks?" he asked.
Her look was that of the old insouciant Lindsey I had fallen for years ago, in her black miniskirt, nose stud, and irreverence that was somehow never cruel. The quarter smile that got the inside joke. The one who would answer him: They'll never know I was there.
Now I knew that within my haunted beauty was her mother's voice telling her she was never good enough, her "Linda Unit" as Robin had called it. I had no question about my wife's skills. But the risks seemed intolerable. There had to be another way.
She looked at Peralta. "You always said I was the best."
"Then do it."
Peralta took Lindsey and Sharon outside while I called Artie Dominguez at the Sheriff's Office.
"How's the best detective in the department?"
His usual ebullient laugh was subdued. "David. Long time, long time. What's it like working one-on-one with the Big Man every day?"
"You can imagine." I asked him how he was. He snorted.
"He's missed," he said. "I might come be a private dick myself soon. You won't believe how fucked up things are. Let's say command these days isn't very friendly if you have a last name like Dominguez. I used to get the best homicides. Lately, I've been on auto theft."
"No shit."
"Real shit, man. Twenty-five years and this is what I get. They're out there playing Border Patrol and everything else has gone to hell. Response times are way down. Serious cases are going untouched. The jail's a mess. Wait until you read about the El Mirage sex cases we're not investigating. But rounding up the campesinos standing outside Home Depot makes the old farts in Sun City and the East Valley feel safer. Sucks."
"Can you run a couple of names through NCIC and ViCAP for me?"
"Sure. It'll take a couple of days so I can do it without my new boss asking questions."
I gave him Larry Zisman and Bob Hunter. He was aggravated with me that I didn't have Social Security numbers and dates of birth. That would mean more work.
"If it makes you feel any better, I have a list of about sixty names with SS numbers that I'd like to email you at home and have you check, too. I know it'll take time."
"Damn, Mapstone. We ought to set you up down here with a desk."
"You know how that would go over with the new guy."
He sighed like a martyr.
"I'll owe you," I added.
"I'll add it to your tab. That it?"
Not quite. I wanted him to check ViCAP—the massive FBI database—for suspicious deaths involving young women falling bound from high places. Extra points if they were high-priced prostitutes. And Claymore mine explosions.
After a pause. "Was that you in San Diego?"
"Yep."
"Fuck me," he said. "I thought you guys were going to be peeping on unfaithful husbands."
"You know Peralta would get bored with that in an hour or less."
"True," he said. "Watch your ass, David."
Then I went into the Danger Room to review the footage of the outside security cameras. I backed it up until it showed a new sedan pull in the dirt beside the south fence. It was a white Chevy Impala. A man got out and looked around. He was young and Anglo with a high-and-tight haircut, shaved on the sides with a weed-like tuft on top. Put him in a military uniform and give him a stolen Claymore and things started to come together. He was no vagrant.
I watched as he climbed on the Impala's roof and expertly vaulted the fence, then walked to the carport. Switching to that camera, I saw him open the Prelude driver's door and lean inside. He popped the seatback forward and climbed into the back. Next, he popped the trunk button and went back there. He was searching for the flash drive. He repeated the move on the passenger side, and then returned to the Impala, looked around again, and got inside.
Switching to the first camera, I saw him back out to leave and expose the license plate. Nevada. I zoomed in, made a screen shot, and printed it out. It was probably a rental car.
Sharon was standing behind me.
"I'm worried about you."
"Me, too." Why deny it?
"You've changed, David. Lindsey feels it, too."
"That's nice. Another excuse for her to leave me."
She's not going to leave you. It would have been nice to hear that, but Sharon didn't say it.
"Mike told me what you went through with the cartels and the old gangster in Chandler," she said. "Nobody could go through that without being changed."
"And Robin being murdered."
Sharon watched me with those big empathetic eyes.
Yes, there was that. And the trial would soon begin. It was another reason I didn't want to read the local newspaper. It wouldn't be covered because the defendant was a drug addict who killed someone. But because the victim was a blond, middle-class woman who lived in a historic district and was the sister-in-law of a former deputy sheriff—that was news. I would have to testify. I dreaded the effect this would have on Lindsey.
"And losing your child," she said. "You two have gone through so much loss in such a short time. But I don't want to see this destroy two people I love. Your child wouldn't want that. Robin wouldn't want that."
I realized my fists were balled up and forced my hands to relax. "We'll never know, now will we?"
"Mike told me how you chose not to kill the woman who shot Robin," she said. "The David I know would have made that choice."
I didn't answer. It was true: I stalked her, found her, but turned her over to the cops. What Sharon didn't know was that I had the woman on her knees with a dishrag in her mouth, and in my hands I held the assassin's .22 caliber pistol with a silencer. I was about to pull the trigger when my cell phone rang and the readout said, "Lindsey." So I didn't pull the trigger. Part of me still regretted it. Nor did Sharon know that the better angels of my nature watched helplessly as I wrapped duct tape around the gangster's mouth and let the Zetas crew carry him out of his Witness Protection Program-funded suburban Chandler house. Or how I rolled the pieces into place for his hit man to be on the receiving end of a hit himself in jail.
I didn't regret those things.
Sharon said, "You have to be willing to give it time. Lindsey loves you. That's why she's here."
Time again. As if I had it.
I said, "I'm really trying."
Sharon hugged me and whispered for me to be good to myself. I didn't know how. We walked back into the office to greet Lindsey and Peralta.
"There's a tracker on his truck, too," Lindsey said.
"She has a very cool scanner." Peralta was like a little kid. He was enamored with gadgets. He was enamored with Lindsey. Who wasn't?
He went on: "It picked the tracker right up. Might be a good idea to check the whole office." He added, "If you don't mind."
Lindsey smiled politely. "This tracking device is identical to the one on the Honda. It's not a logger, the thing people use to follow the movements of a cheating lover. The logger maps out their movements and then you can see where they've been. These are real-time trackers that feed right into a Google map display in a following car. They want to be able to follow at a safe distance and not be detected."
"Are they sophisticated?" Peralta asked.
"Not really," she said. "They're certainly not federal issue. But they're battery operated. The battery might last a month if they track the car an hour or two a day. Less if they track us for more time or the heat really kicks up. Otherwise, they have to replace the batteries." She sighed. "Or they're on a limited timeline so it doesn't matter."
After Lindsey was done, I told Peralta about my review of the security camera. The man with the high-and-tight hair was casing the place.
Peralta sat on the edge of his desk. "It's time to take the war to these assholes."
My anger had been replaced with exhaustion.
"It's over." I held out the truck-stop cell phone. "It's been twenty-four hours since he's called." I was about to say, "The baby is dead," but a look at Lindsey stopped me.
Peralta shook his Easter Island head. "If it was over, that guy wouldn't have been on our property, searching the Prelude. We need to shake things up. Here's how we're going to do it."
An ancient Greek poet wrote, "The fox knows many things but the hedgehog knows one big thing." The philosopher Isaiah Berlin turned that into an influential essay on writers and intellectuals. I was usually a fox. The kidnapping had made me a hedgehog. The single experience defining our lives right now was the kidnapping.
No matter how many law-enforcement agencies were investigating it, I had received the call. I had been bombed with the bloody baby doll—a warning, according to Peralta. It was certainly done in a way that got my attention. The caller told me he had something I wanted. It could only be Tim and Grace's baby. And he said I had something he wanted. That could only be the flash drive. But how did that make sense? He had to know that we would break the encryption and download the client names.
Lindsey wondered if some information was hidden elsewhere on the drive. If so, that could make this particular piece of plastic very valuable, and Mister UNKNOWN was assuming we didn't know the hidden data were there. Finding it was another task for her.
My task was to be bait.
Peralta called the shots. He had investigated hundreds of kidnappings. I had solved only one, from 1940. So I had to follow his lead.
At sundown, I went out alone in the Honda Prelude. Well, not quite alone: for company I had Mister Colt Python and Messrs. Smith and Wesson with the Airlite. And several Speedloaders of extra ammo for each revolver. I also had two cell phones: my new iPhone was plugged into my ears and the truck-stop cell, whose number UNKNOWN had, was on the seat beside me.
I drove east on Camelback Road, a spectacular orange sunset to my back, maddeningly thick traffic ahead of me. It used to be that if you went the speed limit in the city of Phoenix, you would make every light with only a few exceptions. Now the freeway entrances and a few million more people had complicated that, so I ended up missing almost every light. It gave me a chance to see the massive ugliness of a city that had grown so fast it hadn't had time to clean up after itself. Things would be better in full dark. Phoenix was beautiful at night.
Peralta was on the phone. "I'm about half a mile behind you, giving you plenty of room."
"Where's the tracker on your truck?"
"It's sitting on a table at your house, like a good captured tracker."
That made me laugh. I stopped when he told me Lindsey was with him. Not only did she have work to do, most of all I didn't want her in danger if this excursion went sideways. I kept that to myself.
"Where's Sharon?"
"She's renting us a motel room."
That was new. I decided not to ask questions but to focus on my task.
The real estate got nicer at Twenty-Fourth Street, with its alternative downtown of office towers, fancy condos, and the Ritz Carlton. The magical Biltmore Fashion Park had gotten a facelift a few years back and now looked like any suburban mall. Half a mile north was the entrance to the Biltmore resort. Only a few blocks south, the once solidly middle-class neighborhoods had turned over. Now people called it "The Sonoran Biltmore."
I swam the traffic current headed to Scottsdale. If someone were following me, I would never know it. But I deliberately avoided any cute tactics to lose a tail. I wanted a tail. Camelback Mountain loomed straight in front, its head rising first. At Forty-Fourth Street, I turned left and climbed gradually into Paradise Valley.
The road turned east and became McDonald Drive. I wanted to look up and see the Praying Monk formation on the camel's head, but too many headlights intruded. Some toff honked at me for not going the mandatory fifteen miles over the speed limit, then sped around me in his BMW. Phoenicians never used to honk. I used to own a BMW. Patty gave it to me. Lindsey wasn't sorry when some bad guys pushed it out of a parking garage three stories down into Adams Street. I wanted to do the same with this prick.
After the big intersection at Tatum Drive, McDonald calmed down. The area became low-density and very expensive residential, with few streetlights, no sidewalks, and plenty of gates. One would never know that a huge city enveloped this blessed precinct on every side. The road ran to the north of Camelback Mountain. Across Paradise Valley was the mass of Mummy Mountain. I never ceased being moved by these works of nature and how they stood out darker than the night sky. For a few seconds my rearview mirror held no headlights. Then some appeared in the distance. My gut tightened.
Bob Hunter lived in a slummy lot for Paradise Valley, meaning his house was a large, perfectly respectable mid-century ranch. But it was definitely lower end than its neighbors. Most of the similar-age houses along Fifty-Second Place had been torn down and replaced by more impressive mansions. Lush desert landscaping predominated and the land was gentle hills. Paradise Valley had filled in since I was young, but it was still low-density. The properties were spaced far enough apart that a neighbor wouldn't hear a gunshot. A prominent doctor and his wife had recently been bound and shot, and their bodies were only discovered because the meth-addicted killer also set fire to their house.
For my purposes, Bob Hunter's house had an added benefit: no gate. I killed the headlights and slowly came to a stop on the concrete circle in front of the house. Lights were on inside, as well as on a pair of ornamental wrought iron, amber-tinted porch sconces. If someone had seen me, I would know soon enough. Either he would come to the door, or, more likely, I would find the Paradise Valley cops pulling in behind me.
Neither happened. After ten minutes, Peralta called.
"Report."
"I'm sitting here. Nobody has even driven by. The mountains look beautiful."
"We're cruising," Peralta said. "I don't want to get too close."
I told him it was too bad we couldn't reverse the tracker and find out if I was actually being followed.
"I didn't bring that kind of technology home with me, Dave." I heard Lindsey's voice over his speaker. She had said home. That was a good sign, right?
I kept scanning my mirrors and windshield, trying to get as much of a three-hundred-sixty-degree view as possible. Nothing was moving behind the ocotillos and, behind a white wall, the tall stand of oleanders that blocked off the backyard. I tried to imagine Grace growing up here, requiring a car for everything. It was so different from the real neighborhood where I was a child. It was easy to envision her counting the days until she could get away.
It was harder to put together Bob Hunter with his golf buddy Larry Zisman, a friend close enough that he took priority over his own daughter's graduation. San Diego PD had notified Hunter of Grace's death; he had flown there to identify the body. He had known where she had died. They would have asked him if he knew the owner of the condominium from where she fell. And he had lied, to Sanchez and to Peralta. Why? That he had been content to allow the police to classify Grace's death as a suicide ran a dark charge up the back of my neck.
After half an hour, an amazing time for a beat-up Honda to go unnoticed in Paradise Valley, I slid it into gear and slowly coasted out onto the street, then turned north toward Lincoln Drive. I kept my headlights off and drove slowly. Two hundred yards ahead I pulled on the emergency brake and stopped the car without showing taillights. And yet: nothing. If anybody was behind me, he was running without headlights, too.
He could also be a mile away, tracking me on a laptop or a tablet. He didn't have to show himself. But it was better to pretend I was worried about a tail that had me in sight. That way, he could continue to assume we hadn't found the tracking device. So I went ahead with the game.
Lights on again, I sat at the intersection of busy Lincoln Drive. I checked in with Peralta who saw no signs of anyone following me.
"Should we call it off?"
"No," he said. "The guy is out there. He's good. Keep going."
Keeping going meant a drive to Tempe, some ten miles away through heavy traffic. Getting over to the Pima Freeway and zooming south would have gotten me there much faster. But Peralta wanted me on surface streets. So I turned right on Lincoln and took it to Scottsdale Road.
If Central Avenue had been the main commercial thoroughfare from territorial days through the early nineties, Scottsdale Road—and miles of loop freeways containing office "parks"—had taken over that title since then. When I was a child, the intersection with Lincoln had been out in the desert. Now it was deep in the metropolitan blob.
Scottsdale itself was a long, narrow slice between the city of Phoenix and the Salt River Indian Community, the renamed and, thanks to casinos and development beside the freeway, very rich rez. But Scottsdale, oh, Scottsdale, sang of new money, especially up north where it spread east into the McDowell foothills and the people bragged of never coming south of Bell Road, much less to "the Mexican Detroit." Meaning, Phoenix.
Scottsdale was exclusivity and championship golf, celebrities in the wintertime and the weirdness that comes with having more money than brains. It was the capital of plastic surgery: Silicone Valley. City leaders would never allow anything as plebian as light rail. As a result, its traffic was a nightmare, even with most of the wealthy hitting the summer lifeboats for their other homes in the San Juan Islands or other cooler climes. And Scottsdale Road was full of the same schlocky development as the rest of Phoenix, only with some expensive faÇades and more expensively done traffic berms.
Once Scottsdale had been a sweet little add-on to Phoenix, part faux cowboy tourist trap—the West's Most Western Town—part artist's colony. Now it sucked up capital, development, and retail sales from the center city like an Electrolux. Yet it never seemed like a happy place. The politics were poison. Every section and street seemed to vie for the power to look down on everybody else. Scottsdale wanted to be Santa Fe or South Beach, but it was neither artistic nor sexy. Nobody would set a cop show in Scottsdale. A golf or plastic surgery show, maybe.
I suffered the unending traffic jam south past hotels, expensive shopping strips and restaurants, Fashion Square, across the Arizona Canal, and dropping down to Fifth Avenue and Old Scottsdale. Here, a little humanity showed in the scale of the streetscape. A block away was the wonderful Poisoned Pen Bookstore.
South of Old Town, the shopping strips became more downscale and behind them were ordinary tract houses built in the sixties. At Roosevelt, I crossed over into Tempe and the street changed names: Rural Road. It had once been rural. Now all the fields were long gone. The main Arizona State University campus loomed on the right, including the stadium where Larry Zisman had thrown his legendary passes. Then the big new Biodesign Institute. Who knew what they were working on?
By then, I was ready to chew my arm off from the traffic. The average Phoenician made this kind of drive or even longer every day. How did they stand it? The only place I felt comfortable was in the old city. This was my hometown, but it didn't feel like home any more. The Japanese Flower Gardens were gone. The miles of citrus groves were gone. Why did I stay here? I would miss my friends in the old neighborhood, the familiar diorama of mountains, the smell of citrus blossoms in the spring, not much else.
Larry Zisman lived at The Lakes, a series of subdivisions that took over the farm fields south of Baseline Road starting in the seventies. The tract houses were built around little lakes, hence its namesake. Tempe had made a fetish of artificial lakes, most notably Town Lake, contained within dams on the Salt River.
After some wandering along the curvilinear streets, I found Zisman's house. Unlike some of the houses in The Lakes, it lacked any old-growth shade trees. One pitiful little tree was planted on a small, square lawn. Beyond that stood a stucco house with one window, a door through an arch, and the mandatory large garage door and driveway. Above the garage was a second story.
The lights were off. Modest and relatively small, it seemed like an odd home for a one-time football star, but maybe he lost most of his money. Maybe he preferred it here, not far from his college glory days. I pulled directly in front, shut off my lights and engine, and checked in with Peralta. My stomach became a sea of acid. This was as risky as Paradise Valley. Everything about Lindsey's old Prelude screamed "Does Not Belong Here." Signs proclaimed a neighborhood watch. I didn't know how long I dared sit.
Not long.
The Tempe Police cruiser slid in behind me and a spotlight swung white light into the Prelude.
I put my hands on top of the steering wheel and tried to mentally untangle my internal organs. The officer or officers would be looking me over, typing my license plate in for wants and warrants, wondering if the driver was armed. That was my first problem. My second problem: if the person following with the GPS tracker had me in sight, he might misinterpret this interaction. He had ordered me on Sunday to bring no law enforcement. Now here I was, with law enforcement come to me.
"Turn on the overhead light please." A female voice. She was right behind me, in a proper protective stance. I flipped on the dome.
"David Mapstone!"
She came into sight and slid her flashlight into her equipment belt.
"Hey, Amy."
Amy Taylor had been a patrol deputy for the Sheriff's Office. I had worked with her on a number of occasions before she left for a better-paying job in Tempe. She looked the same, attractive and strawberry hair in a tight bun. I glanced over at the truck-stop phone sitting on the passenger seat, willing it to not ring at this moment.
"How's the Sheriff's Office?"
"It sucks."
"That's what I hear. What are you doing?" Her tone was friendly.
So I told her part of the truth. I was working with Peralta now as a private investigator. A young woman had fallen from Larry Zisman's condominium in San Diego, handcuffed and nude, and we have been engaged to find out whether it was a suicide or something more.
"Holy crap!" She put her hands on her hips. "Zisman's married. You know he's a reserve officer in Phoenix?"
"I do. He also owned the handcuffs."
A burst came over her radio and she keyed her mic. I was being saved by a call: a burglar alarm a mile away.
She touched my shoulder. "Gotta roll, David. Call me sometime and we'll catch up. Good luck with Larry. Good guy in my view. Not so much his son."
"Yeah."
"I'm surprised the Army accepted him. Don't tell Larry I said that."
All my senses kicked to a higher gear. The Army. "Of course not. Stay safe, Amy."
In a few seconds she was back in the cruiser, where she executed a U-turn over the rounded curbs and zoomed back out toward the exit of the subdivision. I turned off the dome light and tried to breathe normally again.
I drove back to the center city on surface streets, sick that Peralta's plan didn't seem to be working. My phone was charged and had plenty of time left. It wasn't ringing.
Through downtown Tempe on Mill Avenue, across the Salt River, Galvin Parkway took me through Papago Park, the two iconic buttes backlit by the city, preserved desert all around. I thought about what Amy Taylor had said—not the "call me sometime" part, but about Zisman having a son. That was another new angle. Or it was Occam's Razor and Zisman was the john, even if he wasn't on the flash drive, and Grace had tried to blackmail him exactly as Detective Sanchez had said.
But did that explain why Tim Lewis had been tortured, every finger broken? Somebody thought he had information. Information to kill for. If it were simple blackmail, the problem would have been solved with Grace's supposed suicide. "Death solves all problems," said Joseph Stalin, who had yellow eyes. "No man, no problem." Well, no woman, but there was still a problem. Larry Zisman, former football player, could easily have subdued Grace and thrown her over the balcony. The torturing of Tim Lewis had taken a crew.
At McDowell, I turned left and entered the Phoenix city limits, then drove uphill between the buttes and was greeted by the dense galaxy of lights stretching all the way to the horizon. Phoenix was beautiful at night. On the downhill drive, the iPhone rang.
"I think I've got your tail," Peralta said.
My pulse kicked up. "Do tell."
"A truck followed you though Tempe, made every turn, and then kept going as you went up Galvin through the park and turned on McDowell. He's probably a mile behind you. A black Dodge pickup. California plate. He's got a tag frame that says 'I love Rancho Bernardo,' with a heart thing instead of love, you know."
I did know. It was the truck that had passed me the night I got out of the cab in Ocean Beach, the one I thought was simply looking for a parking space.
"Let's box him in," I said. "Do a felony stop."
After a long pause, Peralta's voice came back on. "No."
"Why?"
"First," he said, "because we're not the cops anymore. Second, because when I hired you many years ago, I hired your whole toolbox, not just the hammer. Since a year ago, all I get is the hammer."
Now it was my turn to be silent. His words stung. His words were accurate.
"So what's the plan?" I asked, and he gave it to me.
"Stay on the phone," he said.
I drove back through downtown and went north on wide, fast-moving Seventh Avenue. Numbered avenues and drives run north and south west of Central; numbered streets and places run north and south east of Central. Now you know how to get around Phoenix. I assumed the pickup driver was learning this from our excursion.
At Northern, I turned west again and after about two miles reached the Black Canyon Freeway, which ran in a trench below grade level. A Motel 6 sat a few blocks up the southbound access road. Getting to it required turning north into the K-Mart parking lot, then passing through the Super 8 parking lot, and finally reaching the Motel 6 parking lot. We didn't even need streets with so many seas of asphalt.
I parked away from the motel building and stepped out into the heat. I had a cell phone in each pocket as I walked the fifty feet to a room on the ground floor right in the middle of the ugly four-story box. It had none of the charm of the old motels that had once lined Grand and Van Buren with their Western themes and neon signs.
Three other cars were parked in the lot, all of them empty.
Precisely as Peralta had said, a key card was slipped into the edge of the door all the way down at ground level. I retrieved it, unsnapped the holster holding the Colt Python but, against my better judgment, left the gun there. I popped the card into the lock and stepped inside.
Nobody shot me.
Turning on the light switch, I surveyed a cheap motel room looking like every other cheap motel room in America. It had been the scene of countless assignations. Bring in an ultraviolet detector, and the pattered orange bedspread would have revealed an army of old semen stains, dead in mid-slither.
I spoke into the headset. "Where's my tail?"
"He's backed off. But don't spend too much time there. I don't have a good feeling about this. Remember, he can track you on a computer. He doesn't have to see you."
I looked at the bed again. The spread looked ruffled, as if a couple had finished and moved on moments before I got there. I sat in a chair and waited for a call on the other cell. The device was a little Sphinx made in a foreign sweatshop.
Then I saw it, sitting on the low chest of drawers. It wasn't a Claymore mine, but somehow it stuck a spike of dread into my throat.
I studied the Zero Halliburton briefcase with its tough aluminum construction. Somewhere I had read this was the brand of case that a military aide carried at all times with the president. Inside was the "nuclear football" containing the launch codes to end the world. And this one looked that sinister.
"What the hell is this?" My voice sounded strange alone in the room.
He knew what I was talking about without describing the flashy case that looked so out of place in the shabby room.
"Sharon bought it today. Open it up." He gave me a code. I dialed open the lock and unlatched it.
Inside were some men's clothes, legal pads and pens, and a shaving kit.
"Look in the socks," he said.
Sure enough, inside one of the rolled-up pairs of socks was a flash drive.
He was inviting them to steal it.
"Is this the real flash drive?"
"Of course not," he said. "But Lindsey encrypted it so it would take even a good techie hours to break in."
"But..."
"Mapstone, why don't you hang there for a few more minutes, then find a place to stash the case, and call me when you're back in the car." He hung up.
The motel room felt close and hot around me. I used the bathroom, checked to make sure the door was locked again, and searched for some artful spot to place the briefcase. The bed was on a solid wood frame, so that wouldn't work. The drawers would be too obvious: better to make them think I was trying to hide it. So I arranged it under the pillows and remade the bed with military neatness.
Back in the car, sweating and worried, I started to go out to the access road, but changed my mind.
Instead, I cruised north through the alley behind the motel, turned around, shut off the headlights, and slowly drove back the way I had come. I nosed out behind the building in time to see another car: a new white Chevy Impala coming around the front of the Super 8. There are thousands of lookalike Impalas. But this one looked exactly like the one that I saw on the security camera earlier in the day outside our office, right down to the Nevada tag.
Wishing the Prelude were not so damned white, I watched as the Impala sped up to the door I had left minutes before. If he noticed me, it didn't show. He was moving so fast, I thought he might ram through the wall. But, no, he slammed to a stop at the last second. If I had the brake-shop monopoly in Phoenix, I would be a rich man.
I dropped the emergency brake enough to slide another couple of feet beyond the edge of the building. The security lighting on the outside of the motel was impeccable. Back where I sat was relative darkness.
Out of the Impala stepped the high-and-tight haircut who had been searching the Prelude earlier in the day. He was wearing jeans and a black T-shirt, carrying something in each hand. One something was a gun. He headed straight to the motel room door without even looking in my direction. If he were a soldier or a former soldier, it was poor situational awareness, but it worked in my favor.
I relayed all this to Peralta on the iPhone.
"He's also got some kind of a crowbar," I said. It was small and black, easy to conceal, and made quick work of the door. "He's inside. I'm going to take him."
Peralta might have had a very clever plan. But this was as close to the suspect as we were likely to get. I felt suddenly cool and comfortable, my breathing even.
Peralta barked at me. "No. This is not the guy who was tailing you. Don't go back to that room, Mapstone..."
"Too bad." I pressed the little red virtual button on the glass screen that said, "end call," and tossed the earbuds onto the seat.
I mapped it out in my head: twenty quick strides to reach the door, keep the Python down against my leg so it wasn't obvious I was packing, pause, assess, and try to quietly ease the door open. No kicking it down. The crowbar had made that unnecessary. Then he and I could have a civil conversation about where the baby was. That is, unless he raised his firearm.
But with my hand on the Honda's door latch, I hesitated. What if the black Dodge Ram suddenly showed up?
High-and-tight almost immediately re-emerged, carrying the Halliburton briefcase. It gleamed in the light. So much for my clever job of hiding it. He quickly got into the Impala and drove toward the access road. I rolled after him, headlights off.
After the third ring, I activated the iPhone.
Peralta's voice came across: "don't follow him."
"Are you nuts? This is the guy who was casing our office."
"The plan is working, Mapstone. Let the plan work."
All I knew was that I had spent several hours I could never get back driving around Phoenix and had nothing to show for it. Still, I reluctantly swung around the other way, back north through the alley, and turned on my headlights.
As I came around the other side of the motel, two Phoenix Police cruisers were sitting driver's door to driver's door. They might have been talking shop or sports or flirting with each other. Or they were watching me. By this time, however, I was only another law-abiding citizen driving through the night.
The Impala driver was long gone.
I muttered profanities.
"Glad you didn't use the hammer, Mapstone?" I could feel the gloat carried across the cell towers. "Sharon left the briefcase when she rented the room. Earlier today she sewed a small tracking device into it. Two can play this game with electronics and ours are better."
I spoke low and slowly, in a rage. "So explain the next move to me, Sheriff."
"Come down to the Whataburger at Bethany Home. Go through the drive-thru. We're in the silver convertible. But don't come over to us."
I did as told, merging into the concrete river of lights that was the freeway and speeding south two miles. After taking the Bethany Home Road exit, I crossed over and made a quick jog up the northbound access road to the restaurant. The building was separated from the traffic by a faux desert berm with a couple of palo verde trees and some creosote bushes. And the drive through, which ran around it like a letter "C." The entrance was at the top of the "C," so I went that way, noticing Sharon's Infiniti parked in one of the spaces to my left, across a gravel-covered berm.
The bad guys knew his pickup, thought they had it rigged with a tracker. In its place, he was driving a silver two-door convertible, starting price sixty grand.
"You're very inconspicuous in that ride," I told Peralta, "especially in this part of town."
"Check it out, Mapstone."
On the left, immediately in front of the restaurant, a black Dodge Ram was parked near the door. Sure enough, his frame hearted Rancho Bernardo. The windows were tinted dark and I couldn't tell if the engine was running.
Better to not linger: I pulled into the drive-thru, anxiously tapping the steering wheel and wondering about the truck's occupant. His partner had probably told him that he had broken into the motel room and taken the briefcase. Now, what would he think if he saw me pulling in? Maybe he was inside, but I doubted it—he would be tracking me from the cab of the truck.
I didn't understand why Peralta was taking the risk of having me drive here. I hoped he believed in coincidences.
"So what's the plan again?"
"Get your order," Peralta ordered. "Pull around to the front, pull in a couple of spaces apart, and eat it where he can see you. Pretend to be dumb."
That part was easy.
By this time, I was actually hungry. So I got a burger, fries, and Diet Coke. Then I parked three spaces south of the Dodge Ram. The tinted windows made it impossible to see if anyone was inside.
Take small bites in case you get in a gunfight, like your grandma taught you.
I was two bites into the cheeseburger when Lindsey stepped out of the convertible and walked toward the restaurant. She was wearing a short khaki skirt and a tight sleeveless top that accentuated her small, pert breasts and very erect nipples. Her ability to look ten years younger than her real age was not diminished by the harsh lights of the parking lot.
She strutted within inches of the Ram driver's door and went inside.
My head throbbed. Over the phone, I demanded, "Are you crazy?"
"No." Peralta was fully in his Zen master mode. I almost preferred the volcano. He was taking a hell of a chance, assuming that my presence would distract the driver. I prayed he hadn't checked me out in enough detail to realize that the woman with the legs that went on for days was Lindsey Faith Mapstone.
Five minutes later, she walked back the way she had come. She paused in front of the Ram's grille and sipped sensually from a drink, paying no attention to me. She turned back as if she were going to return to the restaurant, and then faced forward again, fellating the straw for the occupants of the truck. If they had missed her the first time, they sure didn't now. She stepped off the curb and walked to the convertible, her skirt swinging saucily.
If the truck door opened on the way to grabbing her and hauling her off for rape and ransom, I was going to control and dominate the situation immediately, badge or no badge.
"Fuck!" Lindsey yelled it.
She had spilled her purse on the asphalt behind the truck. She knelt down and slowly gathered up her stuff. Now she was most vulnerable, but neither truck door opened.
After an interminable time spent picking up the contents of her purse and slipping them back in, cursing all the time, she finally made it around the berm and slid into the passenger side of the convertible. Peralta nonchalantly backed up and drove in the opposite direction from the freeway, toward the Big Lots store, and disappeared.
I was left to eat my meal for as long as it took for the Dodge Ram to leave. It consumed a leisurely half hour. They left after twenty minutes but I waited longer before I dared move.
My pulse gradually went down. I called Peralta and reported in.
"So what next?"
"Next," he said, "we go home."
"I thought you were following them?"
"We are, Mapstone. With you there to help distract him, Lindsey inserted a tracking device inside his rear bumper. She also got a good description of him through the windshield."
I'm not sure he needed me there. Lindsey did a fine job of distracting him all by herself.
I was about to turn south on Third Avenue into Willo when the xylophone sound made me jump. Exactly like before, the digital readout said, UNKNOWN.
I answered professionally. "Fuck you."
There was a long pause and I thought he might hang up. Then: "You think you're clever. You think you're putting the pieces together. But you're wrong. You can't solve this case without my help."
"Why would you help me?"
"I thought we could do business."
The past tense didn't give me hope for the baby.
I said, "You're wasting my time."
"Lose anything tonight?"
I was silent.
"You better check, absent-minded professor."
I didn't say a word. Let him think he outwitted us and found where we were hiding the flash drive, in a motel on the freeway.
Finally, I spoke. "I'm tired of games. Drop a baby doll on me? What does that mean to me?"
I feared what it meant. But I didn't say it. Instead, I pushed on. "I used to solve historic cases for a living. There was a mobster in Seattle who liked to dispose of his victims by having them pushed out of an airplane into Elliott Bay, while he watched from a skyscraper downtown. Unless you're him, this call is over."
"You didn't like the airplane? I wanted to get your attention. To get you in a bargaining frame of mind. Where would the fun have been if I had just left the package in the vacant lot for you to find? Anyway, if we can drop a baby doll out of an airplane, we can drop other things, too. Just a simple civilian airplane can be quite lethal. Wait until we steal a drone..."
Taking a chance that he was full of his own grandiosity, I said, "I'm hanging up."
"Wait."
"For what? I bill by the hour. You're not mysterious. You're not scary. You're an ordinary douchebag. You're wasting my time."
"You put up a brave front, professor, but you know it's over. Because of your carelessness, now you have nothing to bargain with. That's a good thing for you. I'll let you and everyone you love live. I got what I want."
Mustering my best acting, having studied theater under Peralta, I filled my voice with surprise. "You son of a bitch!" As if it was only now dawning on me that I had lost the briefcase.
"Don't hang up," he said. "I want you to think about what I've told you about the country. Don't be a traitor to your race."
"What about Tim? What about the guy you shot outside our office? They were white."
I could feel his shrug. "They were in the way of the greater good."
Now I knew he had killed Felix, too.
I asked about Grace.
"She was a whore," he said. "All I wanted was the information she had. She wouldn't give it to me. So we made her give herself up like a whore."
"You raped her before you pushed her off the balcony."
The rich laugh. "Come, come, Professor. We're both men of the world. I had to let my team have some fun. She sounded like an animal being tortured because they wanted her ass, too. I was above any of that nonsense. But boys will be boys. Afterward, I gave her another chance to help herself. She didn't take it."
I was about to call him a baby killer but he cut me off.
"You think I'm a criminal, a terrorist. That's what many contemporaries thought about Washington and the Founders. Soon enough, you'll know that I'm a patriot. Count your blessings tonight, Doctor Mapstone, and sleep well."
The truck-stop cell blinked off, perhaps for good. I pulled over to write down notes on the conversation. The street ahead and behind me was dark and empty.
Robin and I were staying at a beachside resort. It curled around a cove on the Pacific with magnificent scenery but we hadn't left the room. She had never looked more radiant. She didn't have Lindsey's classic beauty and was always aware of that. Indeed, they didn't look much of anything like each other. But her smile was the better of the two sisters and it brought all her features together. Her hair was dirty blond, its wavy tresses hitting three inches below her shoulders.
At the moment, she pushed it out of her face as she told me something important. She held a baby in her lap.
Then she sent me out for something, I don't remember what, and on the way back I couldn't remember the room number. Lindsey was at one of the bars and swiveled her stool to face me. She reached out and we embraced and kissed. But I had to get back to Robin. She had the baby with her. So I told Lindsey I would be back and wandered through the halls, restaurants, and shops trying to find the corridor that led to our room. I would have to explain all this to Lindsey but that would have to wait.
But I couldn't find the room, no matter how many halls I roamed, or stairs I climbed. The resort seemed to be adding new buildings as I walked. The place was full of people and I had to push my way through crowds. Some people seemed to know me. I fished in my pocket for my cell phone to call Robin, but all that I found was a rubber pad that said, FRONT TOWARD ENEMY.
"Dave..."
My eyes came open in a dark room. Our bedroom. Lindsey was standing over me.
My groggy voice came to life. "Do we have a fix on those trackers?"
"We're following them. Remember, Peralta wants to wait and see where they go to nest."
I remembered. It frustrated the hell out of me, but he was no doubt right.
She set her baby Glock on the bedside table, slid out of her clothes, and lay next to me. The skin-on-skin was sublimely visceral.
"Want to see where Grace Hunter's phone went?"
I did.
She opened her new laptop, the bright screen hurting my eyes. I sat up. The clock on the computer read four a.m.
"Have you been up all this time?"
"I couldn't sleep."
It worried me. I didn't like the idea of her perched on the landing above the living room. True, I had checked from the outside. No one could see her through the picture window. But a fresh memory of Robin shot and dying in the back yard shook me.
"We shouldn't be here," I said. "It's not safe."
I didn't give a damn about the assurance I had gotten from the killer.
She said, "We've got an alarm. We've got guns. And we know where the bad guys are. Peralta says we're safe."
"He's not omnipotent, no matter what he thinks."
She nodded to the computer screen.
"Let me distract you. I went back a year, and Grace Hunter never left Ocean Beach, exactly like Tim told you. She would walk down to the market a few blocks away, here on Newport Avenue. All her calls were to Tim, her parents, and her friend, Addison. Now, check out April twenty-second. At two-fifty p.m., she leaves the apartment and walks north. It's like she was going to the store. Maybe for diapers."
I watched as Lindsey brought up a Google maps display.
"Here, at two-fifty-four, she's really on the move." I watched as the red line ran out of O.B. on Narragansett Avenue, turning north on Chatsworth, and east again on Nimitz Boulevard, heading toward downtown.
"Does she have to be making a call for this to show up?"
"Nope," Lindsey said. "People would freak out if they knew how much data were being collected on them every minute. All that needs to happen here is for the phone to be turned on. But look here. At three-oh-five, they stop. Right here."
The map showed the intersection of Nimitz and Locust. It was a nothing little street right before the big stoplight at Rosecrans on the Point Loma Peninsula.
"And that's it. That's where she stays."
I thought about the missing hours.
"Or," Lindsey said, "that's where the phone stays."
"What do you mean?"
"Grace's phone never made it downtown. At four-ten, at Locust and Nimitz, the call was placed on this phone to your office. Grace might have made it. Or, she might have already been in that condo downtown. But at four-seventeen, the phone was turned off at the same location."
I put my arm around her. "So somebody made contact with her on the way to the store. And she got into a vehicle. Somebody she knew. So she got in with him and they drove toward downtown. Toward Zisman's condo. But what happened at Nimitz and Locust..." My voice trailed off. Things didn't track.
Lindsey shook her head, her voice authoritative. "She had a baby waiting at home. She wouldn't leave him for long. And Zisman wasn't one of her johns. So why would she leave the baby and go to his place? No. Somebody snatched her off the street."
I was fully awake now, the dream almost forgotten.
She opened a file. "Here's where things get interesting. There was a call made from that phone a few minutes before the call to your office."
"The San Diego cops didn't have that on their LUDs."
"They wouldn't," she said. "It was placed to a scrambling device. Very advanced, very expensive. It scrubs any of the conventional records of the call, even an incoming call. Only some government agencies and corporate executives use this. You have to know where to go in the cell-company databank to find the trail, then decrypt. But here it is. The call was five minutes long."
"Are you sure nobody knew you were hacking all this?"
"Oh, somebody knew or will know. But what they saw was a low-end data breach coming from the People's Republic of China."
She opened another file: the list of Grace's clients. "The scrambler call was made to this number. It's his private line." Another screen showed me his face on the cover of Fortune magazine. He looked my age yet was making more money in a week than I would make in my lifetime. Why did I need three college degrees?
"He runs one of the top venture-capital funds in the country," she said. "He could afford this kind of security. All these executive types have protection. According to the records, he and Grace saw each other regularly for more than two years."
I took it all in, or thought I did, amazed again at Lindsey's talents.
I stopped myself from tapping my finger on her clean computer screen. "Then the phone was turned off for good, right there on Nimitz?"
"Not exactly. It was turned on again last Friday."
Suddenly, the air conditioning felt too cold.
"Where is it?"
When she gave me the address, I grew colder still. Grace Hunter's cell phone was in evidence storage at the Phoenix Police Department.
She said, "I answered all of Peralta's questions and it hasn't even been twenty-four hours."
I let out a long breath. "You're fast."
She put her hand on my private parts. "I can be."
We were at the Good Egg having breakfast four hours later. Like its neighbor Starbucks at Park Central, it was an institution in Midtown Phoenix. Unlike Sunday, the offices inside the nearby towers were open and the restaurant was busy. The morning was cool enough to sit outside, a dry seventy-nine degrees under the umbrellas, not even hot enough to require the misters. A pleasant dry breeze was coming in from the east. Light-rail trains cruised by on Central, clanging their bells. In her round, nerd-girl sunglasses, Lindsey looked like a spy.
Here we are, I thought, easy targets in assassination range. But the tracker on the Dodge Ram was far away and three Phoenix Police units were in the lot out front, the cops having coffee next door. It would take the bad guys at least a little time to break into the briefcase and even longer to figure out the flash drive.
To figure out they had been played for fools.
A pickup truck did arrive: Peralta's. He was in a suit again and gave us a tiny nod as he walked toward the breezeway and the entrance. I knew it would take time for him to get out on the front patio. He was past his period after leaving office where he didn't want to come here, didn't want to see the assortment of politicos and officials who used the Good Egg for morning meetings. He had shifted his morning routine over to Urban Beans on Seventh Street.
But apparently he was willing to be seen again. I looked back and, sure enough, he was working the room, shaking hands, slapping backs, everyone having a great time. Where were they when he needed them? Now they had a sheriff who was a national embarrassment. He had a long conversation with Henry Sargent, who was sitting at the lunch counter. Henry was a retired honcho from Arizona Public Service.
"Lindsey!" Peralta sat down, full of morning pep. "What have you got for me?"
She went through it as the same waitress who had served him for the past fifteen years poured coffee and went off to place his order.
I read his face: satisfied, impressed, interested, troubled, more interested. An outsider would never know this from his seemingly immobile features, ones that could elicit confessions from criminals or compromises from county supervisors—or, this being Arizona, the other way around. But after so many years, I could see the slight rise of the right eyebrow, the tightening of his mouth, and the easing of a frown which didn't mean his mind was easy. I wondered what troubled him. For me, it was the whole thing.
I asked, "When are we going to interview Zisman?"
He acknowledged me for the first time with a glance of disdain at my Starbucks mocha. "Not yet."
"When?"
"Mapstone, you sound like an annoying child on a trip. 'Are we there yet?'"
"Maybe. That makes you the dad who's lost and is too stubborn to ask for directions."
It was only me and Peralta being ourselves. Lindsey interrupted.
"Boys. I think the targets are definitely in the nest."
She handed over her new iPad, to which she had added Google maps. Peralta studied it, and then handed it to me. Sure enough, both red dots had converged.
"They've been in this same location for several hours," she said.
I worried that they might have discovered the trackers and discarded them at the spot on the map. But Lindsey said she had modified each to send a different signal if anyone fiddled with it.
"What time did they get there?" Peralta handed the tablet back to her.
"Around two a.m. They spent a few hours at a bar in Sunnyslope before that."
He nodded.
The two red dots had nested less than a mile from the bar.
"Excuse me," he said, and walked back inside the restaurant. The next time I caught sight of him, he was in the breezeway, which once held scores of shops when this was a mall. He was leaning against a pillar, his phone to his ear.
Back at the table, he took his time with breakfast. I had no choice but to do the same, even though I wanted to kick down their door an hour ago.
At last, Peralta gave instructions: take the Prelude home and park it. We would ride with him to greet the kidnappers. I hoped they were good and hung over.
As we left Park Central, he was in the cab of his truck, making another call.
Fifteen minutes later, we were northbound on Seventh Street. Lindsey rode on the jump seat of the extended cab, back with the weapons compartment where he kept his heavy metal. Aside from numbered streets to the east and avenues west, the other easy way you knew your way around Phoenix was to look at the mountains. The South Mountains showed you that direction. The Papago Buttes, McDowells, and, on a clear day, Four Peaks stood to the east. West were the White Tanks. We were driving straight toward North Mountain.
Sunnyslope was one of the few places with soul outside the old city, with a real identity that wasn't subsumed in endless subdivisions. It was located beyond the Arizona Canal and outside the oasis, a desert town, a Hooverville from the Great Depression, and a place that retained its own proud, quirky identity even after it had been annexed into Phoenix in the 1950s. The relatively few natives from there my age and older were "Slopers" first, Phoenicians second. From my perspective, it had some interesting unsolved murders.
The place remained unique even though it had filled in with some of the same fake stucco schlock you found everywhere. A couple of its more notorious biker bars remained. You were aware of being higher than downtown, up against the bare, rocky mountains that shimmered in the sun. If the smog hadn't smudged the view to the South Mountains, you'd see you were at about the same elevation as Baseline Road in south Phoenix, where the Japanese Flower Gardens once stood. From both places, the landscape rolled down to the dry Salt River.
Peralta slowed as we approached the five-point intersection with Dunlap and Cave Creek Road. The parking lot of a shabby shopping strip looked like a used-car joint selling black Suburban vans.
"What the hell?" I said.
"Calm down, Mapstone."
He wheeled in and parked.
"Stay here."
He left the engine and air conditioning running and approached the black Suburbans. Out of one stepped a slender man in khakis and an open-collar shirt. Eric Pham, special agent in charge of the Phoenix FBI. Even the head fed wasn't wearing a suit. The New Conformity. They shook hands and talked, and then they walked a ways talking more. Pham was gesticulating, as if laying out a map. Peralta nodded and pointed. Pham nodded.
I asked Lindsey for her iPad and switched the map to a satellite image. The dots had converged at a house at the end of Dunlap, about a mile away. From the photo, it looked like a mid-century modern house. Maybe it was on a little butte; it was hard to tell, but Dunlap rose as it went east before dead-ending at the mountain preserve. That could provide some easy escape routes if they didn't do this right.
Now a couple of Phoenix PD units arrived, along with the huge mobile command post. My stomach was wishing it didn't have breakfast getting in the way of contracting into itself. How long before the news vans and choppers arrived, too?
"Why aren't we doing this ourselves?"
Lindsey put a hand on my shoulder.
"We have to trust him, Dave."
I leaned my face against her hand, hoping she was right. I knew Peralta still had chits to call in and back channels. But I had a local lawman's mistrust of the feds. I had seen how these quasi-military operations could go very wrong.
The door opened and his bulk filled the seat.
"Phoenix PD is closing off streets," he said. "The FBI is preparing to deploy a SWAT team."
"And you explained to Eric Pham that we developed a break in this case...how?"
He took off his sunglasses and rubbed his eyes. "I have my ways, Mapstone."
"I bet."
He slipped the shift into drive and rolled back to Seventh Street.
"Wait!" It was an inane blurt, but it came out anyway. Anything to stop this circus. I knew it was too late, even though I had a bad feeling about going in with so many cops, so much firepower.
"Exactly, Mapstone. Wait. There's a baby in that house. The SWAT boys can't send an undercover to the front door with pizza, toss in a flash-bang grenade, and go in blazing. This is going to take time. They'll have to negotiate these guys to come out. We've got other stuff to do in the meantime."
I looked back with mixed emotions at the gathering army, hoping he was making the right call.
The afternoon sun was cooking toward one-hundred by the time I was waiting for Peralta at the Deer Valley Airport in far north Phoenix, on the other side of the mountains. Since the city had turned Sky Harbor exclusively into a commercial aviation hub, this had become the major general aviation airport. It lacked the cachet of the Scottsdale Airpark, but it was one of the largest general aviation airports in the country. It was also probably the place where UNKNOWN had taken off and landed on his mission to drop the bloody baby doll on me.
But he wasn't unknown now. I had met Artie Dominguez for lunch downtown at Sing Hi. I left the Prelude on Cypress and took light rail downtown. No reason for all my movements to be known. The train was packed as usual. The light-rail system was one of the few elements of progress to arrive in recent years and its popularity made its critics more hysterical in their opposition. I liked it.
It only hurt a little to get out at the stop by the old courthouse. The building was as handsome as ever, although I wouldn't let myself look up to my office. It was a crime that they had ripped out the old palm trees, grass, and shade trees years ago. Downtown needed more shade. And they had added more parking on the south side, more concrete to help make the summers hotter and last longer. For all this, it was the best-looking building downtown. Across Washington Street, a little band protested against the new sheriff.
Sing Hi was two blocks south. Dominguez wasn't worried about being seen with me because the venerable Chinese restaurant had lost a good part of its clientele of deputies and prosecutors to the new restaurants at CityScape, the boring mix-used development to the north. I still liked Sing Hi's chow mein.
He played at being aggrieved over my hurry-up request, but he was clearly interested.
Bob Hunter and Larry Zisman came up pretty clean. Each had accumulated a few speeding tickets. The same was not true of Zisman's son, Andrew. The son had two juvenile arrests for assault and weapons at ages sixteen and seventeen. His father had paid a top criminal lawyer to get him out of both. He joined the Army but was discharged for being part of a white supremacist cell at Fort Hood, Texas, that was blamed for the beating of a black non-com and the rape of a female soldier. Three of his buddies had gone to military prison. Andrew Zisman had been sent back into the civilian population. His last known address was his father's condominium in San Diego but over the past year, he had racked up two moving violations in metro Phoenix.
ViCAP was no help on either anti-personnel mines or women being pushed from balconies.
But I had also emailed Artie the list of Grace Hunter's clients.
"It's like the Forbes 400," he commented.
The list contained chief executives, investment bankers, a venture capitalist, doctors, lawyers, and one Indian chief.
The one exception was named Edward Kevin Dowd, age thirty-six.
Yes, Edward.
"This one has an outstanding federal warrant." Dominguez showed me the intelligence report. "He's suspected of involvement in the theft of anti-personnel mines from Fort Huachuca."
A sheet of paper had never felt so heavy.
"Dowd left the Army six years ago after serving for a decade in Special Forces. He had seen multiple deployments to Iraq and Afghanistan. Then Obama became president and Dowd started recruiting what he called the White Citizens Brigade among other disaffected soldiers. He was no redneck, but a trust-fund baby from back east, attended Andover and Yale. He was a captain. It was two years before the military got a hint of what he was doing on the side and brought him up on charges. But the investigators didn't find any laws broken, yet. So the Army quietly pushed him out."
Dominguez slid a photo across the table. Dowd had a lean face, a full head of reddish-brown hair, a narrow soul patch that looked like a Hitler mustache that had fallen to his chin, and small, mean eyes.
"This guy is a killing machine," Dominguez said. "He's also a licensed pilot."
Killing machine. I thought about what Ed Cartwright had told me.
"I need those back."
I reluctantly slid the material back across the table.
"Did Dowd know Andrew Zisman?"
Dominguez shook his head. "Unknown."
What was known was that Dowd had been a client of Grace's, meeting her a dozen times.
"So Artie, where was Dowd last operating?"
He smiled crookedly. "Phoenix and San Diego. What the hell have you gotten yourself into?"
It was a lethally pertinent question, but when Peralta arrived at the airport terminal we had no time to talk. Two tough, big men in suits came inside and called our names. They led us outside where an imposing Gulfstream jet was waiting on the tarmac.
"I'm going to have to ask for your weapons," one said.
"No," I said. It was one of Peralta's cardinal rules: you never give up your sidearm.
"It's all right, Mapstone." Peralta handed over his Glock. I reluctantly did the same. On a pat-down, they found my last-option knife and confiscated that, too. Peralta glared at me. I glared right back. We stepped up inside the jet, visions of being tossed out in the desert dancing through my head.
"Mike, how the hell are you?"
Mister Fortune Magazine, whose name was Jim Russo, looked older than his photograph, even though he appeared very fit with a golf-course tan. He led us to a sumptuous seating area where a young woman brought us bourbon.
"It's been too long," Russo said. "How's Ed Cartwright doing?"
Now I was confused and paranoid.
"Crazy as ever," Peralta said. To me: "We were all in the same unit."
"That crazy Indian saved my ass more than once," Russo said.
"Mine, too." Peralta savored the bourbon. The small talk continued for an interminable time. He even got around to introducing me.
"I appreciate you flying over here," Peralta said.
Russo stared at the floor. "I'm sorry, Mike. I should have contacted you sooner. But if this had gotten into the financial press..." He shook his head. "After Felix was killed, I didn't know what to do."
He had my attention.
"Felix was the head of my security detail," Russo said. "He was a Navy SEAL who lost a leg in Afghanistan. Won a Silver Star saving his comrades after an IED attack. He was a good man."
"Why did he have multiple driver's licenses?" I asked.
Russo explained that sometimes he needed to check into hotel suites under assumed names. Apparently like many billionaires, he was a target of threats and would be a tempting catch for kidnappers. I tried to pay attention while wondering how the situation in Sunnyslope was progressing.
"Scarlett." Peralta let the name drop ever so lightly.
Russo made a face. "Foolish old rich man, huh? I know what you're thinking."
"I'm not thinking anything, Jim. She was a pretty girl."
"I have a wife and children," Russo said. "But my wife and I drew apart sexually a long time ago. Grace..." He hesitated. "Grace helped me."
"Grace?" I said.
He smiled sadly. "I knew her name. Felix provided a complete dossier on her background for me."
"We're not here to judge," Peralta said. "Hell, I envy you, you horny wop. It might have helped to know you were the one hiring us instead of Felix."
"He did it," Russo said. "I only gave him your name. He didn't trust the police, and he was mindful of my privacy. I thought if anyone could help, you could."
"Was he seeing Grace, too?" That was my fart in church.
"Oh, no," Russo said, "Felix was gay. But he was the one to give Grace a ride to my place in Rancho Santa Fe and back. They got to know one another. Felix did love her, but like a brother. You see, when he was deployed his real sister was abducted and killed. He never forgave himself. He became very protective of Grace, especially after he learned she was being pimped out. He got her out of that situation and back together with her old boyfriend. I hated to have to give her up but she deserved a real life."
We waited. Peralta and Russo received refills.
"In the months that followed, Felix would keep an eye on her. He'd check up from time to time. Of course, her husband didn't know. Grace was very good at keeping secrets and compartmentalizing. About two weeks before her death, she called Felix. She was afraid somebody was stalking her. She didn't know who, or she didn't say. She didn't want to worry Tim, so I'd be surprised if she even told him. Anyway, Felix took a leave of absence, got an apartment in Ocean Beach so he could be close..."
"A guardian angel," I said.
"Exactly." Russo looked me over for the first time. A mixed verdict. "I didn't think it was necessary. Grace was smart and away from that life. But Felix was adamant, and he was a very good employee. He had also served the country. I felt I owed this to him. He gave her a panic button to push if she got in trouble. He was usually about a block away."
Peralta asked what happened on April twenty-second.
"For the first time, Grace pushed the panic button. It had a tracker and Felix was able to get to her..."
"What do you mean?" I was too impatient.
"He ran the car she was in off the road, onto a side street. But it was three against one. They beat him up pretty bad, which would be no easy feat, and they left him there. I didn't realize how bad when he called. He held it together, told me some guys had taken Grace and gotten away from him."
"Why didn't he call the police?"
"I told him to do it. I also gave him your number. I told him Peralta could get results."
He didn't call the cops, but he did phone us, getting the answering machine.
I took a tentative sip. It was very expensive bourbon. "But he called us from Grace's phone."
Russo nodded. "In the fight, he was trying to get Grace. Part of her purse spilled on the street. He ended up with her cell, which looked exactly like his."
Peralta said, "Why wouldn't he know it wasn't his phone when he didn't find you on the favorite calls?"
"My number had to be memorized," Russo said. "Security. Felix didn't realize he had the wrong phone until he had called me and was in the middle of calling you."
But he never made a second call to us. I asked why not.
"He passed out. You've got to understand, he wasn't as physically capable as he once was. He wore a prosthetic leg and was in constant pain. The next thing he knew, he was in the hospital. And Grace was dead. It was two weeks before he could come to Phoenix and see you. He still had her phone." Russo paused and suddenly slammed his fist into his leg."And now he's dead, too."
I went through the names: Larry and Andrew Zisman, Bob Hunter, and Edward Dowd. They drew no reaction from Russo.
Peralta said, "And Felix?"
Russo set his glass down carefully into the brass cup holder on the teak table. "I helped arrange for his mother to bring his body home for a funeral in Indiana. I promise you, she'll never want for anything. She's lost both her children. It's a hell of a thing."
He seemed like as decent a master of the universe as there probably was, and he was at loose ends. Still, I couldn't finish his premium booze. I kept thinking how far ahead we would have been if Peralta's old war buddy had called us sooner. Tim might still be alive. The baby would be safe. It was all I could do to keep from exploding.
He ran a hand down his face and stared at Peralta. "Shit, Mike. How could this happen? I've lost them both. Who would do this? Why?"
I had a pretty good idea who had done it. I didn't know why.
As we drove back and I reported on my lunch meeting, Lindsey texted me that she and Sharon were going shopping—she needed to load up on moisturizers now that she was back in the desert—and she would be home by six. I wanted to go back to Sunnyslope but Peralta vetoed that. So I let him drop me off at home.
With the house to myself, I lay down on the bed and actually started reading the biography of George Frost Kennan that had languished on the table for months. But it did not transport me away from this age of "business casual," tattoos on pretty women, and dunces saying, "No worries!" The perspective it gave me on geopolitics then and now was quickly forgotten. It was not the author's fault.
Too many anxieties hammered on my brain. Edward Kevin Dowd, killing machine, was foremost among them. In these insidious little moments, I noticed Lindsey's suitcases remained in the guest room, only partly unpacked. Was that because of the investigation she had been thrust into, or was her stay here only temporary?
I couldn't stop myself from inconspicuously rifling her bags. I fancied myself a good burglar. I persuaded myself that I was guarding my heart by trying to figure if she was going again. But my breathing was also the fast pant of the voyeur. What did I think I would find? Photos and videos of my wife being impaled by another man? Billets-doux?
I found it inside one zippered compartment: an envelope, addressed to me. It had a stamp, too, but had never been mailed. In fact, it had never been sealed and inside were pages of Crane stationery. Now every electron of good judgment in my body was telling me: Stop, put this away, go no further!
Of course I ignored them. Out came the personalized stationery I had bought for her two Christmases ago. I carefully unfolded it. Hers was not a generation that had been forced to learn and stick with cursive handwriting. "Keyboard proficiency" on a computer mattered more. Instead, Lindsey's block printing was instantly recognizable.
The letter addressed to me was dated May first.
Dear History Shamus,
This is not a "Dear John" letter but it's going to hurt. But please read all the way through. I'm trying to express things I don't know how to say when we're together. You're so good with words and thinking on your feet. I freeze up. So I'm going to try this way.
I said terrible things to you. I don't blame you for what happened to Emma. You know this, right?
As Robin probably told you, I had a baby when I was seventeen. It was probably a cry for help, as they say, from the one who always had to be grown up, always had to be the good girl. Linda called me a slut and put the baby up for adoption and I only got to hold him once. I didn't tell you this when we started dating because I still felt ashamed. And as the years went by, I always wanted to find the way to tell you, but couldn't. Like I said, I don't have your gift of words.
After that, I thought I didn't want to try again. I wouldn't make a good mother. There's madness in my bloodline. But when we conceived Emma, I realized I had been lying to myself. I wanted a child so much, a child with you. A child we could raise with the love and sanity I never had growing up, and the mother and father you never had. And everything inside both of us, good and bad, could go into the future. And maybe that child would remember us kindly and carry that memory with her, too, and pass it on to her children.
When the miscarriage happened, I went crazy. They say people have a "fight or flight" instinct. Mine was flight. So when the governor offered me the job at Homeland Security, I grabbed it and flew. There's no excuse for leaving you. My hope was that Robin would be there as a friend for you and more. I knew she couldn't help herself and neither could you. Did I make you polyamorous, my professor? I didn't realize she would fall in love with you. I didn't know if you would fall in love with her, too. But I figured I deserved it if it happened.
The first time I cheated on you, Emma had been dead exactly one month. I was sitting in a coffee shop near DuPont Circle and saw a man watching me. I smiled at him. He wasn't especially good looking. But he invited me to walk with him and I did. We went two blocks and he pushed me back against a streetlight and kissed me really hard. Then he asked me a question: he wanted to know if I was a slut. The question insulted and stunned me and I didn't answer. "I didn't think so," he said, and pushed me away. He walked off into the evening crowd. I felt so many things. Angry. Guilty. Hurt. Aroused. I liked that kiss. I had missed a man's touch, a man inside me. I wanted to kill this numbness in me. And I thought: yes, I was a slut.
I know this makes no sense to you. We had not had sex in a long time and after the miscarriage, even though you were so gentle and patient, I couldn't be with you. I can't tell you why. I couldn't explain anything, couldn't feel anything but the intensity of my grief. I didn't want five stages or closure or your love. I wanted my baby. And I knew that I could never have another one. That was it. Somehow that anonymous kiss on the street took me away from the pain.
Of course, I never confessed this to you. You wanted me to come back to Phoenix. I couldn't. The thought of it made me ill. So I stayed in Washington and I was a slut. Not with my husband or people I even really knew. Most of the ones I fucked and sucked were men. One was a woman. None of them knew I was mourning my baby. To them, I was a woman who was desirable and impetuous. They loved it that I was marvelously good in bed. The bizarre thing was, I could not come the way I did with you. I won't say it didn't feel good, but my body wouldn't give me a real orgasm. That was all right. Being a slut suspended the ache, the longing for Emma. These lovers didn't know much about me, certainly not the job I did. That only added to the sexual tension and the intensity when we fucked. I was a mystery woman.
Then I met a man at work and settled down. Crazy, huh? Settled down into monogamous infidelity. I figured you were fucking Robin and I kept translating my hurt and guilt over Emma into anger at you. So I became the mistress of a man who was the boss of my boss. He was married, of course, with a pretty wife and children in northern Virginia. He understood that I really needed to fuck and suck. Our encounters were incredibly intense. It took me awhile to realize he knew I needed this passion and riskiness like a drug, to help me forget. Did I write "me"? It wasn't me. None of them knew me. They knew the "not-me" that I became.
It didn't last. Robin's death happened. I sat in the cemetery with you. Remember how the rain started? "Not-me" became me for a while and I was so ashamed and I knew you would never understand or forgive me. How could I ask that? Things were different when I got back to Washington, too. I realized he was tunneling into me, getting past my defenses. But I didn't feel comforted. I felt manipulated. I felt like I was drowning.
Two weeks ago, I broke up with him. I left his apartment in the District and walked back to my place at three a.m. It was raining and I felt as if a very bad cold had suddenly passed. I don't intend to see him again, even though there will probably be consequences. I'm tired of the slut racket.
Dave, I am writing this letter to give you one big honesty dump, and I can only imagine how much it hurts, how mad you are. I want to come home and be with you, try to find us again. I miss us. I know that's a tough request after all that has happened. I don't expect you to agree to it once you read this letter. You will be pissed and jealous. I wouldn't blame you (don't think I didn't feel that way thinking of you with Robin). But I wish you would let me come home and let us find the way to forgive each other and forgive ourselves. You are the love of my life and always will be. When we first got together, you said that I saved your life. But you saved mine, too. I wish we could find a way to save each other one more time. I've written this letter ten times. I don't know if I'll have the guts to send it.
Lindsey
I carefully returned the letter to its envelope and that to the zippered pocket of the rolling suitcase. I rezipped it to exactly the position it had been before my fool's adventure. My hands were shaking, my mind seared. Burglary is a crime and I would pay the penalty alone. She would never know that I had read it.
As if on cue, Lindsey called. She and Sharon were at the Nordstrom at Scottsdale Fashion Square and was I doing okay? I gamely said yes. No, I hadn't heard anything about the situation in Sunnyslope. Did I mind if the two of them had dinner? Of course not. I told her to have fun, hoping my voice didn't betray my knowledge of the fun she had in Washington. After she hung up, I walked to the study, my mind in a soup of queasiness and arousal, barely feeling the old hardwood floor under my feet.
At loose ends, I turned on the television for the first time in probably more than a year. A handsome man holding a microphone was standing in the parking lot where Peralta had talked to the G-man this morning. The reporter had positioned himself so the "S" made out of whitewashed rocks on the slope of a hill above Sunnyslope showed over his right shoulder. I turned up the volume.
"...still no details about the police situation here in Sunnyslope," the reporter said. "Officers have sealed off several blocks around a house on Dunlap, as these aerial images show..."
The screen flipped to an overhead shot of the house, with a long roof and several cars parked in its driveway. The nearest police or FBI vehicles appeared to be at least two hundred yards away. I shut it off. This was turning into the goatfuck that I had feared. I had to get out of the house.
It didn't really matter if the bad guys were tracking me now. They had bigger problems. So I didn't even bother to remove the spy device from the Prelude before I rolled off to catch the freeway system that would take me to Tempe. Fortunately, I saw the disaster of the Papago Freeway before I turned onto the Third Street on-ramp. Rush hour, or hours rather, was not allowing anybody to move at more than a slow crawl, if that. So I settled for the street grid.
By the time I got to Larry Zisman's house at The Lakes, the sun was down and it looked as if most everybody who lived on the cul-de-sac had made it home, closed their garage doors, and were watching television back in their Arizona rooms. Not a single other car was parked at the curb. The lights were still off at Zisman's place, but you never knew. Unlike houses in the historic districts, most of these homes focused activity away from the front and the street. Zisman could well be in his Arizona room watching the "police situation in Sunnyslope" unfold. If so, I could finally ask some questions.
At the moment I didn't give a damn if he was a reserve police officer. I wanted to sit across from him and watch his face and body language as he told me about the night Grace Hunter was pushed off the balcony of his condominium. He wasn't on her client list, so why did he claim she was his girlfriend? Former football star or no, Larry Zisman seemed an unlikely man to attract Grace. I had seen recent photos of Zisman: he had lost his athletic body to a gut and his face was puffy. It wasn't as if Grace needed more chances to, as my once semi-prudish wife puts it, fuck and suck.
I wanted to know who he was covering for: his son, Andrew? Edward Dowd? And where was he when Grace died? Was he really already at his boat? If so, why did he leave her with her murderer at the condo? This was only the beginning of the questions I intended to ask.
But when I set the tip of my toe on the step of the arched entranceway, I knew that he would not be giving me any answers.
The big answer popped me in the nose: that unique, fiendish sulfurous smell I had first encountered as a young deputy, the scent of a body that had been dead for a while. In an un-air-conditioned building in the Arizona heat, it would become noticeable within a day. Air conditioning gave you a little longer. Often mail carriers or neighbors would be knocked down by the odor halfway across the yard.
We called them "stinkers."
A quick scan of the front door showed a mail slot. So no mail was piling up obtrusively outside. Maybe the mailman had a bad sense of smell. No newspapers were accumulating on the doorstep, either, but fewer people subscribed now, a sad thing for democracy.
It was tempting to walk around the house and look through a window. But that would definitely attract a neighborhood watch hotdog. So I walked back down the wide driveway toward the street, looking as if I belonged there in the warm suburban air, stealthily scanning to see if anyone was watching from the neighboring houses. Nobody seemed to be.
Out of the cul-se-sac, I drove north to Baseline Road and found a rare pay phone. There, I called 911 and reported a strange odor coming from the Zisman address. Larry Zip had thrown his last pass. Now I wondered if he had killed himself or become one more loose end for Dowd to tie up.
My phone rang. Peralta.
He was brief. "Get up here as fast as you can."
He didn't have to say where "here" was. I drove as fast as I could to the parking lot on Seventh Street and Dunlap. Peralta's truck was moored beside a dozen marked and unmarked law-enforcement units, plus two giant command vans. All of the television stations had positioned satellite trucks there as well. Bright TV lights were focused ahead of me. I parked the Prelude and pushed my way through a crowd of civilians and cops.
Eric Pham, wearing a vest with "FBI" emblazoned on it, was reading a statement for the cameras. Peralta was standing beside him. He couldn't resist the cameras. He never could.
"....at five p.m., a SWAT team made entry into the home. A brief exchange of fire resulted in one man dead. Our preliminary information indicates he was armed with a semi-automatic rifle and fired on the officers. Five other men were taken into custody. A large cache of weapons was seized, including Claymore anti-personnel mines. We believe these mines were stolen from Fort Huachuca. We also took possession of computers and maps that indicate these individuals were planning to use the mines in attacks on shopping malls and federal facilities in the Phoenix area. We also believe they intended to use shoulder-fired missiles to shoot down an airliner landing or taking off from Sky Harbor Airport..."
His statement contained all the caveats about the early nature of the investigation and how he couldn't disclose further information that might compromise an ongoing probe.
"We believe," he said, "that we have disrupted what could have been a catastrophic domestic terrorism attack."
"Agent Pham!" A female reporter with perfect red hair shouted the question. "We have information that these men were members of the White Citizens Brigade, a domestic terror group. Is that true?"
"This was an organized, anti-government group. Beyond that, I'm not prepared to comment, Megan." He was cool and unruffled as a cascade of further questions followed. What were the names of the suspects? Who was the man who was shot by SWAT? What were the specific targets the terrorists intended to strike? He gave up nothing.
My stomach was an acid bin. No mention of the baby. Why had I expected anything different?
"Is this connected to the explosion in San Diego last week?"
"It's too early for us to draw conclusions, Brahm."
"What is former Sheriff Peralta doing here?" a reporter wanted to know.
Pham nodded knowingly. "Retired Sheriff Peralta is acting as a consultant for the bureau."
When the press conference wrapped up, Peralta worked his way toward me like a slow-moving bulldozer, ignoring the journalists' questions as only he could do. As I had watched countless times over the years, he didn't answer but he worked the crowd. It was showtime all over again. It made me wonder if he intended to run for sheriff again someday, maybe when sanity returned to Arizona.
"Where were you?" He wrapped me in his big arm and steered me toward his truck. I thought: I was rifling my wife's luggage, learning about her fuckathon in the nation's capital while I was sleeping with her younger sister in our marriage bed. A normal family. Any other questions? I said, "I wanted to talk to Larry Zisman."
"How'd it go?"
"He's been dead inside his house for some time. I called Tempe PD anonymously."
"Balls. Get in."
We closed out the noise with a swoosh of the doors and drove slowly out of the lot, turning east on Dunlap. The lights of the cars, streetlights, and houses rocketed by in streams of white and yellow, and ahead was the police roadblock of red and blue. As the road rose, the city lights spread out to my right in an endless jewel.
We are the night detectives. We would never be private investigators peeping on unfaithful husbands. That was not the trouble that we would chase, the trouble that would run us down. I would not write grand history in thrillingly reviewed best-sellers. I am with Gibbon, history being "little more than the register of the crimes, follies, and misfortunes of mankind." I am with Peralta, where we track it down armed. This is the job.
I gingerly fed my curiosity, afraid of what I might learn. "Are you a consultant for the FBI?"
"I guess we are now."
"Are you holding out on me? Have you been playing a side game all along with Pham?"
"Jeez, Mapstone. No."
I asked him what Pham was holding back from the press.
Peralta ticked off points with fingers on the hand that wasn't guiding the steering wheel. "The house was rented three months ago by Edward Dowd, using his own name. He wrote a check for a year's rent on a New York bank and it cleared without any problem. In this economy, the owner was glad to have a tenant who paid ahead. The suspects arrested are all confirmed members of the White Citizens Brigade, all former military. The Brigade is suspected of committing seven bank robberies in Arizona and Southern California over the past two years. It appears they used the money to fund their ordnance purchases, among other things..."
A Phoenix uni who looked about fifteen years old waved us through and we climbed up Dunlap as it narrowed and technically ended, turning into a dirt trail petering out against a metal barrier. Beyond it was the darkness of the Phoenix Mountain Preserve.
One sharp left turn put us at the house I had seen from television. It was built of gray cinder blocks with a wide overhanging roof. The trail made another turn to reach a two-car garage. Black-clad cops from various agencies were milling about, many with nothing to do but try to look busy and officious. The SWAT guys wore helmets, boots, body armor, and, beneath that, T-shirts that were two sizes too small. Dazzling floodlights, running on loud generators, illuminated the scene. A police chopper was hovering overhead, vainly playing its spotlight over the mountain preserve.
I slid to the dirt and walked with him as he laid it out.
The electricity and air conditioning had been shut off early. FBI and ATF negotiators had tried for hours over the landline to persuade the people inside to come out. They had refused. Meanwhile, a SWAT member had been able to snake a tiny night-vision-capable camera into the ventilation system so they could see inside some of the rooms. A robot had scouted the perimeter of the house to make sure it wasn't mined.
They had pumped tear gas into the vents at four-forty-five and then had broken down the front door, tossing in a flash-bang grenade. Only one suspect had returned fire and a tactical officer had put him down instantly with one shot. He had been airlifted to Mister Joe's but was dead when he hit the floor. The others had put down their weapons without a fight.
"It could have been really hairy," Peralta said. It was interesting that he had walked onto so many crime scenes over the years that nobody thought to challenge him now.
Outside the front door, a tarp was spread. Most of it was covered with weapons: AR-15s, pump-action shotguns, assorted varieties of pistols, two shoulder-fired missiles, and enough crates of ammunition to make Ed Cartwright happy. The Claymores were probably safely in the custody of ATF. I barely paid attention.
"Where's the baby?"
"They didn't find him, Mapstone."
"What about Dowd?"
"Him, neither."
I used my hand to stop him at the door, no easy task given his bulk and momentum.
"What are you saying?"
His eyes shone black. "Dowd got away."
"I knew it..." All the cops, all the jurisdictions and expensive toys and command vans and they couldn't make a simple collar. I started a cursing jag notable for its creativity.
He pinched my shoulder until I thought it would fall off and leaned in to whisper. "Play well with others."
I did my best.
Evidence technicians were photographing the living room. The floor had traces of blood and was covered with yellow numbered markers. One marker was on the Halliburton briefcase. A laptop sat on a sofa, drawing another yellow tag. They had probably had plenty of time to realize the flash drive was phony, otherwise Dowd would have taken it with him.
My answer was next to evidence marker forty-two: the flash drive we planted in the expensive briefcase was shattered, as if by an angry boot. My feet felt as if they were sinking into the floor. The remains of the tear gas stung my lungs.
Another tech was taking inventory in the kitchen. The cabinets were fully stocked with canned goods, meals ready to eat, and bottles of water. A bedroom closet held body armor, helmets, and night-vision goggles.
"Dowd told them to make a stand here," Peralta said. "Kill as many police as possible."
"How did he get away?"
"Let me show you."
He led me down a hallway and opened a door that revealed a staircase down. I led the way as he talked.
"The house was built in nineteen sixty-two by a doctor. He put a fallout shelter in the basement. It was the height of the Cold War."
It was the year of the Cuban Missile Crisis, but I kept my mouth shut.
We came into a finished basement with wood paneling and an ancient pool table. He pointed to another, heavier door at the far end of the room. I stepped through that portal into a concrete-encased hallway that slanted down. Bare light bulbs protected by steel frames burned overhead. I started to sweat.
It reminded me of one of my maze dreams as I stepped more slowly, made a turn and went another twenty feet on a slanting concrete floor. Two doors were open. One led through thick walls into a shelter, maybe ten feet by ten feet, looking as if it hadn't been touched since Kennedy was president. A dusty yellow Geiger counter sat on a table. Ed Cartwright would look down his apocalyptic nose at such a primitive set-up.
The other door led outside, where a Phoenix cop stood guard. He greeted Peralta by name, as if the election had never happened.
We were at the bottom of the stubby hill. The house loomed above us.
"This is where Dowd probably got out while we were still staging," Peralta said. "We didn't realize there was this escape route out."
"What's this 'we,' Lone Ranger?" I said sourly. "I said we should go in and do it ourselves instead of setting up the paramilitary show that everybody could see."
"Mapstone, we would have been shot dead."
He was right, of course. But I was still angry. The only benefit was the hot west wind, replacing the tear gas in my lungs with good old Phoenix smog and dust. The sheen of sweat across my chest and belly remained.
"We think Dowd came out here and went into that neighborhood." He pointed to lights two-hundred yards away. "He kidnapped a woman and made her drive him through a checkpoint. Let her go down at Forty-Fourth Street and Camelback. He's probably already ditched her car."
Dowd's black Dodge Ram truck sat ten feet to my right, with its tracker no doubt uselessly attached to the back.
He faced me. "Where are the girls?"
"Shopping in Scottsdale."
"Call. Get them here. Now."
I already had the cell out. I asked Lindsey to bring Sharon and meet us back at Seventh and Dunlap.
He walked out into the darkness, kicking the hard ground, thinking.
"Thoughts? Ideas?" It was as if he were talking to the mountains as much as to me.
I moved toward him, wondering if Dowd was watching with night vision. He could take us out right here with a sniper rifle.
"We can't stay on Cypress." I stated the obvious through a scratchy throat. "Your place in Dreamy Draw is more secure but not secure enough. It's also dangerously isolated."
I had only gotten Lindsey back. Sure, she had left me twice before, but for now it was sweet. The idea of putting her at risk was intolerable, a rocket into my brain. Dowd knew we had defrauded him with the flash drive. He would come to kill us all. And he was the kind of man who would seek out Lindsey first, so my agony would be under way well before he got to me. I would have been responsible for losing them both, Robin and Lindsey.
I said,"You know we've got to find him ourselves. Get him first. You know this, right?"
He nodded.
"But for now," he said, "we need to get out of the Valley. How about San Diego?"
It sounded smart. But one other thing bothered me.
"How many Claymores did they find here?"
"Ten."
"You're sure?"
He shook his head and cursed. He could do the arithmetic as easily as I: a dozen stolen, one used on me in Ocean Beach, ten seized tonight.
One Claymore was still missing and I wagered it was with Dowd.
The call came a little after eight p.m. Only one person had called me on the cheap phone I had bought in El Centro.
"Time's up, Doctor Mapstone."
"For you," I said. "You escaped once, you won't again."
"Did you ever serve in the military? In combat?"
"No."
"Then you don't understand anything. I gave you a chance to serve your country by giving me the list of Scarlett's clients. I appealed to your patriotism. I appealed to your intellectual side. But, no. You refused to obey my orders. You refused to negotiate."
"I'm really sorry about that."
"If you had served in combat, you would know that a soldier can't let his rage get the better of him. It can overwhelm discipline and training. Effectiveness. So I have to push you with some clearer incentives. I've researched you, Doctor Mapstone. I'm going to kill everyone you love. Then I'm going to kill you. And then I'm going to bring the war where it belongs, right here to America."
"Keep talking, General," I said. "The trace is working."
There was no trace.
He laughed as if a private joke had been shared between friends.
"I'm going to start with your first wife, Patricia." He read out an address in La Jolla. It was Patty's address. "I know you're in San Diego. If you come alone and bring the client list, then I'll let her live. I might even be willing to let you live. But you have to come within the next hour. You won't find her there. If I'm satisfied you're alone, I'll call and give you instructions. No cops. No bullshit. This is your last chance to negotiate."
Then I was only holding a useless plastic object to my ear, hearing nothing.
But I was not in San Diego.
I was in Phoenix.
I was in the valley of decision.
Sharon and Lindsey had driven the Prelude to Ocean Beach. Find a parking place and leave it, I had told them. It would be two weeks before the police towed it away. Then they had checked into a hotel downtown.
Peralta and I went to the Hotel Clarendon in Midtown Phoenix to wait for the call I knew would come. The Clarendon was where Arizona Republic reporter Don Bolles had been assassinated by a mobster's bomb in 1976. After its restoration, the new owners put a memorial photo gallery in a hallway.
What I hadn't counted on was him leaving while I was in the shower. "Checking on something with Eric Pham. Back soon," he had scrawled on the hotel stationery. He had known I wouldn't let him go without me. "I'm not an old man," he had barked at me. He had to prove something to himself. At least he was with Pham.
Or so I had thought. In thirty minutes, I had called Pham. He had told me he was held up in a meeting and had canceled on Peralta.
So he had gone on his errand alone. My calls to him went straight to voice mail.
Now, I dressed quickly in black jeans, black running shoes, and black T-shirt. I thought about stopping by the office and unlocking the Danger Room. But, no. There was no time. I didn't even bring the Python. Instead, I carried the Airlite and two Speedloaders. That would be enough or it wouldn't matter. At last, I didn't need the toolbox, only the hammer.
Dowd had let the woman he kidnapped off at Forty-Fourth Street and Camelback. Her car had been recovered at Tatum and Lincoln, in the parking lot by the statue of Barry Goldwater. I made a guess that there was one place nearby where Edward Dowd could hole up: the house of Bob Hunter, Grace's father. Like Larry Zisman, Bob Hunter had become a loose end that needed to be snipped. It was only a few blocks away.
I called a cab. While I waited, I phoned Isabel Sanchez and asked her to check on Patty, if Patty even still lived at that address.
Then I made one more call.
It was nearly ten when the cab let me out on McDonald. I gave him a twenty-dollar tip and hiked into the desert, a ghost passing the million-dollar homes. The night was moonless, a few prominent stars claiming the indigo vault above, and I was profoundly aware of the possibility of snakes. But I didn't move with a heavy step. I walked slowly and carefully, aware of every sound, each scurrying noise of an animal that had been disturbed. The sounds of the city were far away.
I came up on the Hunter house from the south and followed the pale adobe wall toward the front. The air was still and hot. My skin was cool and all my senses were notched up high.
The form on the ground was ten feet ahead. I crouched and watched. It wasn't moving and nobody seemed near it.
Closer, I saw a man prone in the dirt and rocks a few feet off the driveway. He was on his belly and his back contained a messy exit wound the size of a dinner plate. I turned him over carefully. His breathing was shallow and rapid. It was a miracle he was still alive. A bullet had struck him just above the heart. His face looked privileged and tan, even near death: Bob Hunter. He had made his last hike up Camelback. He stared at me without seeing.
"Who's inside?" I demanded it in a whisper.
He opened his lips and mouthed something. My wife? Maybe that was what he said. His eyes might as well have been glass.
The elaborate porch sconces were turned off but the door was cracked open, as if Hunter had left it that way and gone for a stroll. Or tried to make an escape. Once again, I scanned the terrain. The desert landscaping was done so well, too well.
I pushed the door open and entered with the snubnosed revolver out and up.
High-and-tight stood a few feet away, facing toward me and holding a black semi-automatic pistol in his right hand pointed down. This was the same man who had searched the Prelude at the office, the same man who filched the briefcase from the cheap motel on Black Canyon. He looked younger close up. His eyes narrowed as I kept walking.
"Who am I negotiating with? You?"
His gun arm started up and I made the smooth trigger pull of the Airlite. The walls echoed with the gun's quick boom as a dime-sized red hole appeared between his eyes and his head snapped back hard. In nanoseconds, the wadcutter bullet fragmented inside his skull and sent a wide shower of red and gray onto the wall. His body lurched back against an expensive floor lamp and both crashed to the floor.
And I was alone in the large living room. Cowboy paintings hung on taupe walls. But there was little time for art criticism. I swept the dining room and the kitchen, finding each deserted.
"Back here, Doctor Mapstone."
I stepped up into a hallway and followed the voice. It sounded unconcerned.
Edward Dowd was standing in the master suite, unarmed. He appeared ordinary except for the soul patch: medium height, average build, shaved head. The mastermind wore a loose, white Tommy Bahama shirt, shorts, and sandals. The hauteur of his military pretentions didn't extend to his wardrobe tonight. His calves were well defined by muscles.
Close to him on the white comforter of the king bed was an AK-47. I couldn't let even my peripheral vision linger, but the rifle looked lovingly cared-for, its wood stock highly polished. The distinctive curved magazine reminded me of its purpose, which was not to be an objet d'art. Anyway, my view was drawn a little farther. On the other side of the mattress a woman was lying nude as if on a snow bank. She was young and pretty and her lips were dead blue. She was the woman smiling next to Bob Hunter in a photograph in a silver frame on the bedside table.
I moved sideways from the door so I had a clean field of fire in case another bad guy came into the room. Measuring the distance between us, I was careful to make sure Dowd couldn't reach for my gun.
He said, "I wouldn't be quick to shoot again, professor. I'm not quite unarmed."
He slowly raised a hand that clutched a stainless steel cylinder with two small lights, a green one that was dark, and a second burning bright red. It had a button on the top. His thumb was holding down the button.
"You're shrewder than I thought," he said. "I didn't expect you here for some time. I've been trying to extract some information while you're in San Diego protecting your first wife."
"She'll be fine." My tongue felt as if it were covered with sandpaper.
"Well, no plan survives first contact with the enemy." His eyes narrowed.
I kept my voice steady. Dowd was right about one thing: anger would only get in the way of the training and experience that would give me an edge.
I ordered, "Put your arms out and get on your knees, very slowly."
He made no move to comply. "Aren't you going to thank me for my service to our country like every other civilian parasite did?"
"On your knees."
He shifted his weight, nothing more. "I want to show you something."
"Don't move!"
"I'll do it slowly."
I kept the gun on him as he stepped back toward a closet door, continuing to face me. Then he reached behind him and opened it slowly.
Inside, Peralta sat handcuffed to a chair. He'd been beaten badly. Blood was caked around his left eye. The last Claymore was strapped around his middle, with the front of the mine pointed inward.
"Kill him, Mapstone." He sounded groggy.
Dowd held out his other hand, the one with the cylinder. "He'll be dead in one second. This is a panic room, built for the family to hide in if there was a break-in. The walls are thick." He closed the door.
"And this," he indicated the device, "is a detonator for the Claymore. The walls aren't thick enough to block the signal. Right now, the only thing keeping your friend alive is the pressure my thumb is exerting on this detonator. So if you shoot me, the green light goes on and your friend dies. I told you I'd kill every one you love."
I kept the gun on him.
He cocked his head. "All I wanted was the list of Scarlett's clients. You thought you were cute, the expensive case in the motel room, the fake flash drive inside. I should have realized two can play the tracker game. Tonight, when your friend the sheriff showed up to check on her old man, I could have hidden, made daddy pretend everything is fine. But I thought maybe Peralta might have the list. So far, no list. This is really pissing me off. All I want is the list of johns. Why was that so hard for you?"
"I don't care." I didn't recognize my own voice.
His cheek twitched.
"Don't you get it? We'd been robbing banks, but that was too risky. Eventually the illegal government in Washington would have gotten us before we were ready."
He seemed eager to be understood.
I said, "You were going to blackmail Grace's clients."
"Exactly. I could have raised millions to fund the Brigade. Then the fun would have started. By the time we're done, this country will be under martial law, and every target we strike will have evidence that it was done by the hajis and the niggers and the spics who shouldn't be in this country. The Chicano Liberation Army. Al-Qaeda in America. The African Struggle."
"But the groups don't exist, right?"
"People will think they do. I've already got the Web sites reserved, so we can let these groups take credit when a shopping mall blows up. You don't know how savage the American can be. We'll make this a white man's country again."
"I think we're better than that." I nodded to the dead woman on the bed. "Anyway, she looks white to me."
"Collateral damage." He smiled. "Hunter said his slut-nugget daughter didn't have a computer here. If she had, it might have had the client list. Too bad for him she didn't. He had to watch while I humped his young wife a few times. She didn't like it at first, but I won her over. It was awhile since she'd had a real man. Must hurt like a son of a bitch to see another man screw your wife. It'd make me want to kill the motherfucker doing it, but ol' Bob just cried. 'Course, I had him handcuffed. Then I strangled her slowly while he watched. At least he didn't change his story."
He liked the sound of his own voice. I said, "So you knew who Scarlett was."
"I checked her out. She didn't check me out well enough, I guess."
"You had to kill her."
"It didn't start out that way. Look, she was a sweet little lay and I was happy to pay for it. It took me awhile to realize this magnificent piece of ass must have a very affluent group of men she was screwing, and this was going to be our funding source. By that time, she was gone. It took me a long time to find her again."
"But you did find her."
"We had a hero who tried to save her," he said. "It didn't take long to put him down. After that, we drove her around for a long time. I didn't want to hurt her, tried to reason with her just like I did with you. I wanted the list."
"You raped and tortured her," I said. "You waterboarded her in the toilet, right? That's why there was water on the bathroom floor."
"Very good. My thumb is getting tired."
I needed more time. I said, "Then you pushed her off the balcony."
"Young Zisman fucked up," he said. "That's who you shot in the living room. Andrew was supposed to hang her over the edge until she gave it up, but he lost his grip. Every unit has its FUBARs."
I almost pulled the trigger right then.
"Why that condo?"
"It belongs to Andrew's dad. Andrew had the fob to the front entry and the keys to the door. We didn't know his dad would be there, but it didn't take much persuading to get us some privacy. Old man Zisman knew what would happen if he didn't play along." He laughed as if we had shared an arch joke.
"But you eventually killed him, too."
"Not me. Andrew killed his father. Call it a test of loyalty. The unit always comes first. I couldn't take the chance his father would keep silent. Enough of your curiosity, professor. Give me your gun."
"No."
"Just so you know," he said, "whatever happens next, you won't make it out alive. I've got a sniper with a night scope positioned outside. He saw you come onto the property and told me. I let you get this far. Otherwise, you'd be dead. My man was trained as a Marine scout-sniper." He smiled. "I didn't realize you'd shoot Andrew straight off, but he was careless. So you can kill me, and I frag your friend, but you'll be dead, too. Just like Grace's daddy, who thought he could get away. If you do succeed in killing me, another commander will take my place. You can't stop us."
"We can make a start. After I kill you, I'll just call the cops."
His face flushed with anger. "Then you're gonna have a bunch of dead cops from my sniper. He's willing to die to take back his country and he'll take as many enemy with him as he can..."
"So far, all you've killed are white people."
He forced himself to speak in a reasonable tone. "You can give me your gun, I'll put the detonator on safe. We can do it at the exact same time. Then we'll take a ride to get that flash drive. The real goddamned flash drive. If it has the information I want, then I'll let you live..."
Dowd's cheek ticked in surprise. Ed Cartwright spoke behind me and then he was standing beside me.
"Your sniper is incapacitated," he said, cradling a pump shotgun on one arm.
"You killed him?" Dowd's voice shook.
"I just used the Apache Persuasion Hold and handcuffed him. He'll live. Probably."
Cartwright held up a black object that looked like a video-game joystick. He said, "I just made your detonator go limp, asshole. So why don't you slowly get on your knees."
Dowd stared at each of us, mouthed a profanity, lifted his thumb from the detonator in his hand.
Nothing happened.
He threw it at me and in those quick ticks of confusion, I allowed the distance between us to close. Rookie mistake—I had worried he would make a move for the AK on the bed—but it was too late. He dove at me and ferociously grabbed for my revolver. It quickly cost me my balance. We fell together onto the hard tile of the floor and I struggled to keep my panic from overwhelming my training. There was also the danger that Cartwright would use his shotgun on both of us.
Dowd's face was that of a feral dog and he was strong. So strong that he was close to gaining control. We sweated, grunted, and cursed. His face turned dark red. My attempt to knee him in the groin failed. So did his try at head-butting me, but he succeeded in rolling me onto my back and getting astride me. Every muscle in my arms and hands screamed as I watched the gun twist toward me.
That's when I released my left hand and grabbed the last-option knife.
"Ooof." He expelled bad breath in my face as I drove the sure little blade into his abdomen. Blood trickled onto my fingers. He still fought but his strength left him. The revolver came loose in my hands and I fired one shot point blank into his chest.
After an eternity that was probably five seconds, I pushed him off with difficulty. Cartwright just watched.
Grabbing Dowd's shoulders, I shook him hard.
"Where's the baby, you son of a bitch?"
A trickle of blood rolled out the side of his mouth.
"I tried to warn you..."
His eyes flickered and closed. He didn't deserve to die with his eyes shut. I shook and cursed him, but I was just yelling at a cadaver.
Cartwright waited a long time to speak. I realized that I must have had a wild look on my face. I patted down Dowd's body out of habit and forced my breathing down.
"Where'd you get that Airlite, kid?"
I told him: at a gun show.
He held out his hand. I gave it to him.
"You got Speedloaders?"
Digging them from my pocket, I put them in his other hand.
"And the knife."
I rose unsteadily and gave him the knife and sheath.
"Now pay attention," he said. "I did the shooting here, not you. Right?"
I slowly nodded, feeling my senses return to human. The room smelled of discharged ammunition and vaporized blood.
"The Indian's here," he said, "and the cavalry are on the way. So you best be gone."
"Peralta's in there." I indicated the panic room.
"I'll take care of him. It won't be the first time. You go."
His voice stopped me at the door.
"You did okay, kid."
I nodded, then walked back through the house and slipped out into the darkness.
In the ensuing days, the FBI made a dozen more arrests and confiscated more weapons and explosives. It was being called the biggest domestic terror conspiracy in modern American history. Peralta gained major cred with the bureau, which promised it would lead to business for us.
The house on Cypress was back to something resembling normal. Did I dare trust it? Lindsey was reclaiming the gardens, fighting against the rising heat. I was cooking and reading. At the moment, we were both naked in the bedroom and sipping martinis. Coleman Hawkins was on the stereo with perfect synchronicity, Cocktails for Two. Among the things Lindsey had purchased on her shopping trip were two sets of garter belts and sheer stockings: bad-girl black and virginal white. She was wearing the black and draping one leg over me.
"Are you going to stay?" I asked the question that had been metastasizing inside me, fearful of the answer.
She held out her glass. "If you'll take me back, History Shamus."
I clinked my glass against hers. "Gladly."
Oscar Peterson came on. The Maharajah of the Keyboard, as Duke Ellington called him, sealed the deal.
"You're crying." She held my face close and wiped my wet cheeks. "Are they good tears?"
I nodded. But they were, in fact, a mixed bag.
"Happy that you're back," I said. "I want to do everything I can to put us back together..."
"Me, too."
"And I'm sad for all the ones we lost. At least some could have been saved if we'd been faster or smarter. I can't say we covered ourselves in glory on our first case. Grace, Felix, Tim, Larry Zip, Bob Hunter, his wife, all dead. We might have stopped some of it."
"Dave, you can't take all that on yourself."
"The only one who got away was Addison."
Lindsey cocked her head.
"Grace's friend," I explained. "Aside from Tim's parent's, she was the only one Grace and Tim had contact with while they were hiding out in O.B. She left school and went home to Oklahoma they tell me. A good thing. But I can't forget holding that baby after I changed him. Now he's in some hole out in the desert. What a shitty thing."
And I cried.
Lindsey held me close for a long time.
Finally, she said, "Addison is a really bad name."
"That's what I think."
"Mind if I try a hunch?"
We drove east from San Diego through Poway and Ramona on the old Julian Road. Suburbia slipped away and the hills and mountains surrounded us. Ahead were the Anza-Borrego Desert and the little town of Borrego Springs. We climbed around Grapevine Mountain, huge rocks leaning in on us, and then the desert valley emptied beneath.
Patty and I had been here many times. We made a ritual of staying one weekend a year at a little inn at Borrego Springs. It was a single-story speck in the desert surrounded by rocky, bare mountains. I remembered that it had a traffic circle. And I remembered a photo that Patty had taken of me on a hot day, surrounded by barrel cactuses in bloom.
But our trip to the badlands today was not for pleasure. The temperature was over one-fifteen and the town was emptied out of all but the hardiest year-round residents. A room would be cheap this time of year.
The traffic circle was still there: Christmas Circle, and a little beyond was a simple little motel with statues of desert bighorn sheep out front. Patty and I had stayed at the tonier Borrego Valley Inn, with its Southwest architecture and private patios. But I had seen this motel many times, never giving it a second look.
"There," Lindsey said.
She pointed to an older Toyota sedan parked in front of the ranch-style block of rooms. It was the only car in the lot. Peralta parked fifty feet away and we all piled out of the pickup truck.
"Let us go first." By this, Lindsey meant Sharon and her.
Peralta and I were well-armed, but I didn't think we would need firepower today. He nodded, and we watched the two women walk to the door directly in front of the Toyota and knock. They talked to the person who opened it, and after a couple minutes they went inside.
Peralta and I found some shade and waited, saying nothing.
Lindsey had followed her hunch and it pointed true.
Addison Conway's car was not in Oklahoma. It was sitting a few paces from us under the mid-day California sun. Thanks to Lindsey's black magic, the Chinese had hacked the phone company again and tracked Addison's cell phone. Last Friday, it had been in Ocean Beach, at Tim's apartment, an hour after I had left. Then it had taken the same route we had just driven and stayed here.
Sharon stepped out and smiled at us: come on in.
Lindsey sat on one of two double beds cradling little David Lewis in her lap. A young woman sat on the other bed. She turned her face to greet us. She was attractive in a girl-next-door way, no Southern California glamour, none of Grace Hunter's looks hot enough to warm your hands by. She was crying. Lindsey was crying.
"This is Sheriff Peralta," Sharon said, her voice so soothing. "And his partner David Mapstone. You're going to be safe now, Addison." She put an arm on the girl, who leaned into her as if she were a surrogate mother.
Sharon looked at us. "She's been out here with nothing but her fears."
I thought my insides were going to drop out on the floor. I tightened my diaphragm just to make sure it was still there. Lindsey's hunch had been more than rewarded.
Addison Conway spoke with a slight twang and no one would mistake her for a Rhodes Scholar. She had been operating on primal fear these past days, not logic or reason.
"I went to see Tim and Grace," she said. "I hadn't heard a word from Grace and I was worried. I knew about her... You know. I was always afraid it would get her killed. When I got to the apartment, Tim was packing up to leave. He was very scared. He told me what had happened to Grace and I just..."
Sobs took her over and Sharon lightly stroked her hair until she could speak again.
"Tim was getting out, going to hide with his parents." Her voice rose. "It wasn't my fault!"
The baby started crying, and Lindsey expertly rocked him into happy little murmurs.
Sharon told her nobody was blaming her. We just wanted to understand what had happened.
"It happened so fast. Tim told me to take the baby and go down to the car, you know, it was in the covered spaces in back? So I did. He said he was going to pack up a suitcase and come right behind me. Only..."
We waited beneath the sound of the air conditioning and the baby gurgling contentedly.
I spoke for the first time. "What happened next, Addison?"
"They came for him!" She looked at me with a red face, puffy eyes. "Two men. They called out at the door that they were cops, and then they barged in. I heard Tim yell. Something broke inside."
She shivered. Sharon coaxed her to continue.
"Tim yelled, 'Go!' I knew he meant me. I didn't want to leave him. And then David started crying and one of the cops looked out the window."
It was still jarring to hear the baby's name.
She said, "I ducked behind a wall and I got lucky. Right then, a garbage truck turned into the alley and stopped right there. It was making a racket and I went behind it and ran for my car. I was parked a block away and I've never run so fast. I was afraid to look back, but they weren't chasing me. Thank god for that trash truck. I left the city and I drove to the desert. I thought we'd be safe here. Then I saw the television, the explosion at the apartment and Tim dead. They called it terrorism. I didn't know who to call or how I could explain what happened, why I just ran..." Her voice trailed into a pitiful whisper: "How did you find me?"
Nobody answered.
"The next day, I was going to call the FBI, but I got a call. He said he was a San Diego detective but he didn't sound right. He wanted to know where I was. I freaked. I told him I was in Oklahoma..."
I thought: Good old Detective Jones.
Peralta showed her photos of Edward Dowd and Andrew Zisman. "Are these the men you saw going into Tim's apartment?"
"Yes!"
"They're not police. And they'll never bother you again."
I realized that Dowd never had the baby. He thought we did. The baby was gone when he got to Tim's apartment. Dowd's elaborate air show, dropping the baby doll and the blood, had indeed been a threat. But he had never been in a position to carry it out.
She sniffled loudly. "The baby was my priority. I had to keep him safe. I didn't have the phone number for Tim's parents. You've got to believe me."
"We do," Sharon said. "It's going to be all right."
And it would, I supposed. Peralta pulled out his cell phone and slipped outside. I watched my wife cradle the baby with such natural love and wondered what might have been, wondered how she could ever doubt she would make a good mother.
The next morning, Mike and Sharon drove us to the ornate Santa Fe railroad station in downtown San Diego. He was healing quickly from the beating he had received in Paradise Valley. He made sure that I knew he had been ambushed and fought through two Taser shocks before he passed out and they got his gun. Beyond that, I was certain that we would never discuss it. I already knew that no bad guys would ever get his firearms without a hell of a fight.
Lindsey and I had tickets to Los Angeles, where we would catch the Coast Starlight to Seattle. We had a sleeping compartment reserved. I carried a bag full of books.
"You two have fun in the cool weather," Sharon said. After she hugged us, Peralta shook my hand and I saw the gratitude in his eyes. Nothing more needed to be said. Then he slipped his hand into Sharon's and, if even for a moment, everything seemed right with the world.
"When you get back," he said, "we've got work to do."
I had no doubt. I offered my arm, and Lindsey stepped up into the train car. I followed her and we found seats. From inside, we watched our friends wave one more time as the locomotive whistle sounded and we started to move.
I turned to Lindsey.
"Have you ever had sex on a train?"
"Not yet, Dave."
That night, as the train rolled through northern California, I made love with my wife and slept without dreams.
_We hope you enjoyed this book._
_To find out about Jon Talton, clickhere._
_To discover more books by Jon Talton, clickhere._
_For an invitation from the publisher, clickhere._
Paying My Debts
My editor Barbara Peters saw the possibilities in this series from the start. I owe her for encouraging me to keep it going, and especially for the skills, intellect, and inspiration that make her America's top editor of mysteries. She styles herself the Evil Editor, but I have only received the good. The Poisoned Pen Press is a treasure, and I am particularly grateful to Rob Rosenwald, Jessica Tribble, Nan Beams, Annette Rogers and Suzan Baroni.
Cal Lash, retired Phoenix Police detective and a private investigator, once again was exceptionally helpful and patient with my questions. Maricopa County Deputy County Attorney David R. Foster likewise provided valuable assistance.
Even before I finished my previous book, Powers of Arrest, A Cincinnati Casebook, readers wanted to know when the next David Mapstone Mystery would be coming. So I owe you all my biggest debt, whether you started the series at its outset in 2001 or recently got hooked. It's humbling to see how many people are moved by the lives of these characters, including the biggest of all, Phoenix. Thank you.
About this Book
The desert city of Phoenix, Arizona, has its secrets and its not about to give them up easily. Ex-cop Mike Peralta and ex-cold case expert David Mapstone have formed a detective agency to unearth them. And it's started badly...
Moments after hiring them, their first client is gunned down. And the deaths don't stop there. Someone is killing their clients, killing everyone connected with their cases, and is now coming for them. Why? The answer will lead Mapstone and Peralta into the world of human trafficking, corrupt politics, and the white supremacist movement.
Reviews
"A great read!"
_Michael Connelly_
"A strong new voice in contemporary hard-boiled fiction."
_Washington Post_
"Talton has created a richly complex character in Mapstone... a well-crafted, nuanced series."
_Booklist_
"A haunting noir story vividly rendered by Talton's white-hot prose... original... impressively unyielding."
_New York Journal of Books_
"Talton crisply evokes Phoenix's New West ambience and keeps readers guessing with unexpected plot twists."
_Publishers Weekly_
About this Series
**PHOENIX COLD CASES**
David Mapstone: ex-cop, ex-history professor.
As a cop, Mapstone learned never to trust anyone. As a historian, he learned that the past is never past and everything is connected.
Now he's digging into Phoenix, Arizona's secrets. Buried secrets that the city is not about to give up easily.
Just as well he's armed with a .357 magnum Colt Python
_**1.Concrete Desert **_
A young woman's body is found dumped in the desert in circumstances identical to those of an infamous 40-year-old unsolved murder.
_Concrete Desert_ is available here.
_**2.Cactus Heart**_
Mapstone unearths the skeletons of a pair of four-year-old twins, the victims in a notorious Depression-era kidnapping case. But what should be a matter of tying up loose ends quickly becomes something more sinister, more personal... and more deadly.
_Cactus Heart_ is available here.
_**3.Camelback Falls**_
When Sheriff Peralta is shot by a sniper, Mapstone must confront his own past and the deadly consequences of a small-town shoot-out in 1979 that left Peralta and Mapstone standing over four dead bodies.
_Camelback Falls_ is available here.
_**4.Dry Heat**_
Half a century after the unsolved murder of an FBI agent, the victim's missing badge is found on the body of a dead transient. The trail leads Mapstone into the Arizonan desert, and eventually to San Francisco, as he slowly uncovers the bloody secrets surrounding the FBI badge.
_Dry Heat_ is available here.
_**5.Arizona Dreams**_
Mapstone receives a letter confessing to a forty-year-old murder and providing directions to the body. But things are never what they seem in Phoenix. There's a body right where the letter said it would be – but it's only weeks old, not years...
_Arizona Dreams_ is available here.
_**6.South Phoenix Rules**_
Phoenix, Arizona in August. It's 114 degrees in the shade but it's going to get even hotter for cold case investigator David Mapstone when he starts investigating a drug cartel execution.
_South Phoenix Rules_ is available here.
_**7.The Night Detectives**_
Mapstone and Peralta are Phoenix's newest private detective agency. But someone is killing their clients, killing everyone connected with their cases, and is now coming for them. Why?
_The Night Detectives_ is available here.
About the Author
JON TALTON is a fourth-generation Arizonan who grew up in the same Phoenix neighbourhood that David Mapstone calls home. A journalist of more than twenty years, he now lives in Washington state where he is the economics columnist for the _Seattle Times_ and writes the _Rogue Columnist_ blog.
A Letter from the Publisher
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HeadofZeusBooks
The story starts here.
First published in the UK in 2013 by Head of Zeus Ltd
Copyright © Jon Talton, 2013
The moral right of Jon Talton to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act of 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of both the copyright owner and the above publisher of this book.
This is a work of fiction. All characters, organizations, and events portrayed in this novel are either products of the author's imagination or are used fictitiously.
9 7 5 3 1 2 4 6 8
A CIP catalogue record for this book is available from the British Library.
ISBN (E) 9781781851739
Head of Zeus Ltd
Clerkenwell House
45-47 Clerkenwell Green
London EC1R 0HT
www.headofzeus.com
Contents
Cover
Welcome Page
Dedication
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
Chapter 35
Chapter 36
Chapter 37
Chapter 38
Chapter 39
Chapter 40
Paying My Debts
About this Book
Reviews
About this Series
About the Author
An Invitation from the Publisher
Copyright
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{"url":"http:\/\/www.physicsforums.com\/showthread.php?p=3672709","text":"Parallelogram: Sum of the squares of the sides = sum of the squares of the diagonals?\n\nTags: diagonals, parallelogram, sides, squares\n P: 184 1. The problem statement, all variables and given\/known data Apply the formula for the distance between two points to prove the well-known theorem: In a parallelogram the sum of the squares of the sides is equal to the sum of the squares of the diagonals. 2. Relevant equations It gave a hint saying to put one of the parallelogram's vertices at the origin. Also, it hinted to keep in mind that x-values can only be positive in the 1st and 4th quadrants. It says \"upon realizing that, the proof of the theorem reduces to checking a simple algebraic identity. Which?\" Distance formula is \u221a((x1-x2)^2+(y1-y2)^2) 3. The attempt at a solution Okay what I know about a parallelogram is that its opposite sides are parallel and its opposite angles are equal. What do they mean by \"reduces to checking a simple algebraic identity\"? I am lost\n HW Helper Thanks P: 9,681 Draw a picture of a parallelogram. One vertex at the origin, and given the coordinates of two other vertexes, how do you get the coordinates of the fourth vertex? ehild Attached Thumbnails\nP: 184\n Quote by ehild Draw a picture of a parallelogram. One vertex at the origin, and given the coordinates of two other vertexes, how do you get the coordinates of the fourth vertex? ehild\nThanks ehild,\n\nIf vertex A is unknown you can use the equations Ax = Cx-Bx and Ay = Cy-By\nIf vertex B is unknown, use: Bx = Cx-Ax and By = Cy-Ay\nIf vertex C is unknown, use: Cx = Ax+Bx and Cy = Ay+By\n\nThe distance squared to the diagonal from the origin to vertex C is given by Cx^2 + Cy^2\n\nand the distance squared of the other diagonal is given by (Bx-Ax)^2 + (By-Ay)^2\n\nso we have...\n\nCx^2 + Cy^2 + (Bx-Ax)^2 + (By-Ay)^2 = Ax^2 + Ay^2 + Bx^2 + By^2 + (Cx-Ax)^2 + (Cy-Ay)^2 + (Bx-Cx)^2 + (By-Cy)^2\n\nSince (C-A)^2 = B^2 and (B-C) = A^2 we can replace those on the right side of the equation and simplify it to 2A^2 + 2B^2\n\nnow we have to simplify the left side of the equation and we can do that by expanding the (B-A)^2 parts and replacing C with (A+B) and then expanding which will then cause that side to equal 2A^2 + 2B^2 as well. And that is the solution?\n\nI am curious what simple algebraic identity we are checking?\n\nHW Helper\nThanks\nP: 9,681\n\nParallelogram: Sum of the squares of the sides = sum of the squares of the diagonals?\n\nLooking at the problem more carefully, it is totally wrong. The sum of the squares of the diagonals is twice the sum of the squares of the sides.\n\nIt is simpler to derive with vectors. If the origin is one vertex of the parallelogram, $\\vec{a}$ and $\\vec{b}$ are the two sides. One diagonal is $\\vec{d_1}=\\vec{a}+\\vec{b}$, the other diagonal is $\\vec{d_2}=\\vec{a}-\\vec{b}$.\n\nThe square of the magnitude of a vector is equal to its scalar product by itself.\n\n$d_1^2=\\vec{d_1}\\cdot \\vec{d_1}=(\\vec {a}+\\vec {b})\\cdot(\\vec{a}+\\vec{b})=\\vec a^2+\\vec b^2+2\\vec a \\cdot \\vec b$\n\n$d_2^2=(\\vec {a}-\\vec {b})\\cdot(\\vec{a}-\\vec{b})=\\vec a^2+\\vec b^2-2\\vec a \\cdot \\vec b$\n\n$d_1^2+d_2^2=2 a^2+2 b^2$\n\nIf you know the law of cosines, the derivation is even more simple. See: http:\/\/mathworld.wolfram.com\/Parallelogram.html\n\nehild\nHW Helper\nP: 4,534\n Quote by ehild Looking at the problem more carefully, it is totally wrong.\nNo it isn't. You've got yourself confused. Here, you'll need these:\n\n The sum of the squares of the diagonals is twice the sum of the squares of the sides.\nCan something be both right and wrong at the same time?\n\n It is simpler to derive with vectors.\n\nYour working is all correct. You just blundered at interpreting the result.\n $d_1^2+d_2^2=2 a^2+2 b^2$\nLet's rewrite this in a form that gives a clear reminder that a parallelogram has four sides:\n$$d_1^2+d_2^2=a^2 + a^2+ b^2 +b^2$$\nQED\nHW Helper\nThanks\nP: 9,681\n Quote by NascentOxygen Let's rewrite this in a form that gives a clear reminder that a parallelogram has four sides: $$d_1^2+d_2^2=a^2 + a^2+ b^2 +b^2$$ QED\nOhhh! Of course, there are four sides... That happens when one sees only the formula, without thinking of reality.\n\nAnd that simple algebraic identity can be that (a+b)2+(a-b)2=2a2+2b2.\nHW Helper\nP: 4,534\n Quote by ehild And that simple algebraic identity can be that (a+b)2+(a-b)2=2a2+2b2.\nGood, that explains one hint.\n\nAny thoughts on the second?\n Quote by nickadams Also, it hinted to keep in mind that x-values can only be positive in the 1st and 4th quadrants.\nHW Helper\nThanks\nP: 9,681\nDo you think I can read the mind of the unknown maker of the problem? I try:\n\n it hinted to keep in mind that x-values can only be positive in the 1st and 4th quadrants. Upon realizing that, the proof of the theorem reduces to checking a simple algebraic identity.\nI guess he had Cosine Law in mind. The shorter diagonal d1 is opposite to the smaller angle, \u03b8 <pi\/2 in the first or fourth quadrant, and the longer diagonal d2 is opposite to the angle 180-\u03b8. If a and b are the lengths of the sides, the square of the diagonals are\n\n$d_1^2=a^2+b^2-2ab\\cos(\\theta)$\n$d_2^2=a^2+b^2-2ab\\cos(180-\\theta)=a^2+b^2+2ab\\cos(\\theta)$\n\ncos(180-\u03b8) =-cos(\u03b8), therefore\n\nd12+d22=2(a2+b2)\n\nBut then I do not know what is the simple algebraic identity. cos(180-\u03b8) =-cos(\u03b8) is simple but it is not algebraic?\n\nehild\nP: 700\n Quote by ehild Do you think I can read the mind of the unknown maker of the problem? I try: I guess he had Cosine Law in mind. The shorter diagonal d1 is opposite to the smaller angle, \u03b8\n\nHello ehild !\nYou don't have to do so much of calculation. Its just simple coordinate geometry.","date":"2014-03-10 23:14:27","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\": 0.7993810772895813, \"perplexity\": 514.115973963439}, \"config\": {\"markdown_headings\": false, \"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-2014-10\/segments\/1394011042531\/warc\/CC-MAIN-20140305091722-00082-ip-10-183-142-35.ec2.internal.warc.gz\"}"} | null | null |
Milo Historical Society
12 High Street - Milo, Maine
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Sixth Annual Art Show – Jul 27th
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Scheduled Program Postponed
Allen Monroe
The Milo Historical Society program, Milo's Native History, scheduled for April 27 at the Milo Town Hall at 6 PM, has been postponed until a later date due to circumstances beyond our control. Thank you.
The next monthly meeting of the Milo Historical Society will be on Thursday, April 20 at 7 PM at the museum on High Street (next to Bailey Lumber Co.). Anyone with an interest in our community's history is welcomed and encouraged to attend.
Milo, Maine
society@milohistorical.org
175th Commemorative Blanket
The Milo Historical Society Museum is closed for the season. Appointments may be made to visit through the month of October by calling Allen at 943-2268.
Minutes 11-19-15
Welcome to the Curator's Corner
Where does Milo's name come from?
Local Folklore relates that Theophilus Sargent, one of our community's earliest settlers, was given the honor of choosing the town's name. His choice was in all probability based upon the following Greek legend:
Milo (6th Century B.C.)
Milo was a famous Greek athlete in the latter part of the 6th century B.C. It was claimed that he once carried a four-year-old cow through the stadium at Olympia. Afterwards, he ate the whole animal. The story goes that, as an old man, he tried to tear an oak tree in two, but the trunk closed on his hands and pinned him to the tree. While held there, he was attacked and devoured by wolves.
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Copyright © 1996-2019 Three Rivers Community of Maine
© 1996-2019 Milo Historical Society | {
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There are so many complicated diets and crazy "easy" recipes made with the new super food of the week. I know those had the opposite effect on me. I got overwhelmed and then had a bowl of cereal for dinner. Nutrition can be really simple! Start by putting your foods into 4 different categories: Carbs, Protein, Fat, other Vegetables.
So, ok how is this easy? First start with some staples. I try to keep the list below on hand regularly. I actually made a printable grocery list with all of the necessities so I make sure I always remember to get them. You will see that we love dairy in my house!
This is the easiest side for dinners. 90 seconds in the microwave and done!
Veggies (choose your favorites, but try to add variety): Asparagus, broccoli, bell peppers (a few colors), cucumber, baby spinach or kale, celery. I cut up some of the raw veggies (peppers, cucumber, and celery) to keep readily available in the refrigerator, then I just throw a handful on the side of my plate for lunch.
Then, for every meal, pick something from each category (use the appropriate portion size) AND then pile on the veggies. Stay tuned next week for some examples of how I put my meals together. | {
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Q: plotting 3d vectors (arrays) in python Using numpy Packet i produced vectors (arrays) contain x,y,z-coordinates of several atoms in a protein. I would like to plot these vectors.
Does anybody know how to do this?
Since I could't plot the array itself, I tried to plot the coordinations of the atoms with a loop as following:
from matplotlib import pyplot as pot
import matplotlib as mpl
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
for i in range(3):
plt.plot(atom1[i],atom2[i])
In this case I got the following error message: TypeError: object of type 'float' has no len()
I appreciate any help :)
A: The error occurs because plt.plot(x, y) expects x and y to be lists or arrays which have a length while you have given it a float. You could avoid this by enclosing [atom1[i]] in square brackets making it a list.
However, it is usually best to avoid this as it is unclear what is going on. Instead of looping through each atom just stick them all together into one array then plot the columns of the array. You may even find when you create the atoms you can just create them in an array in the first place. Example:
from matplotlib import pyplot as plt
import matplotlib as mpl
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
# Define several atoms, these are numpy arrays of length 3
# randomly pulled from a uniform distribution between -1 and 1
atom1 = np.random.uniform(-1, 1, 3)
atom2 = np.random.uniform(-1, 1, 3)
atom3 = np.random.uniform(-1, 1, 3)
atom4 = np.random.uniform(-1, 1, 3)
# Define a list of colors to plot atoms
colors = ['r', 'g', 'b', 'k']
# Here all the atoms are stacked into a (4, 3) array
atoms = np.vstack([atom1, atom2, atom3, atom4])
ax = plt.subplot(111, projection='3d')
# Plot scatter of points
ax.scatter3D(atoms[:, 0], atoms[:, 1], atoms[:, 2], c=colors)
plt.show()
I added the colors since it is helpful to see which atom is which.
| {
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\section{Introduction}
Partial Differential Equations (PDEs) \cite{PDE_book} are commonly used for the mathematical formulation of the physical behavior of many engineering and physical systems. They capture the continuous dynamics of a system by providing a mathematical relationship between various components of the underlying system by incorporating changes in their associated properties. Due to these distinguishing features, they are broadly used in analyzing many physical phenomena such as, heat or sound propagation, electrodynamics, quantum mechanics and fluid dynamics. For example, they play a pivotal role in the thermal analysis of a system by formulating a general heat equation that can be analyzed using various appropriate boundary and initial conditions \cite{ref_book1}. Similarly, this kind of thermal analysis is a foremost step in the design of many safety-critical applications, such as aerospace, nuclear power plants and automobile engines.
The phenomenon of heat transfer/propagation can occur by three different means, namely, heat conduction \cite{ref_book1}, convection \cite{ref_book2}, and thermal radiation \cite{ref_book4f}. Heat conduction or diffusion is the flow of energy in a system/body from the region of high temperature to the region of low temperature by direct collision of molecules. Whereas, convection refers to the transfer of the energy due to the physical movement of a bulk fluid. Thermal radiation is the transfer of energy in the form of electromagnetic wave. Heat conduction is the most important type of heat transfer and it is commonly used to analyze problems arising in the design and operation of industrial appliances, such as heat exchanger and compressors. The first step for analyzing the heat conduction in a given system/body is to construct a mathematical model of the dynamics of the system, such as heat distribution using the heat equation, which is a PDE. These dynamics provide the variation of the temperature as a function of position/space and time within the heat conducting system/body. The heat distribution (temperature field) usually depends on boundary conditions, initial conditions, material properties, and the geometry of the body. The next step in the heat conduction analysis is to find the solution of the heat equation modeled in the first step that can be obtained by determining a temperature distribution that is consistent with the initial and boundary conditions.
Heat equations are generally analyzed using numerical techniques \cite{ref_book5f} or analytical methods \cite{ref_book6f}. The two most widely used numerical techniques for analyzing PDE based heat equations are Finite Difference \cite{ref_finite_diff} and Finite Element \cite{ref_finite_elem} methods. These methods can solve the complex heat conduction problems by providing the closed-from solutions. However, they involve approximation and rounding of the associated mathematical expressions and thus cannot ensure absolute correctness of the results of the associated analysis. Unlike numerical solutions, the analytical methods for analyzing heat conduction do not involve any approximation of the associated mathematical expressions and thus are preferred on numerical methods for ensuring the correctness of the results. Some commonly used analytical techniques for solving the heat conduction problem are separation of variables \cite{ref_book7f} and transform methods \cite{ref_book8f}.
Conventionally, the heat conduction problem has been analyzed using paper-and-pencil proof and computer based numerical and symbolic methods. However, the former is human-error prone and it is not well-suited for large systems involving extensive human manipulation. Moreover, the required assumptions are not all explicitly mentioned in the analysis, which may lead to inaccurate results. Similarly, the numerical and symbolic methods are based on approximation of the mathematical results due to the finite precision of computer arithmetic. Moreover, the core of the tools involved in the symbolic methods based analysis has a large number of unverified algorithms that puts a question mark on the accuracy of the associated analysis. Given, the safety-critical nature of many systems, these conventional techniques cannot ensure absolute accuracy of the analysis.
As an alternative to related methods and tools, in this paper, we propose to use higher-order-logic theorem proving \cite{ref_book4} for formally analyzing the heat conduction problem in rectangular coordinates and thus overcome the above-mentioned inaccuracy limitations. In particular, we formally model the heat equation using the multivariable calculus theories of the HOL Light theorem prover capturing the heat conduction in the system/body. Next, to formally analyze the heat equation, we use the separation of variables method \cite{ref_book7f} to formally verify the solution of the PDE by incorporating all relevant boundary and initial conditions. One of the primary reasons for choosing HOL Light for the proposed work is the availability of rich theories of multivariable calculus, such as differential, integration, transcendental and real analysis. The HOL Light codes of our formalization is available at \cite{h_light}.
The remainder of the paper is structured as follows: In Section \ref{section2}, we provide an overview of related work on differential equations based formal analysis. Section \ref{section3} describes some fundamentals of the multivariate analysis libraries of the HOL Light theorem prover that are necessary for understanding the rest of the paper. We provide the formalization of the heat equation in rectangular coordinates in Section \ref{section4}. Section \ref{section5} presents the formal verification of the solution of the heat equation. Finally, Section \ref{section6} concludes the paper.
\section{Related Work}
\label{section2}
Many higher-order-logic theorem provers, such as HOL Light\footnote{\url{https://www.cl.cam.ac.uk/ jrh13/hol-light/}}, HOL4\footnote{\url{https://hol-theorem-prover.org/}}, Isabelle/\\HOL\footnote{\url{https://isabelle.in.tum.de/}}, Coq\footnote{\url{https://coq.inria.fr/}} and Mizar\footnote{\url{ http://www.mizar.org/}} have been used for the differential equations based formal analysis of the engineering and physical systems. For instance, Immler et al. \cite{ref_lncs1} used Isabelle/HOL for formally verifying the numerical solutions of Ordinary Differential Equation (ODE). The authors formalized the Initial Value Problems (IVPs) and formally verified the existence of a unique solution of the ODE. Moreover, the authors provide an approximation of the solution using the Euler's method. Immler et al. \cite{ref_article1} presented a formal reasoning support about the flow of ODEs using Isabelle/HOL. In particular, the authors formally verified a solution of ODEs incorporating various initial conditions. They also formalized the Poincaré map and formally verified its differentiability. However, both these approaches rely on approximating the solutions of differential equations representing the dynamical behavior of the underlying system. Guan et al. \cite{ref_article2} used the HOL Light theorem prover to formalize the Euler-Lagrange equation set that is based on Gâutex derivatives. In addition, the authors used their proposed formalization for formally verifying the least resistance problem of gas flow. Similarly, Sanwal et al. \cite{sanwal} formally verified the solutions of the second-order homogeneous linear differential equations using the HOL4 theorem prover. Moreover, they used their proposed formalization for formally verifying the damped harmonic oscillator and a second-order op-amp circuit. Rashid et al. formalized the Laplace \cite{rashid1} and the Fourier \cite{rashid2} transforms using HOL Light and used these formalization for differential equations based analysis of many systems, such as automobile suspension system \cite{rashid2}, unmanned free-swimming submersible vehicle \cite{rashid3} and platoon of automated vehicles \cite{rashid4}. However, the existing formalization of ODEs in HOL4 and HOL Light, respectively, do not provide the formalization of the solution when dealing with separable linear partial differential equations.
Boldo et al. \cite{ref_bol} utilized the Coq theorem prover for formally verifying the numerical solution of one-dimensional acoustic wave equation. The authors used the second-order centered finite difference scheme, commonly known as the three-point scheme for convergence of the result. Similarly, Boldo et al. \cite{ref_article_boldo} mechanically verified the correctness of a C program implementing numerical scheme for the solution of PDE using both automated and interactive theorem provers. Despite important contributions, both these works approximate the solution of acoustic wave equation and did not provide analytical solution. Otsuki et al. \cite{ref_article4} formalized the method of separation of variables and superposition principle and used it for analyzing a one-dimensional wave equation using the Mizar theorem prover. However, they did not extend the solution for the infinite series. In the work we propose in this paper, we provide, for the first time, the formalization in HOL of the heat equation, in the form of a PDE modeling temperature variation for a rectangular solid. We conduct the formal verification in HOL Light of useful properties of the heat equation as well as verify its infinite series solution.
\section{Preliminaries}
\label{section3}
In this section, we provide an overview of some of the fundamental formal definitions and notations of the multivariate calculus theories of HOL Light that are necessary for understanding the rest of the paper. The derivative of a real-valued function is defined in HOL Light as follows:
\begin{definition}
\label{DEF:real_deriv} \emph{\textit{Real Derivative}} \\
{\small\textup{\texttt{$\vdash$ $\forall$f x.
real\_derivative f x = (@f'. (f has real derivative f') (atreal x))}}}
\end{definition}
The function \texttt{real\_derivative} accepts a real valued function \texttt{f} that needs to be differentiated and a real number \texttt{x}, and provides the derivative of \texttt{f} with respect to \texttt{x}. It is formally represented in functional form using the Hilbert choice operator @. The function \texttt{has\_real\_derivative} expresses the same functionality in relational style.
\begin{definition}
\label{DEF:higher_real_deriv} \emph{\textit{Higher Real Derivative}} \\
{\small\textup{\texttt{$\vdash$ $\forall$f x.
higher\_real\_derivative 0 (f:real$\rightarrow$real) (x:real) = f x $\wedge$ \\
\hspace*{0.4cm} (!n. higher\_real\_derivative (SUC n) (f:real$\rightarrow$real) (x:real) = \\
\hspace*{2.2cm} (real\_derivative ($\lambda$x. higher\_real\_derivative n f x) x))}}}
\end{definition}
The HOL Light function \texttt{higher\_real\_derivative} accepts an order \texttt{n} of the derivative, a real-valued function \texttt{f} and a real number \texttt{x}, and provides a higher-order derivative of order \texttt{n} for the function \texttt{f} with respect to \texttt{x}.
The infinite summation over a function \texttt{f}: $\mathbb{N}$ $\rightarrow$ $\mathbb{R}$ is formalized in HOL Light as follows:
\begin{definition}
\label{DEF:real_sums} \emph{\textit{Real Sums}} \\
{\small\textup{\texttt{$\vdash$ $\forall$s f L.
real\_sums (f real\_sums l) s $\Leftrightarrow$ \\
\hspace*{1.2cm} (($\lambda$n. sum (s INTER (0..n)) f) $\rightarrow$ l) sequentially}}}
\end{definition}
The HOL Light function \texttt{real\_sums} accepts a set of natural numbers \texttt{s}: \texttt{$\mathbb{N}$} $\rightarrow$ \texttt{bool}, a function \texttt{f}: $\mathbb{N}$ $\rightarrow$ $\mathbb{R}$ and a limit value \texttt{l}: $\mathbb{R}$, and returns the traditional mathematical expression $\sum\limits_{k = 0}^{\infty}{f(k)} = L$. Here, \textsf{\texttt{INTER}} captures the intersection of two sets. Similarly, \texttt{sequentially} represents a net providing a sequential growth of a function $f$, i.e., $f(k), f(k + 1), f(k + 2), . . . ,$ etc. This is mainly used in modeling the concept of an infinite summation.
We provide the formalization of the summability of a function \texttt{f}: $\mathbb{N}$ $\rightarrow$ $\mathbb{R}$ over \texttt{s}: $\mathbb{N}$ $\rightarrow$ \texttt{bool}, which ensures that there exist some limit value \texttt{L}: $\mathbb{R}$, such that $\sum\limits_{k = 0}^{\infty}{f(k)} = L$ in HOL Light as:
\begin{definition}
\label{DEF:real_summability} \emph{\textit{Real Summability}} \\
{\small\textup{\texttt{$\vdash$ $\forall$s f.
real\_summable s f = ?l. (f real\_sums l)}}}
\end{definition}
Now, we provide a formalization of an infinite summation, which will be used in the formal analysis of the heat conduction problem in Section \ref{section5} of the paper.
\begin{definition}
\label{DEF:real_infsum} \emph{\textit{Real Infsum}} \\
{\small\textup{\texttt{$\vdash$ $\forall$s f.
real\_infsum s f = @l. (f real\_sums l) s}}}
\label{dp5}
\end{definition}
\noindent where the HOL Light function \texttt{real\_infsum} accepts \texttt{\textsf{s}}: \texttt{\textsf{num}} $\rightarrow$ \texttt{\textsf{bool}} specifying the starting point and a function \texttt{f} of data-type $\mathbb{N}$ $\rightarrow$ $\mathbb{R}$, and returns a limit value \texttt{l}: $\mathbb{R}$ to which the infinite summation of \texttt{f} converges from the given \texttt{s}.
An infinite summation of a real-valued function Definition \ref{dp5} can be mathematically expressed in an alternate form as follows:
\begin{equation*}
\sum_{w=0}^{\infty}f_{w}(x) = \lim_{N \to \infty}\sum_{w=0}^{N}f_{w}(x)
\end{equation*}
We proved this equivalence in HOL Light as follows:
\begin{theorem}
\label{THM:alternative_infsum} \emph{\textit {Alternate Representation of an Infinite Summation}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$f k s. real\_infsum s ($\lambda$w. f w x) = \\
\hspace*{1.5cm} reallim sequentially ($\lambda$k. sum (s INTER (0..k))($\lambda$w. f w x))}}}
\end{theorem}
\section{Formalization of the Heat Conduction Problem}
\label{section4}
Heat conduction is a phenomenon of energy transfer that occurs due to differences in temperature in adjacent components of a body/system. The heat is transferred from the high-temperature side to the low-temperature side until the body reaches its thermal equilibrium. The heat conduction or temperature variation can be mathematically defined as a function of space and time. Generally, the heat conduction in a body is three dimensional i.e., the conduction is significant in all three dimensions and a temperature variation in a body can be modeled as $T = T (x, y, z, t)$. The heat conduction is said to be two-dimensional when the conduction is significant in two-dimensions and negligible in the third dimension. Similarly, it is one-dimensional when the conduction is significant in one-dimensional only and the temperature variable can be modeled as $T = T (x, t)$. In this paper, we focus on the formalization of the one-dimensional heat conduction problem. In particular, we formally model the temperature variation in a rectangular slab using a PDE as a heat equation and formally verify its analytical solution by the method of separation of variables based on various boundary and initial conditions.
\subsection{Heat Conduction Problem Formulation}
A heat conduction problem for a rectangular slab having a thickness $L$ is depicted in Figure \ref{figure1}. We consider it as a one-dimensional heat conduction problem. Here, the function $u (x, t)$ provides the temperature in the slab at a point $x$ and time $t$ \cite{ref_book4s}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.4 \textwidth]{nf1.png}
\caption{Heat Conduction Across Thickness of a Slab \cite{ref_book_app}}
\label{figure1}
\end{figure}
We can mathematically express the one-dimensional heat conduction (temperature variation) in the rectangular slab as follows \cite{ref_book_app}:
\begin{equation}
\frac{\partial u(x,t)}{\partial t} = c \dfrac{\partial^{2}u(x,t)}{\partial x^{2}} \quad\quad 0 < x < L, \quad t > 0
\label{1}
\end{equation}
where c is the thermal diffusivity of the slab that depends on the material used for constructing the slab. Equation (\ref{1}) can be equivalently written as:
\begin{equation*}
\frac{\partial u(x,t)}{\partial t} - c \dfrac{\partial^{2}u(x,t)}{\partial x^{2}} = 0
\end{equation*}
Moreover, the solution of the heat equation (Equation (\ref{1})) should satisfy the following initial and boundary conditions.\\
Initial Condition:
\begin{equation}
u (x,t) \mid_{t = 0} = u (x,0) = f(x)
\label{2}
\end{equation}
\hspace*{0.35cm} Boundary Conditions:
\begin{equation}
u (x,t) \mid_{x = 0} = u (0,t) = 0
\label{3}
\end{equation}
\begin{equation}
u (x,t) \mid_{x = L} = u (L,t) = 0
\label{4}
\end{equation}
The heat equation (Equation (\ref{1})) along with Equations (\ref{2}), (\ref{3}) and (\ref{4}) is known as the initial boundary-value problem. It becomes an initial-value problem with respect to time that considers the only initial condition represented by Equation (\ref{2}). Whereas, in the case of its dependence on space only, it represents a boundary-value problem by incorporating the two boundary conditions expressed as Equations (\ref{3}) and (\ref{4}). Next, to formally verify the solution of the heat equation, we need to formalize it in higher-order logic.
\subsection{Formalization of the Heat Equation}
We formalize the heat equation (Equation (\ref{1})) capturing the one-dimensional heat conduction in a rectangular slab in HOL Light as follows:
\begin{definition}\emph{\textit{The Heat equation}}{\small \\
\textup{\texttt{$\vdash$ heat\_equation u(x,t) c $\Leftrightarrow$ heat\_operator u (x,t) c = \&0}}}
\label{d6}
\end{definition}
where \texttt{heat\_equation} accepts a function \texttt{u} of type ($\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R})$, a space variable \texttt{x: $\mathbb{R}$}, a time variable \texttt{t: $\mathbb{R}$} and the thermal diffusivity constant \texttt{c}, and returns the corresponding heat equation. The function \texttt{heat\_operator} is formalized as follows:
\begin{definition}\emph{\textit{Heat operator}}{\small \\
\textup{\texttt{$\vdash$ $\forall$u x t. \\
heat$_{-}$operator u (x,t) c = higher\_real\_derivative 1 ($\lambda$t. u (x,t)) t - \\
\hspace*{3.6cm} c * higher\_real\_derivative 2 ($\lambda$x. u (x,t)) x}}}
\label{d7}
\end{definition}
Next, we verify a few important properties of the \texttt{heat\_operator} Definition \ref{d7} that are required in formally verifying the solution of the heat equation.
\\
\\
\begin{theorem}
\label{THM:linearity} \emph{\textit{Linearity}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$u x t a b.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] ($\forall$t. ($\lambda$t. u (x,t)) real\_differentiable atreal t) $\wedge$
\item [{[A2]}] ($\forall$t. ($\lambda$t. v (x,t)) real\_differentiable atreal t) $\wedge$
\item [{[A3]}] ($\forall$x. ($\lambda$x. u (x,t)) real\_differentiable atreal x) $\wedge$
\item [{[A4]}] ($\forall$x. ($\lambda$x. v (x,t)) real\_differentiable atreal x) $\wedge$
\item [{[A5]}] ($\forall$x. ($\lambda$x. real\_derivative ($\lambda$x. u (x,t)) x) \\
\hspace*{1cm} real\_differentiable atreal x) $\wedge$
\item [{[A6]}] ($\forall$x. ($\lambda$x. real\_derivative ($\lambda$x. v (x,t)) x) \\
\hspace*{1cm} real\_differentiable atreal x)
\end{enumerate}
$\Rightarrow$ (heat\_operator ($\lambda$(x,t). u (x,t) + v (x,t)) (x,t) c =\\
\hspace*{1.0cm}heat\_operator ($\lambda$(x,t). u (x,t)) (x,t) c +\\ \hspace*{1.2cm} heat\_operator ($\lambda$(x,t). v (x,t)) (x,t) c)}}}
\end{theorem}
Assumptions \texttt{A1} and \texttt{A2} ensure that the real-valued functions \texttt{u} and \texttt{v} are differentiable at \texttt{t}, respectively. Assumptions \texttt{A3} and \texttt{A4} assert the differentiability of the functions \texttt{u} and \texttt{v} at \texttt{x}, respectively. Similarly, Assumptions \texttt{A5} and \texttt{A6} provide the differentiability conditions for the derivatives of the functions \texttt{u} and \texttt{v} at \texttt{x}, respectively. The proof of the above theorem is mainly based on the properties of derivative and differentiability of real-valued functions.
\begin{theorem}
\label{THM:scalar} \emph{\textit{Scalar Multiplication}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$u x t a.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] ($\forall$t. ($\lambda$t. u (x,t)) real\_differentiable atreal t) $\wedge$
\item [{[A2]}] ($\forall$x. ($\lambda$x. u (x,t)) real\_differentiable atreal x) $\wedge$
\item [{[A3]}] ($\forall$x. ($\lambda$x. real\_derivative ($\lambda$x. u (x,t)) x)\\
\hspace*{1.2cm}real\_differentiable atreal x)
\end{enumerate}
$\Rightarrow$ heat\_operator ($\lambda$(x,t). a * u (x,t)) (x,t) c = \\
\hspace*{3.5cm} a * heat\_operator ($\lambda$(x,t). u (x,t))(x,t) c}}}
\end{theorem}
Assumptions \texttt{A1} and \texttt{A2} ensure that the real-valued function \texttt{u} is differentiable at \texttt{t} and \texttt{x}, respectively. Assumption \texttt{A3} asserts the differentiability condition for the derivative of the function \texttt{u}.
\section{Formal Verification of the Solution of the Heat Equation}
\label{section5}
To find out the solution of the boundary-value problem, i.e., heat equation alongside the boundary conditions Equations (\ref{1}), (\ref{3}) and (\ref{4}), we use the method of separation of variables that reduces the problem of solving a partial differential equation to a problem of solving the equivalent ordinary differential equations. By this method, we can mathematically express the solution of the heat equation $u (x, t)$ as a separable equation as follows:
\begin{equation}
u(x,t) = X(x)W(t)
\label{5}
\end{equation}
\noindent where $X$ and $W$ are functions of $x$ and $t$, respectively. We formalize Equation (\ref{5}) in HOL Light as follows:
\begin{definition}\emph{\textit{Separable}}{\small \\
\textup{\texttt{$\vdash$ $\forall$u X W t x. separable u x t X W = X(x) * W(t)}}}
\end{definition}
By using Equation (\ref{5}) in the heat equation (Equation (\ref{1})) and after simplification, we obtain the following equation.
\begin{equation}
\dfrac{1}{c}\dfrac{\partial [X(x)W(t)]}{\partial t} = \dfrac{\partial ^{2} [X(x)W(t)]}{\partial x^{2}}
\label{6}
\end{equation}
Next, using the property of the partial derivative of a separable function transforms the above equation as follows:
\begin{equation}
\dfrac{1}{c}\dfrac{dW(t)}{dt}X(x) = W(t)\dfrac{d^{2}X(x)}{dx^{2}}
\label{7}
\end{equation}
\noindent where the operator $\frac{d}{dt}$ captures the simple derivative with respect to $t$. We formally verify the equivalence of the left-hand-sides of Equations (\ref{6}) and (\ref{7}) as the following HOL Light theorem.
\begin{theorem}
\label{THM:equivalence} \emph{\textit{Equivalence of Partial and Simple Derivatives (Left-hand Side)}}\\
\textup{{\small
\texttt{$\vdash$ $\forall$u X x W t.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] (X real\_differentiable atreal t) $\wedge$
\item [{[A2]}] (W real\_differentiable atreal t)
\end{enumerate}
\hspace*{0.2cm} $\Rightarrow$
(real\_derivative ($\lambda$t. separable u x t X W) t) = \\ \hspace*{0.8cm} real\_derivative W t * X x
}}}
\label{t4}
\end{theorem}
Assumptions \texttt{A1} and \texttt{A2} provide the differentiability of the functions \texttt{X} and \texttt{W} at \texttt{t}, respectively. The proof process of the above theorem is mainly based on the properties of derivatives and differentiability of the real-valued functions alongwith some arithmetic reasoning. Similarly, we formally verify the equivalence of the right-hand-sides of Equations (\ref{6}) and (\ref{7}) as follows:
\begin{theorem}
\label{THM:eqv_right} \emph{\textit{Equivalence of Partial and Simple Derivatives (Right-hand Side)}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$u X x W t.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] ($\forall$x. X real\_differentiable atreal x) $\wedge$
\item [{[A2]}] ($\forall$x. W real\_differentiable atreal x) $\wedge$
\item [{[A3]}] ($\lambda$x. real$_{-}$derivative X x) real\_differentiable atreal x
\end{enumerate}
$\Rightarrow$ higher\_real\_derivative 2 ($\lambda$x. (separable u x t X W)) x = \\
\hspace*{3.5cm} W t * higher\_real\_derivative 2 ($\lambda$x. X x) x
}}}
\label{t5}
\end{theorem}
Assumptions \texttt{A1} and \texttt{A2} are very similar to that of Theorem \ref{t4}. Assumption \texttt{A3} ensures that the first-order derivative of the real-valued function \texttt{X} is differentiable at \texttt{x}. The verification of Theorem \ref{t5} is similar to that of Theorem \ref{t4}.
Now, after rearranging various terms, Equation (\ref{7}) can be expressed as follows:
\begin{equation}
\dfrac{1}{c}\dfrac{dW(t)}{dt}\dfrac{1}{W(t)} = \dfrac{1}{X(x)}\dfrac{d^{2}X(x)}{dx^{2}} = -\beta^{2}
\label{8}
\end{equation}
\noindent where the left- and right-hand sides are functions of only $t$ and $x$, respectively. The equivalence of these two functions of different variables is only possible when both are equal to some constant, which is represented by $- \beta^2$ in the above equation.
The above equation can be equivalently represented by the following two ordinary differential equations.
\begin{equation}
\dfrac{d^{2}X(x)}{dx^{2}} + \beta^{2}X(x) = 0
\label{9}
\end{equation}
and
\begin{equation}
\dfrac{dW(t)}{dt} + c. \beta^{2}W(t) = 0
\label{10}
\end{equation}
Now, our problem of solving a boundary-value problem Equations (\ref{1}), (\ref{3}) and (\ref{4}) has been transformed to solving a set of linear homogenous differential equations with constant coefficients Equations (\ref{9}) and (\ref{10}). Moroever, the solution of the heat equation Equation (\ref{1}) can be obtained by multiplying the solution of these two equations. \\
The solution of Equation (\ref{9}) is mathematically expressed as:
\begin{equation}
X(x) = Acos(\beta x) + Bsin(\beta x)
\label{11}
\end{equation}
\noindent where $A$ and $B$ are the arbitrary constants that can be computed by applying the boundary conditions. Similarly, the solution of the second differential equation Equation (\ref{10}) is mathematically described as:
\begin{equation}
W(t) = Ce^{-\beta^{2}ct}
\label{12}
\end{equation}
where $C$ is the constant of integration and can be computed by applying the boundary conditions. \\
We formalize the two differential equations Equations (\ref{9}) and (\ref{10}) in HOL Light as follows:
\begin{definition}\emph{\textit{Formalization of Equation (9)}}{\small \\
\textup{\texttt{$\vdash$ $\forall$X x b.
first\_equation X x b $\Leftrightarrow$ \\
\hspace*{0.3cm} higher\_real\_derivative 2 ($\lambda$x. X (x)) x +
b pow (2) * ($\lambda$x. X(x)) x = 0}}}
\label{d9}
\end{definition}
\begin{definition}\emph{\textit{Formalization of Equation (10)}}{\small \\
\textup{\texttt{$\vdash$ $\forall$W t b c.\\
\hspace*{0.3cm} second\_equation W t b c $\Leftrightarrow$ \\
\hspace*{1.5cm} real\_derivative ($\lambda t.$ W (t)) t + c * b pow (2) * W(t) = 0}}}
\label{d10}
\end{definition}
Similarly, we formalized the solutions of these differential equations in HOL Light as:
\begin{definition}\emph{\textit{Solution of First Differential Equation} }{\small \\
\textup{\texttt{$\vdash$ $\forall$A B x b.
first\_equation\_sol A B x b = A * cos(b * x) + B * sin(b * x)}}}
\label{d11}
\end{definition}
\begin{definition}\emph{\textit{Solution of Second Differential Equation} }{\small \\
\textup{\texttt{$\vdash$ $\forall$C c b t.\\
\hspace*{0.3cm} second\_equation\_sol C c b t = C * exp (--c * b pow (2) * t)}}}
\label{d12}
\end{definition}
Next, we formally verify the solution of the first differential equation Equation (\ref{9}) as the following HOL Light theorem:
\begin{theorem}
\label{THM:sol_diff1} \emph{\textit{Solution of First Differential Equation}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$A B x b.\\
\hspace*{0.3cm} (first\_equation ($\lambda x.$ first\_equation\_sol A B x b)) x b}}}
\label{t6}
\end{theorem}
The proof process of the above theorem is based on Definitions \ref{d9} and \ref{d10} and properties of real derivative alongside some real arithmetic reasoning.\\
Similarly, we formally verify the solution of the second differential equation Equation (\ref{10}) as follows:
\begin{theorem}
\label{THM:sol_diff2} \emph{\textit{Solution of Second Differential Equation}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$C c b t.\\
\hspace*{0.3cm} (second\_equation ($\lambda$t. second\_equation\_sol C c b t))(t) b c}}}
\label{t7}
\end{theorem}
The proof process of the above theorem is based on Definitions \ref{d10} and \ref{12} and properties of real derivative alongside some real arithmetic reasoning.
To find out the values of arbitrary constants $A$ and $B$ of the solution of the ordinary differential equation expressed as Equation (\ref{11}), we apply the corresponding boundary conditions. Applying the first boundary condition Equation (\ref{3}) results into $A = 0$. Similarly, the application of the second boundary condition Equation \ref{4} provides $B sin (\beta L) = 0$. We formally verify values of these arbitrary constants based on the corresponding boundary conditions in HOL Light as follows:
\begin{theorem}
\label{THM:arb_a} \emph{\textit{Verification of the Arbitrary Constant A}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$A B x b.\\
\hspace*{0.8cm} x = \&0 $\wedge$ first\_equation\_sol A B x b = \&0 \\
\hspace*{6.0cm} $\Rightarrow$ A = \&0}}}
\label{t8}
\end{theorem}
\begin{theorem}
\label{THM:arb_b} \emph{\textit{Verification of the Arbitrary Constant B}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$A B x b L.\\
\hspace*{0.8cm} x = L $\wedge$ A = \&0 $\wedge$ first\_equation\_sol A B x b = \&0 \\
\hspace*{0.9cm} $\Rightarrow$ first\_equation\_sol x b A B = B * sin(b * L)}}}
\label{t9}
\end{theorem}
The equation $B sin (\beta L) = 0$ holds if $B = 0$ or $sin (\beta L) = 0$. In case of $B = 0$ alongside $A = 0$, it results into $X (x) = 0$. This further provides $u (x, t) = 0$ as a solution to the heat equation, which is an uninteresting trivial solution. This means that $B$ is equal to some non-zero value, which implies that $sin (\beta L) = 0$. Since $\beta$ can have infinitely many values for which $sin (\beta L) = 0$ holds, namely $\beta = \beta_w = \frac{\omega \pi}{L}$. This results into a non-trivial solution of the boundary-value problem as follows:
\begin{equation}
u (x,t) = u_{w}(x,t) = \left[B_{w}sin\left(\dfrac{w\pi x}{L}\right)\right]e^{-\left(\dfrac{w\pi}{L}\right)^{2}ct}
\label{13}
\end{equation}
Now, assume that the function $f(x)$ in initial condition Equation (\ref{2}) is a linear combination of the function $sin (\frac{w \pi x}{L})$, i.e., Fourier sine series representation as follows:
\begin{equation}
f(x) = \sum_{w=1}^{\infty}B_{w}sin\left(\dfrac{w\pi x}{L}\right) \label{14}
\end{equation}
We can mathematically express the general solution of the heat equation as the following equation since it is a linear combination of the non-trivial solutions of the boundary-value problem that satisfies the initial condition expressed as Equation (\ref{14}).
\begin{equation}
u(x,t) = \sum_{w=1}^{\infty}u_{w} (x,t) = \sum_{w=1}^{\infty}B_{w}sin \left(\dfrac{w\pi x}{L}\right)e^{-\left(\dfrac{w\pi}{L}\right)^{2}ct}
\label{15}
\end{equation}
The constant $B_w$ of the Fourier sine series representation of $f(x)$ can be determined using the orthogonality property of the sine function and is mathematically expressed as follows:
\begin{equation}
B_{w} = \dfrac{2}{L}\int_{0}^{L} f(x)sin \left(\dfrac{w\pi x}{L}\right)dx \quad\quad\quad w = 1,2,3...
\label{16}
\end{equation}
We formalize the Fourier sine coefficient in HOL Light as follows:
\begin{definition}
\emph{\textit{Fourier Sine Coefficient}}{\small \\ \textup{\texttt{$\vdash$ $\forall$f w L.\\
\hspace*{0.3cm} fourier\_sine\_coefficient f w L = \\
\hspace*{0.9cm} 2 / L * (real\_integral (real\_interval [0,L])($\lambda$x. (f x) * \\
\hspace*{1.5cm} sin (\&w * pi * x / L)))}}}
\label{d13}
\end{definition}
where \texttt{fourier\_sine\_coefficient} accepts a function \texttt{f : $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$}, a number \texttt{w} and the width of the slab \texttt{L}, and returns a real number representing the Fourier sine coefficient of the function \texttt{f}.\\
Now, the solution of the heat equation capturing the heat conduction in a rectangular slab can be alternatively expressed as:
\begin{equation}
u (x,t) = \sum_{w=1}^{\infty}u_{w} (x,t) = \sum_{w=1}^{\infty}\left(\dfrac{2}{L}\int_{0}^{L} f(x)sin \left(\dfrac{w\pi x}{L}\right)dx \right) sin \left(\dfrac{w\pi x}{L}\right)e^{-\left(\dfrac{w\pi}{L}\right)^{2}ct}
\label{17}
\end{equation}
We formalize the generalized solution of the heat equation (Equation (\ref{17})) in HOL Light as follows:
\begin{definition}
\emph{\textit{Generalized Solution of the Heat Equation}}{\small \\
\textup{\texttt{$\vdash$ $\forall$f x t c L.\\
\hspace*{0.3cm} heat\_solution f x t c L = real\_infsum (from 1) \\
\hspace*{0.9cm} ($\lambda $w. (fourier\_sine\_coefficient f w L) * \\
\hspace*{1.2cm} exp (--c * ((\&w * pi / L) pow 2) * t) * sin (\&w * pi * x / L))}}}
\label{d14}
\end{definition}
The convergence of the generalized solution of the heat equation depends on the convergence of the infinite series $u_w (x, t)$ and is mathematically expressed as the following bound on $u_w (x, t)$.
\begin{equation}
\vert u_{w} (x,t) \vert \leq M_{w}
\end{equation}
where
\begin{equation}
M_{w} = \left(\dfrac{2}{L}\int_{0}^{L} \vert f(x) \vert dx \right)e^{-\left(\dfrac{w\pi}{L}\right)^{2}ct}
\end{equation}
We compute the upper bound $M_w$ using the upper bound on the Fourier coefficient $B_w$, and the fact that $\left \vert sin\left(\dfrac{w \pi x}{L}\right) \right \vert \leq 1$, along with the following property of the integral:
\begin{equation}
\left \vert \int_{a}^{b} f(x)dx \right \vert \leq \int_{a}^{b} \vert f(x)\vert dx.
\end{equation}
Next, we formally verify the convergence of the generalized solution of the heat equation as the following HOL Light theorem.
\begin{theorem}
\label{THM:conv_gen} \emph{\textit{Convergence of the Generalized Solution}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$f x c L t.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] \&0 < L $\wedge$ [A2] \&0 < t $\wedge$ [A3] \&0 < c $\wedge$
\item [{[A4]}] f absolutely\_real\_integrable\_on real\_interval [\&0, L]
\end{enumerate}
$\Rightarrow$ (($\lambda$w. fourier\_sine\_coefficient f w L * \\
\hspace*{1.7cm}exp (--c * (\&w * pi / L) pow 2 * t) sin (\&w * pi * x / L)) \\
\hspace*{3.7cm}real\_sums heat\_solution f x t c L) (from 1)}}}
\label{t10}
\end{theorem}
Assumptions (\texttt{A1-A3}) ensure that the length \texttt{L}, the time \texttt{t} and the constant \texttt{c} are positive real values. Assumption (\texttt{A4}) provides the absolute integrability of the function \texttt{f} over the interval \texttt{[0,L]}. The conclusion presents the convergence of the generalized solution of the heat equation. The verification of above Theorem \ref{t10} is mainly based on the following two important lemmas about the summability of the bound $M_w$ and the generalized solution alongside some real arithmetic reasoning.
\begin{lemma}
\label{THM:sum_bound} \emph{\textit{Summability of the Bound $M_{w}$}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$f c L t.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] \&0 < L $\wedge$ [A2] \&0 < t $\wedge$
[A3] \&0 < c
\end{enumerate}
$\Rightarrow$ real\_summable (from 1) ($\lambda$w. \&2 / L * real\_integral \\
\hspace*{2.9cm}(real\_interval [\&0,L]) ($\lambda$x. abs (f x)) * \\
\hspace*{4.8cm} exp (--c * ((\&w * pi / L) pow 2) * t))}}}
\label{l1}
\end{lemma}
Assumptions (\texttt{A1-A3}) are the same as those of Theorem \ref{t10}. The conclusion of the above lemma provides the summability of the upper bound $M_w$. The verification of Lemma \ref{l1} is mainly based on the Ratio test \cite{calculus} along with some real arithmetic reasoning.
\begin{lemma}
\label{THM:sum_gen} \emph{\textit{Summability of the Generalized Solution}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$f x c L t.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] \&0 < L $\wedge$ [A2] \&0 < t $\wedge$ [A3] \&0 < c $\wedge$
\item [{[A4]}] f absolutely\_real\_integrable\_on real\_interval [\&0, L]
\end{enumerate}
$\Rightarrow$ real\_summable (from 1)($\lambda$w. fourier\_sine\_coefficient f w L * \\
\hspace*{1.3cm}exp (--c * (\&w * pi / L) pow 2 * t) * sin (\&w * pi * x / L))}}}
\label{l2}
\end{lemma}
Assumptions (\texttt{A1-A4}) are the same as those of Theorem \ref{t10}. The verification of Lemma \ref{l2} is mainly based on the Comparison test \cite{calculus} and Lemma \ref{l1} along with some real arithmetic reasoning. More details about the verification of these lemmas and the convergence of the generalized solution of the heat equation can be found in our HOL Light script \cite{h_light}.
Next, we formally verify some interesting properties involving the derivatives of the general solution with respect to position $x$ and time $t$ that capture the heat conduction (variation of temperature) in the rectangular slab with respect to position and time.
\begin{theorem}
\label{THM:deriv_gen} \emph{\textit{Derivative of the Generalized Solution with Respect to Time}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$f x t c L u u'.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] ($\forall$t. (($\lambda$w. (fourier\_sine\_coefficient f w L) * \\
\hspace*{0.3cm}exp (--c * (\&w * pi / L) pow 2 * t) * sin (\&w * pi * x / L)) \\
\hspace*{0.3cm}real\_sums u(x,t)) (from 1)) $\wedge$
\item [{[A2]}] ($\forall$t. (($\lambda$w. --c * (\&w * pi / L) pow 2 * (fourier\_sine\_coefficient \\
\hspace*{0.2cm} f w L) * exp (--c * (\&w * pi / L) pow 2 * t) * \\
\hspace*{0.2cm} sin (\&w * pi * x / L)) real\_sums u'(x,t)) (from 1)) $\wedge$
\item [{[A3]}] (($\lambda$t. u(x,t)) has\_real\_derivative u'(x,t)) (atreal t)
\end{enumerate}
$\Rightarrow$ real\_derivative ($\lambda$t. heat\_solution f x t c L ) t = \\
\hspace*{0.6cm} real\_infsum (from 1) ($\lambda$w. --c * (\&w * pi / L) pow 2) * \\
\hspace*{0.9cm}(fourier\_sine\_coefficient f w L) * \\
\hspace*{1.2cm} exp (--c * (\&w * pi / L) pow 2) * t) * sin (\&w * pi * x / L))}}}
\label{t11}
\end{theorem}
\vspace{-0.2cm}
Assumption \texttt{A1} provides the condition that the infinite series converges to the function $u (x, t)$. Similarly, Assumption \texttt{A2} asserts that the derivative of the infinite series with respect to $t$ converges to the derivative of function $u (x, t)$, i.e. $u'(x, t)$. Assumption \texttt{A3} ensures the function $u$ has derivative $u'(x, t)$ at point $t$. The verification of the above theorem is mainly based on swapping the operation of differentiation and infinite summation alongwith properties of the infinite summation and derivatives.
\begin{theorem}
\label{THM:first_deriv} \emph{\textit{First Derivative of the Generalized Solution with Respect to Space}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$f x t c L u u'.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] ($\forall$x. (($\lambda$w. (fourier\_sine\_coefficient f w L) * exp (--c * (\&w * pi / L) \\
\hspace*{0.3cm}pow 2 * t) * sin (w * pi * x / L)) real\_sums u(x,t)) (from 1)) $\wedge$
\item [{[A2]}] (($\forall$x.(($\lambda$w. (fourier\_sine\_coefficient f w L) * exp (--c * (\&w * pi / L) \\
\hspace*{0.1cm} pow 2 * t) * (\&w * pi/ L) * cos (\&w * pi * x / L)) real\_sums \\
\hspace*{0.1cm} u'(x,t)))(from 1) $\wedge$
\item [{[A3]}] (($\lambda$x. u(x,t)) has\_real\_derivative u'(x,t)) (atreal x)
\end{enumerate}
$\Rightarrow$ real\_derivative ($\lambda$x. heat\_solution f x t c L) x = \\
\hspace*{0.5cm} real\_infsum (from 1)($\lambda$w. (fourier\_sine\_coefficient f w L) * \\
\hspace*{1.55cm}exp (--c * ((\&w * pi / L) pow 2) * t) * (\&w * pi / L) * \\
\hspace*{6.55cm}cos (\&w * pi * x / L)) }}}
\label{t12}
\end{theorem}
The proof process of Theorem \ref{t12} is very similar to that of Theorem \ref{t11}.
\begin{theorem}
\label{THM:second_der} \emph{\textit{Second Derivative of the General Solution with Respect to Space}} \\
\textup{{\small
\texttt{$\vdash$ $\forall$f x t c L u u' u''.
\begin{enumerate}
\setlength\itemindent{0.23cm}
\item [{[A1]}] ($\forall$x. (($\lambda$w. (fourier\_sine\_coefficient f w L) * exp (--c * (\&w * pi / L)\\
\hspace*{0.1cm} pow 2 * t) * sin (\&w * pi * x / L)) real\_sums u (x,t)) (from 1)) $\wedge$
\item [{[A2]}] (($\forall$x.(($\lambda$w. (fourier\_sine\_coefficient f w L) * exp (--c * (\&w * pi / L) \\
\hspace*{0.1cm} pow 2 * t) * (\&w * pi/ L) * cos (\&w * pi * x / L)) real\_sums \\
\hspace*{0.1cm} u'(x,t)))(from 1) $\wedge$
\item [{[A3]}] (($\lambda$x. u(x,t)) has$_{-}$real$_{-}$derivative u' (x,t)) (atreal x) $\wedge$
\item [{[A4]}] (($\forall$x. (($\lambda$w. (fourier\_sine\_coefficient f w L) * exp (--c * (\&w * pi / L) \\
\hspace*{0.1cm} pow 2 * t) * (\&w * pi / L) pow 2 *
--sin (\&w * pi * x / L)) real\_sums \\
\hspace*{0.1cm} u''(x,t))) (from 1) $\wedge$
\item [{[A5]}] (($\lambda$x. u'(x,t)) has\_real\_derivative
u''(x,t)) (atreal x)
\end{enumerate}
$\Rightarrow$ higher\_real\_derivative 2 ($\lambda$x. heat\_solution f x t c L) x = \\
\hspace*{0.5cm} real$_{-}$infsum (from 1)
($\lambda$w. (fourier\_sine\_coefficient f w L) * \\
\hspace*{0.7cm} exp (--c * ((\&w * pi / L) pow 2) * t) * ((\&w * pi / L) pow 2) *\\
\hspace*{0.9cm} --sin (\&w * pi * x / L))}}}
\label{t13}
\end{theorem}
\vspace{-0.1cm}
The verification of the above theorem is mainly based on Theorem \ref{t12} and properties of derivatives along with some arithmetic reasoning.
\vspace{0.1cm}
\subsection* {Discussion}
The distinguishing feature of our proposed formal analysis of the heat conduction problem, as compared to traditional analysis techniques, is that all verified theorems are of generic nature, i.e., all functions and variables involved in these theorems are universally quantified and thus can be specialized based on the requirement of the analysis of a rectangular slab with any width and corresponding boundary and initial conditions. Another advantage of our proposed approach is the inherent soundness of the theorem proving technique. It ensures that all the required assumptions are explicitly present along with the theorem, which are often ignored in conventional simulation based analysis and their absence may affect the accuracy of the corresponding analysis. One of the major difficulties in the proposed formalization was the swapping of the infinite summation and the differential operator that is used in the verification of Theorems \ref{t11}-\ref{t13}. The mathematical proofs available in the literature for this swap operation were very abstract and we developed our own formal reasoning. In addition, to the best of our knowledge, this is the first formal work on the formalization of a one-dimensional heat equation and the verification of its infinite series solution.
\section{Conclusion}
\label{section6}
In this paper, we proposed a HOL theorem proving based approach for formally analyzing the one-dimensional heat conduction in a rectangular slab. We formalized the heat equation and formally verified its linearity and scaling properties. Moreover, we used the separation of variables method for formally verifying the solution of the heat equation incorporating the corresponding boundary and initial conditions. Next, we formally verified convergence of the generalized solution of the heat equation. Finally, we verified some interesting properties regarding the derivatives of the generalized solution of the heat equation that provide useful insights to the variation of the temperature in the body. In future, we plan to formally verify the uniqueness of the generalized solution of the heat equation and its uniform convergence. Another future direction is to formally analyze the heat transfer in composite slabs \cite{composite}, thermal protection systems \cite{analytical} and heat transfer through various thermoelectric devices, such as thermoelectric generator and thermocouple \cite{thermo} that are widely used in many safety-critical systems.
\bibliographystyle{splncs03_unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,013 |
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Caroline Luigini, anomenada Câline, (Lió, 2 de novembre de 1873 - París, 5 de juliol de 1968) va ser una arpista i professora d'arpa francesa d'origen italià, descendent d'una família de músics de Mòdena; son pare va ser el compositor i director d'orquestra Alexandre Luigini.
Es va formar al Conservatori de Lió, i en el concurs anual organitzat per aquest centre ja obtingué el 1892 el primer premi per unanimitat del jurat en la modalitat d'arpa. Un dels seus mestres va ser després Camille Saint-Saëns. Essent auxiliar d'Alphonse Hasselmans, professor d'arpa en el Conservatori de París, tingué entre els seus alumnes a Germaine Tailleferre, de qui es va fer amiga i que va escriure per a ella Le petit livre de harpe de Madame Tardieu (1913-1917), un recull de divuit peces breus per a arpa que ella va utilitzar per als seus concerts i les seues lliçons. Diversos altres compositors, entre els quals Marcel Tournier, van escriure també peces per a arpa expressament per a ella.
Es casà el 2 de juliol de 1902 amb el pintor Victor Tardieu, i va ser mare de l'escriptor Jean Tardieu, que l'evoca tendrament en les seues memòries.
Referències
Músics lionesos
Arpistes francesos
Morts a París
Persones del Roine | {
"redpajama_set_name": "RedPajamaWikipedia"
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Q: is it possible to get insert id without commit in postgresql Now I want to insert a article record into two PostgreSQL(version 13) table, one table contain the article base info, and the other table contains the article content(maybe move to another db in the future). Here is the problem that sometimes one save sql failed make the data lose.
now I want the two action both success or both failed, but the content table must get the base table's id to insert. is it possible to get the base table insert id before commit? so that I could commit at one places. This is my python 3 code:
def save_single(self, guid, pub_time, title, author, content, feed_url):
if content is not None and len(content) > 0:
article = ArticleService()
article = article.save(title, guid, author, pub_time)
content = ArticleContentService()
if article.id is not None:
content.save(article.id, content)
else:
logger.error("article content is null,title:" + title)
what should I do to make data keep consistant? the database orm library is sqlalchemy.
A: No. Most likely your id is autoincrement and is generated when you commit to the database.
As for doing both things at once, yes. If you have a parent:child relationship set up, you can append the child to the parent and just commit the parent.
| {
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Q: How to access highest role based on role priority Hi I have a synerio in that user mapped to two roles and roles have priority set like 1 and 2 ...my question is how to access highest code and role only ...
For exp
Code Name Role RolePriority
1 Rehman Abc 2
2 Rehman Def 3
3 Neha Des 1
So i want Rehman with Def role and Neha with Des role
A: You can use ROW_NUMBER() OVER() to select highest priority
DECLARE @test AS TABLE(Code INT,Name VARCHAR(100),Role VARCHAR(100),RolePriority int)
INSERT INTO @test VALUES(1,'Rehman','Abc',2),(2,'Rehman','Def',3),(3,'Neha','Des',1)
SELECT * FROM(
SELECT *,ROW_NUMBER() OVER(PARTITION BY Name ORDER BY RolePriority desc) AS
rw FROM @test)res
WHERE rw = 1
| {
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SportsFootball
Rams hold off Cowboys to reach NFC Championship Game
Todd Gurley of the Los Angeles Rams scores a 35-yard touchdown in the second quarter against the Dallas Cowboys in the NFC Divisional Playoff game on Jan. 12, 2019. Photo Credit: Getty Images/Harry How
By The Associated Press Updated January 12, 2019 11:39 PM
LOS ANGELES — C.J. Anderson rushed for 123 yards and two touchdowns, and Todd Gurley rushed for 115 more yards and another TD in the Los Angeles Rams' first playoff victory in 14 years, 30-22 over the Dallas Cowboys in the divisional round on Saturday night.
Gurley and Anderson punished the Cowboys' normally sturdy run defense and sent the second-seeded Rams (14-3) to the NFC championship game for the first time in 17 years. Los Angeles racked up a franchise playoff-record 273 yards on the ground — also the most ever allowed in the postseason by the Cowboys, who were playing in their NFL-record 63rd postseason game.
The long-struggling Rams had won only one postseason game since their last trip to the Super Bowl in February 2002, but 32-year-old coach Sean McVay has added his first playoff victory to his spectacular two-season franchise turnaround.
Ezekiel Elliott rushed for a TD and Amari Cooper caught an early TD pass for the Cowboys (11-7), who still haven't won a playoff game on the road in 26 years. After winning the NFC East and beating Seattle last week, Dallas lost in the divisional playoff round for the sixth consecutive time and fell short of its first trip to the NFC championship game since January 1993.
Next weekend, the Rams will face the winner of the other divisional playoff game in New Orleans between the top-seeded Saints and the defending Super Bowl champion Philadelphia Eagles.
The Rams are one win away from another Super Bowl trip after McVay's high-flying, inventive offense largely kept it on the ground, methodically punishing the Cowboys' normally sturdy run defense with their unlikely running back tandem.
"It's scary," Anderson said. "We've got two different styles, and we can keep teams off balance."
The Cowboys, who largely shut down Seattle's league-best rushing attack last week, hadn't allowed two 100-yard rushers in a playoff game since the NFL-AFL merger.
Dak Prescott passed for 266 yards and rushed for a TD with 2:11 to play, but the Cowboys couldn't climb out after falling into a 23-7 hole midway through the third quarter. Elliott managed just 47 yards on 20 carries as Dallas lost for just the second time in its last 10 games.
Jared Goff passed for 186 yards and spent much of the night handing off, but the gangly quarterback improbably scrambled 11 yards for a first down with 1:51 to play, essentially wrapping up his first playoff victory.
"Incredible what they did today," Goff said of his offensive line. "We know this defense and we prepared for it well all week."
It was also the first postseason victory for the anchor of that line, 37-year-old left tackle Andrew Whitworth, and veteran defensive lineman Ndamukong Suh.
Gurley and Anderson became the fourth set of teammates in NFL history to rush for 100 yards apiece in a postseason game, an improbable development just a few weeks ago. Gurley is the Rams' offensive centerpiece and one of the NFL's elite running backs, but Anderson is a veteran playing only his third game with the Rams after signing last month when Gurley was struggling with a knee injury.
They both ran with equal verve while Los Angeles racked up more yards than the Cowboys had allowed on the ground all season. Dallas hadn't given up 200 yards in a playoff game since Eric Dickerson did it for the Rams in 1986.
But with the Rams' veteran offensive line cutting holes in the Cowboys' defense, Anderson became the third Rams player with more than 100 yards and two rushing TDs in a game, joining Dickerson and Marshall Faulk.
Three years to the day after the NFL approved the Rams' return to Los Angeles, the franchise earned its first playoff victory since coming home from 21 years in St. Louis.
The Rams chewed up the ground and the clock on four lengthy scoring drives in the first half. They kept it up after halftime, particularly on a decisive 12-play, 65-yard drive in the fourth quarter capped by Anderson's bold 1-yard TD plunge on fourth down with 7:16 to play.
Los Angeles' first two drives stalled into field goals by Greg Zuerlein, and the Cowboys briefly led in the first quarter on Cooper's 29-yard TD catch. But the Rams' defense corralled Elliott while LA's run game steady rolled ahead.
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Elliott scored on a 1-yard plunge and the Cowboys got a 2-point conversion to trim the Rams' lead to 23-15 in the third quarter.
A sellout crowd assembled in the cavernous Coliseum, with a significant portion of those fans cheering on America's Team. The sideline was dotted with celebrities, and LeBron James hosted several Lakers teammates in a field suite behind the east end zone.
Cowboys: WR Cole Beasley had one catch for 15 yards, and tight end Blake Jarwin had two catches for 17 yards. They both missed practice all week with ankle injuries.
Rams: Gurley appeared a bit hesitant early in his first action in four weeks, but his knee seemed to hold up well. ... CB Aqib Talib went to the locker room in the third quarter to be evaluated for a concussion. He returned for the fourth quarter.
Cowboys: Season over.
Rams: Either a trip to New Orleans or a visit from Philadelphia for the NFC championship game. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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World now views Bangladesh as role model of development: PM
* Affluent people urged to serve distressed humanity * PM opens, lays foundation stones of 31 projects * RP Saha Gold Medal handed over to four personalities
March 15, 2019 00:00:00
TANGAIL, Mar 14 (UNB): Prime Minister Sheikh Hasina urged on Thursday the affluent people of the country to come forward for the welfare of the distressed people following the example set by Ranada Prasad Saha through his philanthropic activities.
"He (Ranada Prasad) had founded many service-providing institutions in such a remote village.
What he did was to serve people. So, I think, there're many wealthy people in our country who can follow the example the Kumidini Welfare Trust (founded by him) set for the promotion of women education and services of distressed people."
The Prime Minister said this while addressing a function arranged to distribute the 'Danveer Ranada Prasad Saha Memorial Gold Medal' at Kumudini Complex in Mirzapur of the district.
The function was arranged on the occasion of the 86th founding anniversary of Kumudini Welfare Trust.
The Prime Minister said now no one can neglect Bangladesh treating it as a poor country, rather the whole world considers Bangladesh as a development role model.
"We could bring these changes in the last 10 year," she said adding that Bangladesh is marching towards development.
Hasina said her government wants to develop the country and society further.
Noting that today's children are the future citizen of the country, she said the government wants to build a decent life and bright future for them.
Sheikh Rehana, younger daughter of Father of the Nation Bangabandhu Sheikh Mujibur Rahman, was present as a special guest at the function organised by Kumudini Welfare Trust of Bengal (BD).
The Prime Minister handed over 'the Danveer Ranada Prasad Saha Memorial Gold Medal' to four eminent persons this year.
The four persons are legendary political leader and former Prime Minister of erstwhile Pakistan Huseyn Shaheed Suhrawardy (posthumous), National Poet Kazi Nazrul Islam (posthumous), Nazrul expert and researcher National Professor Rafiqul Islam and internationally renowned artist Shahabuddin Ahmed.
On behalf of Suhrawardy, Sheikh Rehana received the gold medal from the Prime Minister, while Khilkhil Kazi, granddaughter of Poet Nazrul Islam, received the medal on behalf of the National Poet.
Chairman and Managing Director of the Kumudini Welfare Trust of Bengal (BD) Ltd Rajiv Prasad Saha also spoke at the event presided over by its director and language movement veteran Protiva Mutsundi. Another director of the trust Srimoti Shaha delivered the welcome speech.
At the function, two separate documentaries were screened over the life and works of Ranada Prasad Saha and the 86-year journey of Kumudini Welfare Trust.
Earlier, the Prime Minister opened and laid the foundation stones of 31 development projects in the district. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
Active fluids constitute a fascinating class of nonequilibrium systems and include numerous examples such as bacterial suspensions and artificial nano-/microswimmers, where constituent particles, being suspended in a base fluid, display the ability to self-propel by making use of internal mechanisms and the ambient free energy \cite{Ramaswamy,Bechinger,Marchetti,Julicher,Vicsek,Needleman,Elgeti_2,Chate,Cates,Lauga,Saintillan,Bertin}. Being of mounting theoretical and experimental interest in recent years, active fluids are found to exhibit many peculiarities, not paralleled with analogous properties in thermodynamic equilibrium \cite{Bar,Li,Bialke,Chen,Mallory,Klamser}.
While some of the key properties of active particles, such as their excessive nonequilibrium accumulation near confining boundaries \cite{Hernandez,Spagnolie,Schaar,Nash,Elgeti_3,Jamali} or their shear-induced behavior in an imposed flow \cite{Anand,Kaya,Rusconi,Fu,Asheichyk,Shabanniya,Ezhilan_2,Nili}, can be captured within noninteracting models, their collective properties and phase behaviors are largely determined by particle interactions, including steric and hydrodynamic interactions \cite{Burkholder,Solon_3,Singh}. Phenomenological models with polar alignment interactions, which tend to align the directions of self-propulsion of neighboring active particles, have also emerged as an important class of models, describing development of long-range orientational order in active systems, as first proposed by Vicsek et al. \cite{Vicsek_2,Vicsek}. Coherent collective motion, pattern formation and large-scale traveling structures, such as clusters and lanes, are among the notable phenomena observed in Vicsek-type models \cite{Chate_2,Gregoire,Martin,Sese,Grossmann}. In suspensions of active particles, steric interactions, near-field hydrodynamics and active stresses \cite{Delfau,Yoshinaga,Lushi,Aranson} play significant roles in engendering interparticle alignment.
Geometric confinement has also emerged as an important factor, determining various properties of active systems, such as active flow patterns and stabilization of dense suspensions into spiral vortices \cite{Wioland,Wioland_2,Ravnik,Doostmohammadi}. While many interesting behaviors (e.g., capillary rise of active fluids in thin tubes, shape deformation of vesicles enclosing active particles, upstream swim in channel flow, etc; see Refs. \cite{Li,Wysocki,Kantsler} and references therein) have been reported to occur due to the near-surface behavior of active particles, geometric constraints have also been utilized in useful applications, such as steering living or artificial microswimmers in a desired direction \cite{Das_2,Ostapenko,Popescu}.
Active particles can lead to intriguing attractive and repulsive effective interactions \cite{Leite,Mokhtari,Zaeifi,Zarif,Sebtosheikh} between passive colloidal objects and rigid surface boundaries as well, with interaction profiles that exhibit rather complex dependencies on system parameters, including the distance between the juxtaposed surfaces. The latter arises due partly to the aforementioned boundary accumulation of active particles that can lead to pronounced near-surface particle layering \cite{Zaeifi,Zarif,Sebtosheikh,Ni,Ye}. The surface accumulation of active particles has been associated with a number of different factors, including hydrodynamic particle-wall couplings \cite{Elgeti,Elgeti_3,Hernandez,Li_5,Spagnolie} and, on a more basic level, the persistent motion of the particles \cite{Schaar,Wysocki_2}, causing prolonged near-surface detention times. A remarkable manifestation of the foregoing effects is the so-called swim pressure produced by active particles on confining boundaries, a notion of conceptual significance in describing the steady-state properties of active fluids via possible analogies with equilibrium thermodynamics that has attracted much interest and debate in the recent past \cite{Takatori,Fily_2,Chen,Winkler,Solon_2,Junot,Marconi,Speck_2,Solon_3,Yan,Yan_2,Ginot,Patch,Marconi_2,Solon,Smallenburg,Ezhilan}. Swim pressure and effective coupling between active particles and curved boundaries are known to result in a diverse range of phenomena, including negative interfacial tension \cite{Bialke,Speck_3}, bidirectional flows and helical vortices in cylindrical capillaries \cite{Ravnik}, microphase separation \cite{Singh,Tung} , and reverse/anomalous Ostwald ripening \cite{Jamali,Tjhung},
with the latter mechanism enabling stabilized mesophases of monodispersed droplets (see also other related works on active droplets in Refs. \cite{Zwicker,Zwicker_2,Weber,Golestanian}).
In this work, we study surface accumulation and swim pressure of active, nonchiral and chiral, Brownian particles within a minimal two-dimensional model by combining the three main ingredients noted above, i.e., the steric and polar interparticle alignment interactions, geometric confinement and boundary curvature. The two latter factors are brought in by adopting a ring-shaped (annulus) confinement. This geometry is particularly useful in that it allows one to examine the interplay between convex and concave curvatures introduced by its circular inner and outer boundaries, respectively. The system is described in the steady state using a Smoluchowski equation, governing the joint position-orientation probability distribution function of active particles, which is numerically analyzed to derive quantities such as particle density and polarization profiles, chirality-induced current and swim pressure on the inner and outer circular boundaries of the annulus. We thus report a surface-population reversal, whereby active particles accumulate more strongly near the convex inner boundary of the annulus rather than its concave outer boundary. This contrasts the conventional picture implying preferential accumulation (due to longer detention times) of active particles near concave boundaries relative to the convex ones, a behavior that is found for noninteracting particles within the present model as well. We thus show that the said population reversal is a direct consequence of both alignment interactions and boundary curvature and will be absent in the absence of either of them. The implications of this effect for the chirality-induced current and swim pressure of active particles are then explored in detail. We describe our model and its governing equations in Sections \ref{II} and \ref{III}, discuss our results in Section \ref{IV}, and conclude the paper in Section \ref{V}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.95\textwidth]{fig1}
\caption{
(a) Left: Active particles are described by their position and orientation vectors parametrized in polar coordinates as ${\mathbf r}=r(\cos\theta, \sin\theta)$ and $\hat{\mathbf{u}}=(\cos\varphi,\sin\varphi)$; Right: They are confined within a ring-shaped confinement, or annulus, with impermeable circular boundaries of radii $R_1$ and $R_2$ from the origin O. (b) and (c) Steady-state PDF, $\tilde{\cal P}_0(\tilde r, \psi)$ of active particles within the shown confined geometry is plotted in the $\psi-\tilde r$ plane for nonaligning (panel b, $U_0=0$) and interaligning particles (panel c, $U_0=10$) with fixed parameter values are $\tilde R_1=10$, $\tilde R_2=40$, ${\rm Pe}=10$, $\tilde \rho_0=0.1$ and $\Gamma=1$.
}
\label{fig1}
\end{figure*}
\section{Model}\label{II}
We consider a two-dimensional minimal model of active Brownian particles \cite{Marchetti_2} confined between two impermeable, concentric, circular boundaries with radii $R_{1}$ and $R_{2}>R_1$, forming a ring-shaped (annulus) confinement; see Fig. \ref{fig1}a. Each particle has an intrinsic, constant, self-propulsion speed $v>0$ along its instantaneous direction of motion, determined by the unit vector $\hat{\mathbf{u}}$, and may also possess an intrinsic, constant, in-plane angular velocity (chirality) of signed magnitude $\omega$; see, e.g., Refs. \cite{Liebchen,Ai,Mijalkov,Nourhani,Lowen}. Particles are assumed to interact via pair potentials of two main types: (i) a phenomenological alignment interaction of strength $U_0>0$ to be modeled through the dot-product of particle orientations as $-U_0\,\hat{\mathbf{u}}_i\cdot\hat{\mathbf{u}}_j$ for the pair of particles labeled by $i$ and $j$ \cite{Fazli}; (ii) a local steric delta-function potential $U_1\,\delta({\mathbf r}_i-{\mathbf r}_j)$ for the given pair. The latter provides a formally straightforward route to incorporate a finite excluded area $U_1$ for each particle in our later continuum formulation. Thus, the total pair interaction energy ${\mathcal U}_{ij}\equiv{\mathcal U}({\mathbf r}_i, \hat{\mathbf{u}}_i; {\mathbf r}_j, \hat{\mathbf{u}}_j)$, is
\begin{equation}
\frac{{\mathcal U}_{ij}}{k_{\mathrm{B}}T }=-U_0\,\hat{\mathbf{u}}_i\cdot\hat{\mathbf{u}}_j+U_1\,\delta({\mathbf r}_i-{\mathbf r}_j).
\end{equation}
Particles interact with the circular boundaries of the confinement with the repulsive simple harmonic potential
\begin{equation}\label{harm_pot}
\!{\mathcal V}_i\!=\!\frac{k_{1}}{2}\!\left(r_i-R_{1}\right)^2\!\Theta(R_{1}-r_i)+\frac{k_{2}}{2}\!\left(r_i-R_{2}\right)^2\!\Theta(r_i-R_{2}),
\end{equation}
where ${\mathcal V}_i\equiv {\mathcal V}(r_i)$ and $r_i$ is the radial distance of the $i$th particle from the origin, see Fig. \ref{fig1}, $\Theta(\cdot)$ is the Heaviside step function, and $k_{1}=k_{2}$ are the respective harmonic constants. These constants are fixed at a sufficiently large value to establish nearly hard boundaries, being also assumed to be torque-free (no dependence of ${\mathcal V}_i$ on the particle orientation); hence, the swim pressure on them is expected to be a state function \cite{Solon_2,Fily_2}. It shows no dependence on the interfacial potential strengths.
\section{Governing equations}\label{III}
The translational and rotational dynamics of active particles are described by the Langevin equations
\begin{eqnarray}\label{Lant}
&&\dot{{\mathbf r}}_i=v\hat{\mathbf{u}}_i-\mu_{\mathrm{t}}\nabla_i\left( {\mathcal V}_i+\sum_{j\neq i}{\mathcal U}_{ij}\right)+\sqrt{2D_{\mathrm{t}}}\boldsymbol\eta_i(t),\\
\label{Lanr}
&&\dot{\varphi}_i=\omega-\mu_{\mathrm{r}}\partial_\varphi\sum_{j\neq i}{\mathcal U}_{ij}+\sqrt{2D_{\mathrm{r}}}\zeta_i(t),
\end{eqnarray}
with the shorthand notations $\nabla_i\!=\!\partial/\partial {\mathbf r}_i$ and $\!\partial_\varphi\!=\!\partial/\partial \varphi$. Here, ${\boldsymbol\eta}_i$ and $\zeta_i$ are the translational and rotational white noises of zero mean and unit variances, with $D_{\mathrm{t}}$ and $D_{\mathrm{r}}$ being the single-particle translational and rotational diffusivities, respectively. To ensure that the system behavior reduces to that of an analogues equilibrium system, when the active sources are put to zero, $v=\omega=0$, the diffusivities are assumed to fulfill the Smoluchowski-Einstein-Sutherland relations $D_{\mathrm{t}}=\mu_{\mathrm{t}}k_{\mathrm{B}}T$ and $D_{\mathrm{r}}=\mu_{\mathrm{r}}k_{\mathrm{B}}T$, where $k_{\mathrm{B}}T$ is the ambient thermal energy scale and $\mu_{\mathrm{t}}$ and $\mu_{\mathrm{r}}$ are the translational and rotational (Stokes) mobilities, respectively \cite{Happel}.
Equations (\ref{Lant}) and (\ref{Lanr}) can standardly be mapped to a time-evolution equation involving one- and two-point PDFs, with the latter originating from the two-particle interactions. Without delving into further details, we adopt a mean-field approximation as a closure scheme for such an equation by neglecting interparticle correlations and assuming ${\cal P}({\mathbf r},\varphi, {\mathbf r}',\varphi'; t)={\cal P}({\mathbf r},\varphi; t){\cal P}({\mathbf r}',\varphi'; t)$, which is expected to hold in sufficiently dilute suspensions. This leads to an effective Smoluchowski equation, involving an effective translational flux velocity $-\mu_{\mathrm{t}}(\nabla \overline{{\mathcal U}})$ experienced on average by each particle due to its interactions with other particles, where the averaged two-particle interaction $\overline{{\mathcal U}}=\int {\mathrm{d}} {\mathbf r}'{\mathrm{d}} \varphi'\,{\mathcal U}\,{\cal P} ({\mathbf r}',\varphi')$. Such an equation is nonlocal and nonlinear and can be expressed as
\begin{eqnarray}\label{eq:SE}
&&\partial_t{\cal P}=-\nabla\cdot\left[\left(v\hat{\mathbf{u}}-\mu_{\mathrm{t}}\nabla ({\mathcal V}+\overline{{\mathcal U}})\right){\cal P}-D_{\mathrm{t}}\nabla{\cal P}\right]\nonumber\\
&&\qquad\quad -\partial_\varphi\left[\left(\omega-\mu_{\mathrm{r}}\partial_\varphi\, \overline{{\mathcal U}}\right){\cal P}-D_{\mathrm{r}}\partial_\varphi{\cal P}\right].
\end{eqnarray}
Our focus will be on the steady-state solution $\tilde{{\cal P}}_0({\mathbf r},\varphi)$. Because of the rotational symmetry in the present geometry, the angular dependence of the PDF can be assumed to occur only through the relative angle $\psi=\varphi-\theta$, allowing the relations $\partial_\varphi{\cal P}=-\partial_\theta{\cal P}=\partial_\psi{\cal P}$ and $\int{\mathrm{d}} {\mathbf r}'{\mathrm{d}} \varphi'\equiv\int r' {\mathrm{d}} r'{\mathrm{d}} \psi'$ to be used, where necessary. In this case, the mean-field interaction energy reads
\begin{equation}\label{mean_int}
\frac{\overline{{\mathcal U}}}{k_{\mathrm{B}}T }\!=\!-U_0\!\!\int\!\! r'\! {\mathrm{d}} r'\!{\mathrm{d}} \psi'\!\cos(\psi'\!-\psi){\cal P}_0 (r'\!,\psi')+U_1\!\!\int\!\!{\mathrm{d}} \psi'{\cal P}_0(r,\psi').
\end{equation}
Equation \eqref{eq:SE} is supplemented by no-flux boundary conditions in the radial direction on the two circular boundaries at $r=R_1$ and $R_2$ and periodic boundary condition over the angular coordinate $\psi$ for $r\in [R_1, R_2]$. Since we have assumed a closed system with a fixed number $N$ of active particles within the annulus, the normalization condition for the PDF can be expressed using the mean number density $\rho_0=N/[\pi({R}_2^2-{R}_1^2)]$ of particles as
\begin{equation}
\label{eq:norm}
\int_{ R_1}^{ R_2}\!\int_{-\pi}^{\pi} r\,{\mathrm{d}} r \,{\mathrm{d}}\psi \,{\cal P}_0( r,\psi)=\pi\rho_0\left(R_2^2-R_1^2\right).
\end{equation}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.85\textwidth]{fig2}
\caption{
(a) Rescaled number density profile of active particles, $\tilde \rho(\tilde r)$, as a function of the radial distance inside the annulus in the absence ($U_0=0$) and presence ($U_0=10$) of alignment interactions. Inset shows the corresponding polarization profiles, $\tilde m_1(\tilde r)$. Here, we have fixed $\Gamma=1$ and other parameters are as in Fig. \ref{fig1}. (b) Same as (a) but plotted for different values of chirality strength, as indicated on the plot both without (inset, $U_0=0$) and with (main set, $U_0=10$) alignment interactions.
}
\label{fig2}
\end{figure*}
\subsection{Dimensionless representation}
We proceed by rescaling units of length/time with the characteristic length/time $a=({D_{\mathrm{t}}}/{D_{\mathrm{r}}})^{{1}/{2}}$
and $1/D_{\mathrm{r}}$, respectively, and the units of energy with $k_{\mathrm{B}}T$; thus, e.g., the boundary interaction energy is rescaled as $\tilde{{\mathcal V}}(\tilde r)={\mathcal V}(a \tilde r)/({k_{\mathrm{B}}T})$. The PDF is suitably rescaled as $\tilde{{\cal P}}_0(\tilde{r}, \psi)={\cal P}_0(a\tilde{r}, \psi)/\rho_0$. The parameter space is thus spanned by the set of dimensionless parameters defined by the {\em (swim) P{\'e}clet number} and the {\em chirality strength}
\begin{equation}\label{peclet}
{\rm Pe}=\frac{v}{aD_{\mathrm{r}}},\quad\Gamma=\frac{\omega}{D_{\mathrm{r}}},
\end{equation}
respectively, the rescaled interaction parameters $U_0$, $\tilde{U}_1={U_1}/{a^2}$, and $\tilde{k}_{1,2}={k_{1,2} a^2}/({k_{\mathrm{B}}T })$, the rescaled radii $\tilde{R}_{1,2}={R_{1,2}}/{a}$, and the rescaled mean particle density $ \tilde \rho_0 = \rho_0a^2$. The steady-state Smoluchowski equation can then be expressed in dimensionless units as
\begin{equation}\label{gov}
\frac{1}{\tilde{r}}\partial_{\tilde{r}}\left(\tilde{r}\tilde{{\cal J}}_{\tilde{r}}\right)+\frac{1}{\tilde{r}}\partial_{\psi}\tilde{{\cal J}}_\psi=0,
\end{equation}
where we have defined the rescaled spatial and angular probability flux densities $\tilde{{\cal J}}_{\tilde{r}}$ and $\tilde{{\cal J}}_\psi $, respectively, as
\begin{widetext}
\begin{eqnarray}\label{j1_s}
&&\tilde{{\cal J}}_{\tilde{r}}=-\partial_{\tilde{r}}\tilde{{\cal P}}_0+\left({\rm Pe}\cos\psi-\partial_{\tilde{r}}\tilde{{\mathcal V}}\right)\tilde{{\cal P}}_0-\tilde{U}_1\tilde{\rho}_0\left(\partial_{\tilde{r}}\!\!\int\!{\mathrm{d}} \psi'\tilde{{\cal P}}_0(\tilde{r},\psi')\right)\tilde{{\cal P}}_0,\\
\label{j2_s}
&&\tilde{{\cal J}}_\psi =-\frac{1}{\tilde{r}}\left(1+\tilde{r}^2\right)\partial_\psi\tilde{{\cal P}}_0+\left(-{\rm Pe}\sin\psi+\Gamma\tilde{r}\right)\tilde{{\cal P}}_0+\frac{U_0\tilde{\rho}_0}{\tilde{r}}\left(1+\tilde{r}^2\right)\left(\int{\tilde{r}}'{\mathrm{d}} {\tilde{r}}'{\mathrm{d}} \psi'\sin(\psi'-\psi)\tilde{{\cal P}}_0(\tilde{r}',\psi')\right)\tilde{{\cal P}}_0,
\end{eqnarray}
\end{widetext}
where
$\int_{\tilde R_1}^{\tilde R_2}\int_{-\pi}^{\pi}\tilde r{\mathrm{d}}\tilde r {\mathrm{d}} \psi \tilde{\cal P}_0(\tilde r,\psi)=1$. Equation \eqref{gov} is an integrodifferential PDE, which is solved using finite-element methods within the annulus subject to the aforementioned boundary conditions. Other rescaled steady-state quantities such as the particle density profiles within the annulus and swim pressure on the boundaries follow directly from the numerically obtained PDF \cite{Jamali}. The choice of dimensionless parameter values and their relevance to realistic systems is discussed in Appendix \ref{app:parameters}. Since Eq. \eqref{gov} remains invariant under chirality and orientation angle reversal, $\Gamma\rightarrow -\Gamma$ and $\psi\rightarrow -\psi$, we restrict our discussion to only positive (counterclockwise) values of $\Gamma$, bearing in mind that chirality-induced currents will be reversed for negative values of $\Gamma$.
\section{Results}\label{IV}
\subsection{Particle distribution and radial density profile}
\label{subsec:PDF}
In Figs. \ref{fig1}b and c, the steady-state PDF of active particles inside the annulus is plotted in the $\psi-\tilde r$ plane in the absence (panel b, $U_0=0$) and presence (panel c, $U_0=10$) of alignment interactions between the particles for a representative set of parameter values. As seen, most of active particles accumulate near the boundaries (represented by larger probability densities appearing in yellow and red colors in the plots), a well-known consequence of the persistent motion of particles \cite{Elgeti}. Also, in both the absence and the presence of alignment interactions, the typical orientations of particles on the outer (inner) boundaries are expectedly in the outward (inward) directions pointing away (toward) the origin of the annulus, respectively. The PDF of nonchiral active particles \cite{Jamali} would be peaked exactly at $\psi=0$ and $\psi=\pm\pi$ on the outer and inner boundaries, respectively, reflecting the normal-to-boundary orientations of the particles. In the plots, however, we have taken a relatively small, counterclockwise, particle chirality ($\Gamma=1$) that produces a shift of the probability-density peaks from the noted angular orientations to the first and third angular quadrants near the outer and inner boundaries, respectively.
Also, as seen in Fig. \ref{fig1}b, nonaligning particles are more strongly accumulated near the outer boundary than the inner one, in accord with the standard paradigm that active particles spend longer `detention' times at concave rather than convex boundaries \cite{Fily_3,Nikola,Paoluzzi}. Remarkably, we find a reversed situation in the presence of interparticle alignment (panel c) with larger particle probabilities appearing near the inner boundary. This appears to suggest an alignment-induced crossover between two different configurational states with active particles primarily attracted to only one of the boundaries. Such a behavior, which shall examine more closely later, can also be discerned from the local density of active particles inside the annulus defined as
\begin{equation}
\tilde\rho(\tilde r)=\int_{-\pi}^{\pi}{\mathrm{d}} \psi \, \tilde {\cal P}_0(\tilde r,\psi).
\end{equation}
This quantity being plotted in Fig. \ref{fig2}a indicates a reversal in the relative accumulation of active particles near the two boundaries, when the alignment interaction strength $U_0$ is changed from $U_0=0$ (dotted curve) to $U_0=10$ (solid curve). The figure also shows that the wide plateau present in the density profile of nonaligning particles changes to a linear stretch of particle density between the inner and outer peaks, when the alignment interaction is switched on. The active particles within this linear region of the density profiles turn out to show no specific orientational order, as it can be verified using the first angular moment of the PDF defined as
\begin{equation}
\tilde m_1(\tilde r)=\int_{-\pi}^{\pi}{\mathrm{d}} \psi\cos(\psi) \tilde{\cal P}_0(\tilde r,\psi).
\label{eq:m1}
\end{equation}
Being shown in the inset of Fig. \ref{fig2}a, $\tilde m_1(\tilde r)$ reflects the expected outward- (inward-) pointing mean polarization of the particles near the outer (inner) boundaries but no mean orientational order ($\tilde m_1\simeq 0$) elsewhere across the central regions, neither for nonaligning nor for interaligning particles.
When the chirality strength $\Gamma$, is increased, as shown in Fig. \ref{fig2}b, the near-boundary accumulation of active particles is suppressed in both the absence (inset, $U_0=0$) and the presence (main set, $U_0=10$) of alignment interactions. One eventually finds a homogeneous density profile at elevated $\Gamma$ (see the red dotted curve for $\Gamma=100$), consistent with known results \cite{Ao,Li_2}, indicating that the behavior of chiral active particles reduces to that of passive particles in the limit of infinite chirality (see Ref. \cite{Jamali} for a systematic derivation). At intermediate $\Gamma$, however, our results reveal nontrivial variations in the density profile, with the density peak at the inner (outer) boundaries followed (preceded) by a shallow dip, indicating partial depletion of active particles (green dot-dashed curve for $\Gamma=3$). This is a consequence of the fact that chirality effects in suppressing the persistent motion of active particles are more apparent near the boundaries and the density peaks are suppressed more strongly than the linear bridge connecting them, as $\Gamma$ is increased.
\begin{figure}[t!]
\centering
\includegraphics[width=0.85\linewidth]{fig3}
\caption{Mean fraction of active particles found within the inner and outer semi-annuli, $Q_1(\tilde{R}_c)$ and $1-Q_1(\tilde{R}_c)$, respectively (see the text for definitions), as functions of the alignment interaction strength for fixed parameter values $\tilde{R}_1=10$, $\tilde{R}_2=40$, ${\rm Pe}=10$, $\tilde{\rho}_0=0.01$, $\Gamma=1$. Symbols are numerical data and curves are guides to the eye.
}
\label{fig3}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.83\linewidth]{fig4}
\caption{(a) Dependence of $U_{0c}$, the alignment strength giving the maximum population reversal, on the mean number density (main set) and chirality strength (inset) of active particles for fixed $\tilde{R}_1=10$, $\tilde{R}_2=40$, ${\rm Pe}=10$, and with $\Gamma=1$ (main set) and $\tilde{\rho}_0=0.01$ (inset). (b) Dependence of $U_{0c}$ on the radii of the inner/outer boundaries of the annulus for fixed $\tilde{\rho}_0=0.01$, ${\rm Pe}=10$, $\Gamma=1$, and with $\tilde{R}_2=100$ (main set) and $\tilde{R}_1=10$ (inset). Symbols are numerical data and curves are guides to the eye. }
\label{fig4}
\end{figure}
\subsection{Surface-population reversal}
\label{subsec:Q}
Further insight into the alignment-induced effects can be obtained by conventionally defining the inner and outer fractions $Q_1(\tilde R_c)$ and $1-Q_1(\tilde R_c)$ of active particles as those found in radial distances $\tilde r<\tilde R_c$ and $\tilde r>\tilde R_c$, respectively, where $\tilde{R}_c=(\tilde R_1+\tilde R_2)/2$ is the mean radius of the annulus. We thus have
\begin{equation}
Q_1( \tilde{R}_c)=\frac{\int_{\tilde{R}_1}^{\tilde{R}_c}\,\tilde{r}{\mathrm{d}} \tilde{r}\tilde{\rho}(\tilde{r})}{\int_{\tilde{R}_1}^{\tilde{R}_2}\,\tilde{r}{\mathrm{d}} \tilde{r}\tilde{\rho}(\tilde{r})}.
\label{eq:Q}
\end{equation}
These fractions are shown as functions of the alignment interaction strength in Fig. \ref{fig3}. In the nonaligning case ($U_0=0$), a larger fraction of particles is found in the outer semi-annulus, which, as noted before, is because self-propelling particles accumulate more strongly near concave than convex boundaries due to their lingered detention times. As seen in the figure, the inner fraction $Q_1(\tilde R_c)$ increases with $U_0$ and $1-Q_1(\tilde R_c)$ decreases with it until they are equalized at a certain value of $U_{0\ast}$ (here, $U_{0\ast}\simeq 5$). This particular value corresponds to the onset of a counterintuitive {\em surface-population reversal}, beyond which active particles are more strongly accumulated by the {\em convex} inner boundary with the {\em smaller} radius of curvature rather than the concave outer boundary with the larger radius of curvature. The maximum population reversal is achieved at a slightly larger value of $U_{0c}$ (here, $U_{0c}\simeq 8$), where $Q_1(\tilde R_c)$ displays a global maximum and $1-Q_1(\tilde R_c)$ a global minimum. Beyond this point, both quantities level off and gradually tend toward 1/2, indicating an even distribution of particles developing in the infinite $U_0$ limit within the annulus (not shown).
The designated values $U_{0\ast}$ and $U_{0c}$ are typically found to be close and, to examine the dependence of the population reversal phenomenon on other system parameters, we concentrate on the latter quantity.
Figure \ref{fig4}a (main set) shows the dependence of the maximum population reversal plotted as a function of the mean number density of particles inside the annulus. As seen, a larger (smaller) mean density, $\tilde{\rho}_0$, necessitates a weaker (stronger) alignment interaction strength, $U_{0c}$, to achieve the maximum population reversal. The monotonically decreasing trend fits closely with a functional dependence of the form $U_{0c}\sim \tilde{\rho}_0^{-\alpha}$ with $\alpha\simeq 2$, a remarkable scaling-like behavior that remains to be understood. Figure \ref{fig4}a (inset) shows that $U_{0c}$ increases almost linearly with chirality strength, which is plausible as, for larger chirality strengths, particle directions change more rapidly, requiring a larger value of alignment interaction strength to establish surface-population reversal.
Figure \ref{fig4}b, on the other hand, shows that population reversal is more easily established as the difference between the radii of the inner and outer boundaries increases. At a fixed value of the outer boundary radius (here $R_2=100$, main set) or at a fixed value of the inner boundary radius (here $R_1=10$, inset), $U_{0c}$ increases as the radial width of the annulus is reduced and vice versa. This is indicative of the fact that in a narrower ring-shaped confinement, active particles tend distribute more evenly within the confinement and easily interchange between the two circular boundaries, necessitating a stronger alignment interaction to produce population reversal.
Our numerical results (not shown) indicate that $U_{0c}$ does not significantly vary with ${\rm Pe}>1$ for which self-propulsion dominates particle diffusion (for ${\rm Pe}<1$, particles are mostly dispersed nearly uniformly within the confinement and varying the alignment strength does not effectively change particle distribution).
\subsection{Chirality-induced current}
\label{chiral_point}
Chiral active particles can generate net rotational currents near circular boundaries. This effect emerges as a result of the deviation of the most probable orientation of chiral particles from the normal-to-surface direction (see Section \ref{subsec:PDF}), creating a tangential velocity component on the surface. To examine the steady-state chirality-induced current in the present context with alignment and steric interactions between particles (see Ref. \cite{Jamali} for the special case of noninteracting active particles), we begin by integrating the Smoluchowski equation over the rotational degree of freedom, $\varphi$, which gives a relation as $\nabla\cdot\tilde{\mathbf{J}}=0$, where the current density, $\tilde{\mathbf{J}}=({\tilde{J}}_{\tilde{r}}, {\tilde{J}}_{\theta})$, has the following two components
\begin{equation}
{\tilde{J}}_{\tilde{r}}=\int_{-\pi}^{\pi}{\mathrm{d}} \psi\,{\tilde{A}}_{\tilde{r}}(\tilde{r},\psi),\,\,\, {\tilde{J}}_{\theta}=\int_{-\pi}^{\pi}{\mathrm{d}} \psi\,{\tilde{A}}_{\theta}(\tilde{r},\psi),
\end{equation}
where
\begin{equation}
\tilde{A}_{\tilde {r}}=\left[{\rm Pe}\cos\psi-\partial_{\tilde{r}}\tilde{{\mathcal V}}\right]\tilde{{\cal P}}_0
-\partial_{\tilde{r}}\tilde{{\cal P}}_0-\tilde{U}_1\tilde{\rho}_0\tilde{{\cal P}}_0\,\partial_{\tilde{r}}\int{\mathrm{d}} \psi'\tilde{{\cal P}}_0,
\end{equation}
\begin{eqnarray}
\tilde{A}_{\theta}&=&{\rm Pe}\sin\psi\tilde{{\cal P}}_0-\frac{1}{\tilde{r}}\partial_{\psi}\tilde{{\cal P}}_0
\\
&+&\frac{U_0\tilde{\rho}_0}{\tilde{r}}\tilde{{\cal P}}_0\int \tilde{r}'{\mathrm{d}} \tilde{r}'{\mathrm{d}} \psi'\,\tilde{{\cal P}}_0\sin(\psi-\psi').
\nonumber
\end{eqnarray}
\begin{figure}[t!]
\centering
\includegraphics[width=0.85\linewidth]{fig5}
\caption{Main set: Rescaled chirality-induced current, $\tilde{J}_\theta$, as a function of radial distance for active particles with ($U_0=40$) and without ($U_0=0$) alignment interactions. Inset: $\tilde{J}_\theta$ for a selected set of $U_0$ values (here, $U_{0c}=8$). Other parameters fixed as $\tilde{R}_1=10$, $\tilde{R}_2=40$, ${\rm Pe}=10$, $\tilde{\rho}_0=0.01$ and $\Gamma=1$.
}
\label{fig5}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.85\textwidth]{fig6}
\caption{(a) Rescaled swim pressures, $\tilde{P}_1$ and $\tilde{P}_2$ , on the inner and outer boundaries, respectively, as functions of ${\rm Pe}$ with ($U_0=10$) and without ($U_0=0$) alignment interactions for fixed $\tilde{R}_1=10$, $\tilde{R}_2=40$, $\tilde{\rho}_0=0.1$ and $\Gamma=1$. (b) Same as (a) but plotted for swim pressures as functions of alignment interaction strength, $U_0$, for fixed $\tilde{R}_1=10$, $\tilde{R}_2=40$, ${\rm Pe}=10$, $\tilde{\rho}_0=0.01$ and $\Gamma=1$. Symbols are numerical data and curves are guides to the eye. }
\label{fig6}
\end{figure*}
The radial current density component, $\tilde{J}_{\tilde{r}}$, is zero for all parameter values.
The angular component, $\tilde{J}_\theta$, represents the chirality-induced current and is always nonzero on the boundaries, see Fig. \ref{fig5}, while it falls off to zero as one moves away from the boundaries, where the net current density of particles passing through a particular point vanishes \cite{Jamali}. Figure \ref{fig5} (main set) also shows this quantity for different values of the alignment interaction strength for a representative set of parameter values for which the maximum population reversal occurs at $U_{0c}=8$ (being nearly equal to its onset).The results represented in Fig. \ref{fig5} are typical behaviors obtained for the chirality strength $\Gamma=1$ that are shown for the sake of illustration and we can observe similar results for other representative values of $\Gamma$ near their corresponding population reversal. Thus, while in the absence of alignment interactions ($U_0=0$, red dashed curve), the inner population shows clockwise (negative current) of smaller magnitude relative to the outer population that shows counterclockwise (positive current) of larger magnitude, the situation is reversed for active particles with strong alignment interactions ($U_0=40$, blue solid curve).
For the rotational current on the outer boundary, increasing $U_0$ only results in a smaller current, one that never changes sign on this boundary as it diminishes with strengthening the alignment interaction. For the current on the inner boundary, we find a more complex behavior. In the inset of Fig. \ref{fig5} we show $\tilde{J}_\theta$ near the inner boundary for a few different values of alignment strength chosen around the onset of population reversal. As seen, by increasing $U_0$ from $U_0=4$ (purple dashed curve) to $U_0=U_{0c}=8$ (green dotted curve), the shallow dip with negative rotational current close to the inner boundary turns to a region of positive current with a nonmonotonic profile with a pronounced peak. On increasing $U_0$ further to $U_0=9$ (orange dot-dashed curve) and then $U_0=10$ (pink dashed curve), the positive current profile is suppressed, even though it can now exhibit regions with both positive and negative currents ($U_0=9$) and a pronounced dip with a strong negative current ($U_0=10$) near the inner boundary. On increasing $U_0$ further, one only finds a negative current on this boundary, whose magnitude is further enhanced.
The generation of rotational current profile of varying sign near the population reversal are reminiscent of particle layering that develops at the onset of flocking transition in Vicsek-type models \cite{Gregoire,Caussin}, even though the ring-shaped confinement is expected to have a dominant role in the present context.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.85\textwidth]{fig7}
\caption{(a) Rescaled swim pressures, $\tilde{P}_1$ and $\tilde{P}_2$, on the inner and outer boundaries, respectively, as functions of the inner radius, $\tilde{R}_1$, with ($U_0=10$) and without ($U_0=0$) alignment interactions for fixed $\tilde{R}_2=100$. (b) Same as (a) but plotted for swim pressures as functions of the outer radius, $\tilde{R}_2$, for fixed $\tilde{R}_1=10$. Other parameters are fixed in the plots as ${\rm Pe}=10$, $\tilde{\rho}_0=0.1$ and $\Gamma=1$. Symbols are numerical data and curves are guides to the eye.
}
\label{fig7}
\end{figure*}
\subsection{Swim pressure}\label{pressure}
Other interesting quantities we can explore in this system are swim pressures on the inner and outer boundaries, denoted by $\tilde{P}_1$ and $\tilde{P}_2$, respectively, which can be calculated by integrating the force density exerted by active particles on the boundaries as
\begin{equation}\label{pre}
\tilde{P}_1=\int_\Lambda^0{\mathrm{d}} \tilde{r}\,\tilde{\rho}(\tilde{r})\partial_{\tilde{r}}\tilde{{\mathcal V}},\quad
\tilde{P}_2=\int_\Lambda^\infty{\mathrm{d}} \tilde{r}\,\tilde{\rho}(\tilde{r})\partial_{\tilde{r}}\tilde{{\mathcal V}}.
\end{equation}
where $\Lambda$ is a radial distance away from the two boundaries and inside the annulus (where $\partial_{\tilde{r}}\tilde{{\mathcal V}}$ vanishes), which we arbitrarily set equal to the mean radius of the annulus.
The swim pressures $\tilde{P}_1$ and $\tilde{P}_2$ both turn out to be monotonically increasing functions of P\'eclet number in the presence and absence of alignment interactions, as seen in Fig. \ref{fig6}a (for the given set of fixed parameter values in the figure, the maximum population reversal occurs around $U_{0c}\simeq 0.08$; hence, the two cases shown occur are far from the onset of the reversal). For the case with aligning particles, $\tilde{P}_1$ is larger than its corresponding value in the nonaligning case (compare blue squares and triangle-downs), but $\tilde{P}_2$ shows the reverse property (compare red circles and triangle-ups). Thus, alignment interactions increase the swim pressure on the inner boundary and decrease it on the outer one. At small $\rm Pe$, $\tilde{P}_1$ and $\tilde{P}_2$ converge and tend to zero, as expected. $\tilde{P}_1$ and $\tilde{P}_2$ diverge as soon as $\rm Pe$ increases beyond $\rm Pe=1$. Also, for all nonvanishing $\rm Pe$, the difference between $\tilde{P}_1$ and $\tilde{P}_2$ is larger in the aligning case, which is a direct consequence of population reversal and particles departing from outer to the inner boundary.
Figure \ref{fig6}b shows $\tilde{P}_1$ and $\tilde{P}_2$ as functions of the alignment strength (for the given set of fixed parameter values, $U_{0c}\simeq 8$). As seen, $\tilde{P}_1$ first increases rapidly to a maximum value and then smoothly drops to finite values at large $U_0$. $\tilde{P}_2$ behaves in an opposite way, as it first drops to almost zero and then slowly approaches a small and finite value at large $U_0$. For $U_0=0$, the swim pressure on the inner boundary is slightly smaller than the outer one. Upon switching on the alignment interaction, particles start to migrate to the inner boundary and this causes $\tilde{P}_1$ to rapidly increase and $\tilde{P}_2$ to decrease with $U_0$ until they reach their maximum and minimum, respectively, at $U_{0c}$, with the overall trends occurring in accord with Fig. \ref{fig3}. In Appendix \ref{app1}, we derive an analytical expression for the swim pressure in the presence of alignment interactions, corroborating the foregoing discussions.
In Fig. \ref{fig7}, we show $\tilde{P}_1$ and $\tilde{P}_2$ as functions of radii of the circular boundaries (for all values of $\tilde R_1$ and $\tilde R_2$ shown in the figure, $U_0=10$ is larger than $U_{0c}$).
In the absence of alignment interactions, $\tilde{P}_1$ and $\tilde{P}_2$ vary weakly with both $\tilde{R}_1$ and $\tilde{R}_2$ and the difference between $\tilde{P}_1$ and $\tilde{P}_2$ is small. In the presence of alignment interactions, we find a different behavior. Figure \ref{fig7}a shows that the swim pressure on the inner boundary (blue squares) decreases monotonically as a function of $\tilde{R}_1$, attaining its maximum value at the smallest inner radii $\tilde{R}_1=10$, corresponding to the largest confinement, given that $\tilde R_2=100$ is fixed. This is reflective of stronger active-particle accumulation on the inner boundary as the system is deep inside the population-reversal regime. In this case, the swim pressure on the outer boundary (red triangle-ups) varies only weakly with $\tilde{R}_1$. The aforementioned behaviors are corroborated by those in Fig. \ref{fig7}b, depicting $\tilde{P}_1$ and $\tilde{P}_2$ as functions of $\tilde{R}_2$. Also, as seen in Fig. \ref{fig7}a, all four curves are found to converge to a common value as the ring-shaped confinement becomes narrower, with $\tilde{R}_1$ tending toward $\tilde{R}_2$. This is due to the fact that in extremely narrow annuli, active particles tend to distribute almost evenly within the confinement with equal probabilities of interacting with either of the boundaries, regardless of their alignment interactions.
\section{Summary}\label{V}
In this paper, we study a two-dimensional system of nonchiral and chiral active Brownian particles constrained to move in a circular ring-shaped confinement (annulus) with impermeable confining boundaries. The active particles are assumed to have constant intrinsic, linear (self-propulsion) and angular, velocities and interact through alignment as well as steric pair potentials. The alignment interaction between particles is assumed to have a dot-product form in a way that it tends to align the self-propulsion directions of a particle pair. The steady-state properties of active particles are analyzed using a probabilistic Smoluchowski equation, which is solved numerically. This equation takes the form of an integrodifferential equation due to nonlocal coupling terms generated by the particle interactions that are then treated using a mean-field approximation.
While active particles are typically known to accumulate more strongly at concave rather than convex boundaries due to their longer near-boundary detention times in the former case \cite{Fily_3,Nikola,Paoluzzi,Jamali}, we show that the presence of alignment interactions causes a reverse phenomena to take place in the present setting; hence, a larger fraction of active particles are found to accumulate at the inner boundary of the annulus, which is convex and has a smaller radius of curvature. Such a {\em surface-population reversal} is both an alignment-induced and a curvature-induced effect and will be absent in the absence of alignment interactions and/or in a planar confinement. The effect is quite robust in the sense that it emerges over a wide range of moderately large alignment interaction strengths, $U_0$, and moderately large P\'eclet numbers (with the latter required to be only so large as to facilitate surface accumulation of active particles against particle diffusion into the bulk). The population reversal is typically maximized at an alignment strength $U_{0c}$ that is only slightly larger than the onset of the reversal, indicating a rapid crossover at the onset, followed by a nonmonotonic behavior, i.e., a relatively sharp hump and then decay of the surface populations down to certain saturation values, as $U_0$ is increased. We also find that the wider the annulus (the larger the difference between the outer and inner radii) the weaker will be the alignment strength required to cause the reversal, and vice versa. Similar results are found for the dependence of the population reversal on the mean particle number density, as in a more dilute (denser) system the reversal is realized for a stronger (weaker) alignment strength.
We study the implications of the aforementioned effect for the chirality-induced current at and swim pressure on the inner and outer boundaries of the annulus. As generally expected, particle chirality leads to suppression of activity-induced effects such as boundary-accumulation of active particles and, as such, chiral active particles require a larger alignment strength to establish population reversal relative to nonchiral active particles. A remarkable effect emerges in the case of near-boundary chirality-induced currents. While for alignment strengths sufficiently far from the onset of population reversal, we find rotational current of a single well-defined sign (clockwise or counterclockwise) forming near each of the inner and outer boundaries (albeit with opposing signs on the inner relative to the outer boundary), the situation turns out to be more complex near the onset of the population reversal; hence, the currents near the two boundaries can take similar signs and thus rotate in the same direction, and multiple `layers' of rotating currents can be seen, especially for weak chirality strengths.
As the surface-population reversal is directly caused by the presence of interparticle alignment, it may be tempting to compare it with the bulk flocking transition in Vicsek-type models \cite{Vicsek}. Our results indicate that the confinement and boundary curvature play key roles in regulating the population reversal, making it possible to compare it also with surface-induced transitions, e.g., in wetting \cite{Sepu,Joanny} and capillary \cite{Wysocki,Kne} systems, and in counterion condensation phenomena \cite{Naji,Naji_2}.
Our model treats the problem at hand on the level of a minimal model of active Brownian particles \cite{Marchetti_2}. Hence, it neglects several other important factors that could be considered for a more comprehensive analysis in the future. These include the roles of hydrodynamic interactions between particles and between particles and the boundaries (see, e.g., Refs. \cite{Li_3,Elgeti}). In these contexts, an interesting problem would be that of the so-called pusher and puller microswimmers with dipolar flow fields (see, e.g., Refs. \cite{Ishikawa,Lauga,Pooley}) and how the ensuing active stress due to these markedly different types of active particles might influence the behaviors predicted with the current setting, especially at elevated area fractions. Since the far-field hydrodynamic interactions may be screened by confinement effects \cite{Delfau}, the near-field hydrodynamics will be of particular interest in the analysis of active particle distributions within an annulus. The interparticle alignment due to such hydrodynamic effects may thus compete or cooperate with the dot-product alignment model considered here, paving the way for more intriguing possible scenarios. When the rodlike nature of active particles is accounted for, particle interactions with the boundaries will not be torque-free anymore and the swim pressure can vary depending on the type of surface potentials \cite{Solon_2,Wang}. Active rods in circular geometries \cite{Van,Vladescu} can also be subjected to imposed shear flows, constituting another potential direction of research that can be explored in the future.
\section{Conflicts of interest}
There are no conflicts of interest to declare.
\section{Acknowledgements}
Z.F. thanks A. Partovifard and M. R. Shabanniya for useful discussions and comments. A.N. acknowledges partial support from the Associateship Scheme of The Abdus Salam International Centre for Theoretical Physics (Trieste, Italy). We thank the High Performance Computing Center of the Institute for Research in Fundamental Sciences (IPM) for computational resources.
\section{Author contributions}
Z.F. performed the theoretical derivations and numerical coding, generated the output data and produced the figures. Both authors analyzed the results, contributed to the discussions and wrote the manuscript. A.N. conceived the study and supervised the research.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,013 |
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# Blockquote Indentation
> [ESLint rule][eslint-rules] to enforce Markdown blockquote indentation in JSDoc descriptions.
<section class="intro">
</section>
<!-- /.intro -->
<section class="usage">
## Usage
```javascript
var rule = require( '@stdlib/_tools/eslint/rules/jsdoc-blockquote-indentation' );
```
#### rule
[ESLint rule][eslint-rules] to enforce Markdown blockquote indentation in JSDoc descriptions.
**Bad**:
<!-- eslint-disable stdlib/jsdoc-blockquote-indentation, stdlib/jsdoc-no-paragraph-content-indent, stdlib/jsdoc-markdown-remark -->
```javascript
/**
* Beep.
*
* > This is a blockquote.
*
* @returns {string} a value
*
* @example
* var str = beep();
* // returns 'boop'
*/
function beep() {
return 'boop';
}
```
**Good**:
```javascript
/**
* Beep.
*
* > This is a blockquote.
*
* @returns {string} a value
*
* @example
* var str = beep();
* // returns 'boop'
*/
function beep() {
return 'boop';
}
```
The [rule][eslint-rules] may be configured using the same options as supported by [remark][remark-lint-blockquote-indentation]. By default, the rule enforces an indentation of `2` characters, which includes the `>` symbol.
</section>
<!-- /.usage -->
<section class="examples">
## Examples
<!-- eslint no-undef: "error" -->
```javascript
var Linter = require( 'eslint' ).Linter;
var rule = require( '@stdlib/_tools/eslint/rules/jsdoc-blockquote-indentation' );
var linter = new Linter();
var result;
var code;
// Generate our source code:
code = [
'/**',
'* Beep boop.',
'*',
'* > This is an indented blockquote.',
'*',
'*',
'* @param {string} str - input value',
'* @returns {string} output value',
'*',
'* @example',
'* var out = beep( "boop" );',
'* // returns "beepboop"',
'*/',
'function beep( str ) {',
'\treturn "beep" + str;',
'}'
].join( '\n' );
// Register the ESLint rule:
linter.defineRule( 'jsdoc-blockquote-indentation', rule );
// Lint the code:
result = linter.verify( code, {
'rules': {
'jsdoc-blockquote-indentation': [ 'error', 2 ]
}
});
console.log( result );
/* =>
[
{
'ruleId': 'jsdoc-blockquote-indentation',
'severity': 2,
'message': 'Remove 3 spaces between blockquote and content',
'line': 4,
'column': 5,
'nodeType': null,
'source': '* > This is an indented blockquote.',
'endLine': 13,
'endColumn': 3
}
]
*/
```
</section>
<!-- /.examples -->
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<section class="related">
</section>
<!-- /.related -->
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<section class="links">
[eslint-rules]: https://eslint.org/docs/developer-guide/working-with-rules
[remark-lint-blockquote-indentation]: https://github.com/remarkjs/remark-lint/tree/19150d94f89f7a0d94d083417890236d11839641/packages/remark-lint-blockquote-indentation
</section>
<!-- /.links -->
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,246 |
\section{Introduction}
The motion of an incompressible inviscid fluid in a Riemannian manifold $(M, g)$ is described by a time-dependent vector field $u_t$ satisfying the \emph{Euler equations}
\begin{equation}\label{euler}
\partial_t u_t+\nabla^{LC}_{u_t} u_t= -\nabla p ,\,\,\,\,\, \Div u_t=0 \,.
\end{equation}
Here $\Div$, $\nabla$, and $\nabla^{LC}$ denote the divergence, gradient and covariant derivative on $M$, defined with the metric $g$; and $p$ is a (time-dependent) function called the pressure, also an unknown in the equations.
These equations define a dynamical system of an infinite number of degrees of freedom: they can be interpreted as a first order ODE (which we will call the Euler system) in the infinite-dimensional linear space $\mathfrak{X}^{m}(M)$ of $m$-times differentiable divergence-free vector fields on the domain $M$.
We will denote by $\Phi_t(v):=u_t$ the flow of this ODE starting at the field $v \in \mathfrak{X}^{m}$. Small time existence and uniqueness hold for $M$ closed (compact without boundary) and non-integer $m>1$ \cite{EB}; moreover, the solution $u_t$ is $C^{1}$ on the variable $t$.
Global in time existence of solutions, however, is only known in 2 dimensions \cite{Wo}; in higher dimensions, whether or not there are initial conditions for which the flow $\Phi_t$ ``blows-up'' in finite time (i.e $||\Phi_t(v)||_{C^{m}(M)} \rightarrow \infty$ as $t \rightarrow T < \infty$) is a well-known open problem.
As with dynamical systems of a finite number of degrees of freedom, besides existence and uniqueness of solutions we would like to understand the qualitative properties of the flow $\Phi_t$, when it exists. Examples of these properties are the number of equilibrium points and their stability, the periodicity and almost periodicity of trajectories, and the geometry and dynamics of more complex invariant subsets (other qualitative properties of the Euler flow $\Phi_t$ for which some results are known are mixing \cite{KKPS1, KKPS2} and wandering of solutions \cite{Na, Sn}, but our results will imply nothing about these).
So far, the only invariant sets whose existence has been established are the simplest: stationary solutions (i.e, zeros), heteroclinic and homoclinic trajectories between zeroes, periodic orbits, and quasiperiodic invariant tori \cite{CF, Taoquadratic}; in these invariant sets, of course, the flow $\Phi_t$ is well defined for all times.
Observe that, in the finite-dimensional invariant manifolds of $\Phi_t$, the evolution of the fluid velocity in time can be completely described by the evolution of a finite set of parameters; in other words, the Euler equation reduces to a finite dimensional first order ODE on the invariant manifold. The previous paragraph implies that the Euler equation is only known to reduce to the simplest finite dimensional ODEs, conjugate to linear periodic or quasiperiodic flows on tori.
The goal of this paper is to show that, in fact, almost any finite dimensional smooth dynamics can be found in the phase space of the Euler system, in some Riemannian manifold. More precisely, our main result states:
\begin{theorem}\label{main}
Let $N$ be any closed (compact without boundary) manifold. Given any $C^{\infty}$ vector field $X$ on $N$, and two positive numbers $\epsilon$ and $m$, there is
\begin{enumerate}
\item a $C^{\infty}$ vector field $Y$ in $N$ satisfying
\[
||X-Y||_{C^{m}(N)} \leq \epsilon \,,
\]
\item a compact Riemannian manifold $M$, together with a finite-dimensional linear subspace $E \subset \mathfrak{X}^{\infty}(M)$
\item a $C^{\infty}$ embedding $\Theta: N \rightarrow E$
\end{enumerate}
such that for any point $p \in N$, we have that the time-dependent divergence-free field $u_t=\Theta(\phi^t_{Y}(p))$ (here $\phi^t_{Y}$ stands for the flow of $Y$) is the solution to the Euler equation \eqref{euler} on $M$ with initial velocity $u_0=\Theta(p)$.
\end{theorem}
\begin{remark}
The embedding $\Theta$ in Theorem \ref{main} is $C^{\infty}$ for any $C^{k}$ ($k\in [0, \infty]$) topology in the space $\mathfrak{X}^{\infty}(M)$ of smooth divergence-free vector fields on $M$. In what follows, by smooth we will always mean $C^{\infty}$.
\end{remark}
We will deduce Theorem \ref{main} from Theorem \ref{main2} below, whose proof will constitute the core of the paper:
\begin{theorem}\label{main2}
Let ${\mathbb{M}}$ denote the $n$-sphere $\SS^{n}$ or the $n$-dimensional torus $\mathbb{T}^n=(\mathbb{R}/\mathbb{Z})^n$, for any $n\geq 1$. Let $X$ be any vector field on ${\mathbb{M}}$ that can be expressed as a trigonometric polynomial (in the case of $\mathbb{T}^n)$ or as a polynomial vector field in $\mathbb{R}^{n+1}$ tangent to $\SS^n$. Then, there is:
\begin{enumerate}
\item a compact Riemannian manifold $M$, together with a finite-dimensional linear subspace $E \subset \mathfrak{X}^{\infty}(M)$
\item a smooth embedding $\Phi: {\mathbb{M}} \rightarrow E$
\end{enumerate}
such that for any point $x \in {\mathbb{M}}$, the time-dependent divergence-free field $u_t=\Phi(\phi^t_{X}(x))$ is the solution to the Euler equation \eqref{euler} on $M$ with initial condition $u_0=\Phi(x)$.
\end{theorem}
\begin{remark}
The dimension of the manifold $M$ in Theorem \ref{main2} can be computed exactly and depends only on $n$ and on the degree of the vector field $X$; we refer to Subsection \ref{dimension2} for more details. In the case of Theorem \ref{main}, the dependence of the dimension of $M$ is more complex, we discuss it and find an upper bound in Subsection \ref{dimension}.
\end{remark}
Let us present rough sketches of the proofs.
To prove Theorem \ref{main2}, the key is to construct a homogeneous quadratic ODE in $\mathbb{R}^{d}$, for $d$ big enough, having an invariant manifold diffeomorphic to ${\mathbb{M}}$ where the flow of the ODE is conjugate to the flow of the vector field $X$. This ODE, moreover, will be shown to preserve the standard euclidean inner product (in other words, its trajectories are always tangent to spheres), so we can embed it into the Euler equations on the manifold $SO(d)\times \mathbb{T}^{d}$ with a certain metric, applying a Theorem of Tao \cite{Taoquadratic}.
To prove Theorem \ref{main}, we embed the manifold $N$ in a sphere $\SS^{n}$ of high enough dimension, then extend the vector field $X$ on $N$ to $ \SS^n$ in a way that guarantees structural stability of $N$ under sufficiently small perturbations of the extension of $X$. This ensured, we approximate the extension of $X$ by a polynomial vector field, and apply Theorem \ref{main2}.
\begin{remark}
The results above can be interpreted as a universality result for the (time-dependent) Euler equation. Let us remark that this is different from the recent universality result obtained for steady Euler in \cite{CMDP}. There it is shown, using techniques from contact geometry, that any vector field on a compact manifold can be embedded, without perturbation, into a stationary solution of the Euler equations on a Riemannian manifold of higher dimension.
\end{remark}
\subsection{Applications of Theorems \ref{main} and \ref{main2}}
The results above can be used to construct solutions to the Euler equations with dynamical structures that were previously unknown, or at best conjectural. In particular, there are finite dimensional invariant submanifolds in the phase space of the Euler equation where the evolution is chaotic:
\begin{theorem}\label{chaos}
There is a Riemannian manifold $(M, g)$ of dimension $21$, for which the Euler dynamical system is chaotic. More precisely, there is a 3-dimensional torus $\Sigma$ inside $\mathfrak{X}^{\infty}(M, g)$ such that
\begin{enumerate}
\item the solutions to the Euler equation with initial condition in $\Sigma$ exists for all time, and remain in $\Sigma$ (i.e, $\Sigma$ is an invariant torus of the Euler dynamical system)
\item the Euler dynamical system on $\Sigma$ has a compact invariant set where the dynamic is chaotic (presence of transverse homoclinic intersections, horseshoes, and positive topological entropy); as well as other regions foliated by 2-dimensional invariant tori.
\end{enumerate}
\end{theorem}
We will prove this Theorem in Section \ref{s4}, as an application of Theorem \ref{main2}.
We can also use Theorem \ref{main} to prove that some submanifolds of phase space are filled with strange attractors of hyperbolic type (for example, Smale-Williams solenoids).
Indeed, let $X$ be a vector field on some manifold $N$ having a hyperbolic strange attractor $A$. This means that any sufficiently small $C^m$ perturbation of $X$ also has a hyperbolic strange attractors $A'$ close to $A$, on which the flow is topologically conjugate to the flow of $X$ in $A$. Thus we have
\begin{corollary}[Hyperbolic strange attractor]
On the space of smooth divergence-free vector fields on some Riemannian manifolds $M$, we can find invariant finite-dimensional subsets on which the Euler equation has solutions for all time that converge to an hyperbolic strange attractor.
\end{corollary}
\begin{remark}
The Lorenz attractor is not hyperbolic, so the previous corollary does not apply to it. However, there is a sense in which we can still find it in the Euler system. We refer to Section \ref{sf}.
\end{remark}
Another consequence is that we can find finite dimensional families of trajectories that are Anosov, since Anosov flows are stable under small $C^{1}$ perturbations:
\begin{corollary}[Anosov flows]
On the space of smooth divergence-free vector fields on some Riemannian manifolds $M$, we can find finite-dimensional manifolds on which the Euler equation has solutions for all time and the dynamics is Anosov.
\end{corollary}
As a explicit example, we can consider a small perturbation of the geodesic flow on the unit tangent bundle of a genus $g\geq 2$ Riemannian surface with constant negative curvature.
More generally, Theorem \ref{main} implies that any dynamics displayed by a vector field on a closed manifold that is structurally stable under small $C^m$ perturbations is found in the Euler equations.
Finally, let us point out that, by virtue of Theorem \ref{main2}, certain well-known types of steady state solutions of the Euler and Navier Stokes equations in $\mathbb{T}^2$, $\mathbb{T}^3$ and $\SS^3$ can be embedded exactly into the phase space of the Euler equations in a higher-dimensional manifold $M$. That is, the lagrangian trajectories of the fluid particles in these steady solutions correspond exactly to eulerian trajectories of non-steady vector fields obeying the Euler equation on $M$. More specifically, we are referring to Kolmogorov steady flows in $\mathbb{T}^2$ (steady solutions of the Navier-Stokes equations under forcing) and Beltrami fields in $\mathbb{T}^3$ and $\SS^3$, as they have finite Fourier (or spherical harmonics) expansions.
The paper is organized as follows. The proof of Theorem \ref{main2} on embedding dynamics in $\SS^n$ and $\mathbb{T}^n$ is given in Section \ref{s1}, with the key Proposition (Proposition \ref{main}) proven in Section \ref{s11}. Section \ref{s2} proves Theorem \ref{main}. We apply these results to prove Theorem \ref{chaos} on chaotic dynamics in Section \ref{s4}. We conclude with some further constructions and corollaries.
\section{Proof of Theorem \ref{main2}}\label{s1}
The proof is broken down into two steps.
\subsection{Step 1}
In the following Proposition, we will call ``polynomic'' those vector fields on $\SS^n$ and $\mathbb{T}^n$ that can be written as finite sums of sines and cosines (in the case of $\mathbb{T}^n$) or as polynomial vector fields in $\mathbb{R}^{n+1}$ that are tangent to $\SS^n$. We define the degree of the polynomic vector field to be the modulus squared of the highest frequency (in $\mathbb{T}^n$) or the degree of the polynomial in $\mathbb{R}^{n+1}$.
\begin{proposition}\label{aux1}
Let $X$ be a polynomic vector field on ${\mathbb{M}}$ (=$\mathbb{T}^{n}$ or $\SS^{n}$). There is a smooth embedding $\Psi: {\mathbb{M}} \rightarrow \mathbb{R}^{d}$ (with $d$ depending only on $n$ and on the degree of the vector field) and a homogeneous quadratic ODE on $\mathbb{R}^{d}$
\[
\frac{d y_i}{dt}=B_{ijk} y_j y_k \text{ for $i=1,..,d$}\,,
\]
satisfying $B_{ijk}=-B_{kji}$, such that for any point $x \in {\mathbb{M}}$, $y(t)=\Psi(\phi^{t}_{X}(x))$ is the solution of the quadratic ODE with initial condition $y(0)=\Psi(x)$.
Moreover, if the vector field $X$ is divergence-free in ${\mathbb{M}}$ (with respect to the round metric in $\SS^n$, or to the flat metric in $\mathbb{T}^{n}$), then the corresponding quadratic vector field
\[
V(y)=\sum_{ijk} B_{ijk} y_j y_k \frac{\partial}{\partial y_i}
\]
is also divergence-free in $\mathbb{R}^d$ with the standard Euclidean metric.
\end{proposition}
We prove this proposition in Section \ref{s11}. The fact that the vector field in $\mathbb{R}^{d}$ is divergence-free when $X$ is will not be needed in what follows, but we found it might be of interest.
\subsection{Step 2}
The next and final step consists in embedding the quadratic ODE obtained in step 1 into the phase space of the Euler equation in some manifold $M$. We apply the following special case of Theorem 1.1 in \cite{Taoquadratic}, which we restate here in a slightly different wording, adapted to our setting:
\begin{theorem}[T. Tao \cite{Taoquadratic}]\label{tao}
Let
\[
\frac{d y_i}{dt}= V_i(y)=\sum_{j, k=1}^{d} \tilde{B}_{ijk} y_j y_k \text{ for $i=1,..,d$}
\]
be a homogeneous quadratic ODE on $\mathbb{R}^{d}$, with $\tilde{B}_{ijk}=\tilde{B}_{ikj}$ and
\begin{equation}\label{conditiontao}
\sum_{i, j, k=1}^{d} \tilde{B}_{ijk} y_i y_j y_k=0\,.
\end{equation}
Then there is a Riemannian manifold $M$ (that can be explicitly taken to be $SO(d) \times \mathbb{T}^{d}$) and a linear injective map $T: \mathbb{R}^{d} \rightarrow \mathfrak{X}^{\infty}(M)$ such that, for any $y \in \mathbb{R}^d$, the time-dependent vector field $u_{t}:=T(\phi^{t}_{V}(y))$ is a smooth solution to the Euler equation on $M$ with initial condition $u_0=T(y)$.
\end{theorem}
Observe that the coefficients $B_{ijk}$ in Proposition \ref{aux1} do not directly satisfy the hypothesis in Theorem \ref{tao}, because they are not symmetric in the $j, k$ indices. Nevertheless, setting
\[
\tilde{B}_{ijk}:=\frac{1}{2}(B_{ijk}+B_{ikj})
\]
we see that the coefficients $\tilde{B}_{ijk}$ and $B_{ijk}$ define the same ODE. Moreover, by virtue of Proposition \ref{aux1} we have that $B_{ijk}=-B_{kji}$, so
\[
\sum_{i, j, k=1}^{d} \tilde{B}_{ijk} y_i y_j y_k=0 \,.
\]
Thus we can apply Theorem \ref{tao} to the ODE given by the coefficients $\tilde{B}_{ijk}$. The embedding $\Phi$ in the statement of Theorem \ref{main2} is then given by $\Phi=T \circ \Psi$.
To conclude the proof of Theorem \ref{main2}, it remains to be shown that $\Phi=T \circ \Psi$ is smooth for any $C^{k}$ topology (with $k\in [0, \infty]$) in the space of smooth divergence-free fields $\mathfrak{X}^{\infty}(M)$.
For finite $k$ this is obvious, because the embedding $\Psi: {\mathbb{M}} \rightarrow \mathbb{R}^{d}$ is smooth and $T$ is a linear injective map. Indeed, let $\{e_{1},...,e_{d}\}$ denote the standard basis of $\mathbb{R}^{d}$; the images $\{w_{\mu}=T(e_{\mu})\}$ are a basis of the finite dimensional linear subspace $E \subset \mathfrak{X}^{\infty}$ where we embed ${\mathbb{M}}$. The $C^k$ norms of the basis elements are bounded, $||w_{\mu}||_{C^{k}(M)}\leq C(k)$, so the smoothness of $\Psi: {\mathbb{M}} \rightarrow \mathbb{R}^{d}$ immediately implies that of $\Phi=T \circ \Psi: {\mathbb{M}}\rightarrow E$.
To see that $\Phi$ is smooth for the $C^{\infty}$ topology in $\mathfrak{X}^{\infty}(M)$, we must show that the $C^{k}$-norms of the vector fields $w_{
\mu}=T(e_{\mu})$ are bounded independently of $k$. This requires a more detailed discussion of the nature of the map $T$ in Tao's Theorem.
The map $T$ has the form
\[
T(e_{\mu})=\Pi^{*} U_{\mu}+\sum_{\nu=1}^{d} \Pi^{*} F_{\mu \nu} \frac{\partial}{\partial t_{\nu}}
\]
where $\Pi$ is the projection map $\Pi: SO(d) \times \mathbb{T}^{d} \rightarrow SO(d)$, $\{\frac{\partial}{\partial t_{\nu}}\}_{\nu=1}^{d}$ is the standard basis of $T \mathbb{T}^{d}$, and $U_{\mu}$ and $F_{\mu \nu}$ are vector fields and functions on $SO(d)$, respectively, that we will describe below: our goal is to show that their $C^{k}$ norms are bounded uniformly in $k$, so that the vector fields $w_{\mu}=T(e_{\mu})$ have the same property.
Consider $SO(d)$ as a submanifold in the linear space Mat$(d)$ of $d\times d$ matrices, so that we can identify the tangent space $T_{Q} SO(d)$ at any orthogonal matrix $Q$ with a linear subspace of Mat$(d)$. As defined in \cite{Taoquadratic}, Section 5, for any given index $\mu=1,...,d$, $U_{\mu}$ is the right invariant vector field on $SO(d)$ whose value at any $Q \in SO(d)$ is given by
\[
U_{\mu}(Q)= S_{\mu} Q
\]
where $S_{\mu}$ is a matrix in $\mathfrak{so}(d)$ that depends only on the coefficients $B_{ijk}$ of our quadratic ODE. As for the functions $F_{\mu \nu}: SO(d)\rightarrow \mathbb{R}$, they are defined as
\[
F_{\mu \nu}(Q)= e_{\mu} \cdot Q e_{\nu}=Q_{\nu \mu} \,.
\]
To see that the $C^{k}$-norms of the functions and the vector fields are bounded, consider the standard basis $\{E_{\alpha}\}$ of the Lie algebra $\mathfrak{so}(d)$ (consisting of antisymmetric matrices having only two non-zero entries, one equal to $1$ and the other to $-1$).
The derivative of the function $F_{\mu \nu}$ in the direction given by the tangent vector $Q E_{\alpha_1}$ is given by
\[
\frac{d}{dt}\bigg|_{t=0} F_{\mu \nu}(Q e^{tE_{\alpha_1}})=e_{\mu} \cdot Q E_{\alpha_1} e_{\nu}
\]
Iterating this, we see that the derivatives of order $k$ at the point $Q$ of the function $F_{\mu \nu}$ have the form
\[
e_{\mu} \cdot Q E_{\alpha_k}... E_{\alpha_1} e_{\nu}
\]
The products of the matrices $E_{\alpha}$ have at most one non-zero element in any row or column, equal to $1$ or $-1$. Thus the $k$-derivatives are always equal to some single entry $Q_{\rho \lambda}$ of the matrix $Q$, hence the $C^{k}$ norms of the functions $F_{\mu \nu}$ are bounded independently of $k$.
As for the $C^{k}$-norms of the right-invariant vector fields $U_{\mu}$, it is enough to check them at the identity. In other words, it is enough to prove that any $k$-iterated commutators
\[
[E_{\alpha_1},[E_{\alpha_2},[....,[E_{\alpha_k}, S_{\mu}]...]]
\]
have norms (for any metric on $\mathfrak{so}(d))$ that can be bounded independently of $k$. This easily follows from the fact that the commutators of the basis elements $\{E_{\alpha}\}$ are all of the type $[E_{\alpha}, E_{\beta}]=\pm E_{\gamma}$.
\subsection{The dimension of the manifold $M$}\label{dimension2}
As the proof of the Proposition \ref{aux1}, given in Section \ref{s11} below, will make manifest, the parameter $d$ in $SO(d)\times \mathbb{T}^{d}$ is at most the dimension of the linear space of trigonometric polynomials of degrees up to the degree, say $D$, of $X$ (when ${\mathbb{M}}=\mathbb{T}^n)$; or, in the case of $\SS^n$, at most the dimension of the space of harmonic polynomials of degrees up to one minus the degree of $X$.
More explicitly, in the case of the torus, this dimension is equal to the number of points in $\mathbb{Z}^n$ that lie in the (closed) ball of radius $\sqrt{D}$; while in the case of $\SS^n$, it is equal to
\[
d=\sum_{j=0}^{D}\binom{j+n-1}{n}\frac{2j+n-1}{j+n-1}=\binom{D+n}{n}\frac{D(2Dn+n^2+1)}{n(n+1)(D+n)}
\]
The dimension of $M=SO(d)\times \mathbb{T}^{d}$ grows then as the square of these numbers.
\section{Proof of Proposition \ref{aux1}}\label{s11}
\subsection{Proof for ${\mathbb{M}}=\mathbb{T}^{n}$}
The field $X$ can be written as a sum of the form
\[
X(x)=\sum_{k \in \mathbb{Z}^n, \,|k| \leq \Lambda} a_{k} \sin(2 \pi k \cdot x)+b_{k} \cos(2 \pi k \cdot x) \,\,
\]
with $a_{k}, b_k \in \mathbb{R}^n$ and $a_{k}=-a_{-k}$, $b_{k}=b_{-k}$.
Let $d(\Lambda)$ be the number of integer lattice points contained in the $n$-ball of radius $\Lambda$. Consider the map $\Psi: \mathbb{T}^{n} \rightarrow \mathbb{R}^{2d(\Lambda)}$ given by
\[
\Psi(x)=\{\sin(2 \pi k \cdot x), \, \cos(2 \pi k \cdot x)\}_{k \in \mathbb{Z}^{n}\cap B^{n}(0, \Lambda)}
\]
where we represent the points $(q, p) \in \mathbb{R}^{2d(\lambda)}$ as $(q, p):=\{q_{k}, p_k\}_{k \in \mathbb{Z}^{n}\cap B^{n}(0, \Lambda)}$.
\begin{lemma}\label{emb}
For $\Lambda \geq 1$, the map $\Psi$ is an embedding.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{emb}]
It suffices to prove the claim for $\Lambda=1$. Denote by $B$ the $n$-dimensional ball of radius 1. The frequencies $k \in \mathbb{Z}^{n}\cap B$ are the ones of the form
\[
k=\pm(0, 0,...,1,0,...,0)
\]
plus the zero vector, so $d(\Lambda)=2n+1$.
First we show that $d\Psi$ is injective. Suppose that at some point $x$ there is a vector $v$ in the kernel of the differential :
\[
d_x \Psi (v)= \sum_{k} 2 \pi v \cdot k \cos(2 \pi k \cdot x) \frac{\partial}{\partial q_{k}}-2 \pi v \cdot k \sin(2 \pi k \cdot x)\frac{\partial}{\partial p_{k}}=0
\]
this means that $v \cdot k=0$ for all $k \in \mathbb{Z}^n \cap B$. But the frequencies $k$ span the whole $\mathbb{R}^n$, so we must have $v=0$.
It remains to be shown that for any two distinct points $x, y \in \mathbb{T}^n$ we must have $\Psi(x)\neq \Psi(y)$. This is easy to see, for if $\Psi(x)=\Psi(y)$,
\[
\cos(2 \pi k \cdot x)=\cos(2 \pi k \cdot y) \,,\,\,\,\sin(2 \pi k \cdot x)=\sin(2 \pi k \cdot y)
\]
for integer frequencies $k$ spanning $\mathbb{R}^n$. This is only possible if $x=y +2 \pi m$ for some $m \in \mathbb{Z}^n$, that is, if $x$ and $y$ label the same point in $\mathbb{T}^n$.
\end{proof}
\begin{remark}\label{harmonics}
If the Fourier expansion of $X$ does not have a constant ($k=0$) component, we can define the map $\Psi$ as
\[
\Psi(x)=\{\sin(2 \pi k \cdot x), \, \cos(2 \pi k \cdot x)\}_{k \in \mathbb{Z}^{n}\cap B^{n}(0, \Lambda)\setminus\{0\}}\,.
\]
It can be readily checked that this does not affect the assertion in Lemma \ref{emb}, nor any further step in the proof.
\end{remark}
Consider now the vector field $\Psi_{*}(X)$ in $\Psi(\mathbb{T}^n)$. It has the form
\begin{align*}
d_x \Psi(X)= \sum_{k, k' \in \mathbb{Z}^{n}\cap B^{n}(0, \Lambda)} 2\pi \bigg( a_{k'} \cdot k \sin(2 \pi k' \cdot x) \cos(2 \pi k \cdot x)+ \\
+b_{k'} \cdot k \cos(2 \pi k'\cdot x) \cos(2 \pi k \cdot x) \bigg) \frac{\partial}{\partial q_k} \\ -2 \pi \bigg( a_{k'} \cdot k \sin(2 \pi k' \cdot x) \sin(2 \pi k \cdot x)+ \\
+b_{k'} \cdot k \cos(2 \pi k'\cdot x) \sin(2 \pi k \cdot x) \bigg) \frac{\partial}{\partial p_k}
\end{align*}
Consider as well the following vector field on the whole space $\mathbb{R}^{2d(\lambda)}$
\[
V(q, p)=2 \pi \sum_{k} \sum_{k'} (a_{k'} \cdot k \,q_{k'} +b_{k'} \cdot k \, p_{k'}) p_k \frac{\partial}{\partial q_k}-(a_{k'} \cdot k \, q_{k'} +b_{k'} \cdot k \, p_{k'}) q_k \frac{\partial}{\partial p_k}\,.
\]
We see that $d_x \Psi(X)=V(q, p)$ when $(q, p)=(q(x), p(x)) \in \Psi(\mathbb{T}^{n})$, that is, $V$ is tangent to $\Psi(\mathbb{T}^{n})$ and there, it is equal to $\Psi_{*} X$.
The vector field $V$ defines the homogeneous quadratic ODE:
\[
\frac{d q_k}{dt}=\sum_{k'} (a_{k'} \cdot k\, q_{k'} +b_{k'} \cdot k\, p_{k'}) p_k
\]
\[
\frac{d p_k}{dt}=- \sum_{k'} (a_{k'} \cdot k\, q_{k'} +b_{k'} \cdot k\, p_{k'}) q_k
\]
To check that this ODE satisfies the antisymmetry condition in Proposition \ref{aux1}, let us relabel the coordinates in $\mathbb{R}^{2 d(\Lambda)}$ as $x_{\alpha}$, with $\alpha \in \{1,..., 2d(\Lambda)\}$. We see that the coefficients $B_{\alpha \beta \gamma}=0$ unless $x_{\alpha}=q_{k}$ and $x_{\gamma}=p_{k}$ or viceversa. In that case, we have either
\[
B_{q_k, q_{k'}, p_{k}}= a_{k'}\cdot k\, q_{k'}=-B_{p_k, q_{k'}, q_k}
\]
or
\[
B_{q_k, p_{k'}, p_{k}}= b_{k'}\cdot k\, p_{k'}=-B_{p_k, q_{k'}, q_k}
\]
Thus $B_{\alpha \beta \gamma}=-B_{\gamma \beta \alpha}$, that is, the coefficients of the ODE are always antisymmetric under exchange of the first and last index, as we wanted to show.
It remains to be shown that, if $X$ is divergence-free, $V$ is also divergence-free, i.e
\[
\Div V=\sum_k \frac{\partial {V_{q_k}}}{\partial q_k}+\frac{\partial {V_{p_k}}}{\partial p_k}=0
\]
Indeed, we have
\[
\frac{\partial {V_{q_k}}}{\partial q_k}=a_{k} \cdot k\, p_k
\]
\[
\frac{\partial {V_{p_k}}}{\partial p_k}=b_{k} \cdot k\, q_k
\]
and if $X$ is divergence-free, the coefficients in its Fourier expansion satisfy $a_k \cdot k=b_k \cdot k=0$.
\subsection{Proof for ${\mathbb{M}}=\SS^{n}$}
Let $\{A_{\mu}\}_{\mu=1}^m$ be a basis of the Lie algebra $\mathfrak{so}(n+1)$ of $(n+1) \times (n+1)$ traceless antisymmetric matrices (so here $m=\frac{n(n+1)}{2}$). It is easy to check that the vector fields in $\mathbb{R}^{n+1}$ given by
\begin{equation}\label{hopf}
h_{\mu}(x)=A_{\mu} \cdot x
\end{equation}
(where $A_{\mu} \cdot x$ denotes matrix multiplication of $A_{\mu}$ and the vector $x\in \mathbb{R}^{n+1}$) are tangent to $\SS^{n} \subset \mathbb{R}^{n+1}$ and, moreover, for any $x \in \SS^{n}$,
\[
\text{ span }\{h_1(x),...,h_m(x)\}=T_{x} \SS^{n}
\]
The vector field $X$ in $\SS^{n}$ can thus be written as
\[
X=\sum_{\mu=1}^{m} f_{\mu} h_{\mu}
\]
where $f_{\mu} : \SS^{n} \rightarrow \mathbb{R}$ are smooth functions.
For any given $N\in \mathbb{N}$, let $\{Y_{\alpha}\}_{\alpha=1}^{d(N)}$ be a $L^2$-orthonormal basis of the space of spherical harmonics in $\SS^{n}$ of degree up to $N$. We recall that this is the linear space spanned by the eigenfunctions of the Laplace-Beltrami operator (defined with the round metric) of eigenvalues up to $N(N+n-1)$. Equivalently, they are obtained as the restrictions to $\SS^{n}$ of the homogeneous harmonic polynomials in $\mathbb{R}^{n+1}$ of degrees up to $N$. We label the elements of the basis in increasing degree, so that if the degree of $Y_{\alpha}$ is less than the degree of $Y_{\beta}$, $\alpha < \beta$.
Note that the components of the vector fields $\{h_{\mu}\}$ are homogeneous polynomials of degree $1$. Since the vector field $X$ is the restriction to $\SS^{n}$ of a polynomial vector field in $\mathbb{R}^{n+1}$, its components $f_{\mu}$ must be given by a finite sum of spherical harmonics
\[
f_{\mu}= \sum_{\alpha=1}^{d(N)} c^{\alpha}_{\mu} Y_{\alpha}
\]
up to some degree $N$.
Define a map $\Psi: \SS^{n} \rightarrow \mathbb{R}^{d(N)}$ as
\[
\Psi(x)=(Y_{1}(x),...,Y_{d(N)}(x)) \,.
\]
As in the case of the torus, our first goal is to prove that the map $\Psi$ is an embedding. It suffices to do so for $N=1$, where the spherical harmonics are just restrictions to $\SS^n$ of affine functions on $\mathbb{R}^{n+1}$, so the map is simply
\[
\Psi(x)=(1, x_1(x), x_2(x), ....,x_{n+1}(x))
\]
or some coordinate permutation and rotation of the above, depending on our choice of basis of spherical harmonics. This is clearly an embedding.
\begin{remark}\label{harmonics2}
As in the case of the torus, if the functions $f_{\mu}$ have zero mean (i.e, they do not have projection into the constant functions on $\SS^n$), we can define the map $\Psi$ as
\[
\Psi(x)=(Y_2(x), Y_3(x),...,Y_{d(N)}(x))\,.
\]
\end{remark}
We have
\[
d_x \Psi(X)=\sum_{\mu=1}^{m} \sum_{\alpha=1}^{d(N)} c^{\alpha}_{\mu} Y_{\alpha}(x) ( h_{\mu}(Y_{1})(x),...,h_{\mu}(Y_{d(N)}(x)))\,.
\]
Here $h_{\mu}(Y_{\beta})(x)$ represents the derivative of the eigenfunction $Y_{\beta}$ in the direction of the vector field $h_{\mu}$. We now claim that we can express these derivatives as linear combinations of the original spherical harmonics $\{Y_{\alpha}\}$, with $\alpha \leq d(N)$. In other words, there are explicit coefficients $\theta^{\gamma}_{\mu \beta}$, with $\mu=1,...,m$, $\gamma, \beta=1,...,d(N)$ so that
\[
h_{\mu}(Y_{\beta})(x)=\sum_{\gamma=1}^{d(N)} \theta^{\gamma}_{\mu \beta} Y_{\gamma}(x)\,.
\]
These coefficients will depend on the basis of harmonics $\{Y_{\alpha}\}$, but not on the point $x$ (and actually, they will be zero unless the degree of $Y_{\gamma}$ matches that of $Y_{\beta}$, but we will not use this property).
The existence of such coefficients is a straightforward consequence of the relationship between the spherical harmonics in $\SS^n$ and the representations of the group $SO(n+1)$, but in order to keep the article as elementary and self-contained as possible, we will give here a simple proof.
Recall that the vector fields $h_{\mu}$ are given by
\[
h_{\mu}(x)=A_{\mu} \cdot x\,.
\]
For any $t \in \mathbb{R}$, the matrices $\Lambda^{t}_{\mu}:=\exp (t A_{\mu})$ are elements of $SO(n+1)$. Thus, if $P(x)$ is an harmonic polynomial of degree $\alpha$, $Q(x):=P(\Lambda^{t}_{\mu} x)$ is also an harmonic polynomial of the same degree.
Hence the spherical harmonics, being restrictions to $\SS^n$ of the harmonic polynomials, inherit this invariance under the action of $\Lambda^{t}_{\mu}$, so that for any basis element $Y_{\beta}$ we have
\[
Y_{\beta}(\Lambda^{t}_{\mu} x)=\sum_{\gamma=1}^{d(N)} \Theta^{\gamma}_{\beta}(\Lambda^{t}_{\mu}) Y_{\gamma}(x)
\]
where the coefficients $\Theta^{\gamma}_{\beta}(\Lambda^{t}_{\mu})$ do not depend on the point.
Now observe that
\[
\frac{d}{dt}\bigg|_{t=0} Y_{\beta}(\exp(t A_{\mu}) x)= h_{\mu}(Y_{\gamma})(x)\,,
\]
so defining
\[
\theta^{\gamma}_{\mu \beta}=\frac{d}{dt}\bigg|_{t=0} \Theta^{\gamma}_{\beta}(\exp(t A_\mu))
\]
our claim follows.
Hence
\[
d_x \Psi(X)=\sum_{\mu=1}^{m} \sum_{\alpha=1}^{d(N)} \sum_{\gamma=1}^{d(N)} c^{\alpha}_{\mu} Y_{\gamma}(x) Y_{\alpha}(x) (\theta^{\gamma}_{\mu 1},...,\theta^{\gamma}_{\mu d(N)} )\,.
\]
This means that the quadratic vector field in $\mathbb{R}^{d(N)}$ defined as
\[
V=\sum_{i=1}^{3} \sum_{\alpha, \beta, \gamma =1}^{d(N)} \theta^{\gamma}_{i \beta} c^{\alpha}_{\mu} y_{\gamma} y_{\alpha} \frac{\partial}{\partial y_{\beta}}
\]
is tangent to $\Psi(\SS^{n})$, and coincides there with $d \Psi(X)$.
Furthermore, the coefficients $\theta^{\gamma}_{\mu\beta}$ are antisymmetric in $\gamma, \beta$.:
\[
\theta^{\gamma}_{\mu \beta}=\int_{\SS^{n}} h_{\mu}(Y_{\beta})(x) Y_{\gamma}(x) d \Omega(x)=-\int_{\SS^{n}} h_{\mu}(Y_{\gamma})(x) Y_{\beta}(x) d \Omega(x)=-\theta^{\beta}_{\mu \gamma}
\]
(here we have integrated by parts and used the fact that the vector fields $h_{\mu}$, being infinitesimal generators of isometries, are divergence-free); so the coefficients
\[
B_{\beta \alpha \gamma}:= \sum_{\mu=1}^{m} \theta^{\gamma}_{\mu\beta} c^{\alpha}_{\mu}
\]
satisfy $B_{\beta \alpha \gamma}=-B_{\gamma \alpha \beta}$, as we wanted to show.
Finally, we are left to prove that, assuming the vector field $X$ in $\SS^{n}$ is divergence-free with respect to the round metric, $V$ in $\mathbb{R}^{d(N)}$ is divergence-free with respect to the Euclidean metric.
In terms of the expansion in spherical harmonics, the divergence-free condition reads
\[
\Div_{\SS^{n}} X=\sum_{\mu=1}^{m} \sum_{\alpha=1}^{d(N)} c^{\alpha}_{\mu} h_{\mu}(Y_{\alpha})=\sum_{\mu=1}^{m} \sum_{\alpha, \beta=1}^{d(N)} c^{\alpha}_{\mu} \theta^{\alpha}_{\mu \beta} Y_{\beta}=0\,.
\]
Thus we conclude that, if $X$ is divergence-free, the coefficients $c^{\alpha}_{\mu}$ satisfy, for any $\beta$:
\begin{equation}\label{div}
\sum_{\mu=1}^{m} \sum_{\alpha=1}^{d(N)} c^{\alpha}_{\mu} \theta^{\alpha}_{\mu \beta}=0\,,
\end{equation}
so
\[
\Div_{\mathbb{R}^{d(N)}} V= \sum_{\mu=1}^{m} \sum_{\alpha, \beta, \gamma =1}^{d(N)} \theta^{\gamma}_{\mu \beta} c^{\beta}_{\mu} y_{\gamma} +\theta^{\beta}_{\mu \beta} c^{\alpha}_{\mu} y_{\alpha}=0
\]
(here we have also used the fact that $\theta^{\beta}_{\mu \beta}=0$ because of antisymmetry).
\section{Proof of Theorem \ref{main}}\label{s2}
The idea of the proof is as follows: first the manifold $N$ is embedded into a sphere $\SS^n$ of suitable dimension, and the push-forward of the vector field $X$ is extended to the whole $\SS^n$. By constructing this extension in a suitable way, we can ensure that any other vector field close enough to the extended vector field has an invariant manifold diffeomorphic to $N$, on which it is very close to $X$. We then take a polynomic approximation of the extension of $X$ and apply Theorem \ref{main2} to conclude.
More precisely, let $F: N \rightarrow \SS^{n}$ be an embedding of $N$ into $\SS^n$. Provided we take the dimension of the sphere high enough, such an embedding always exists.
Our aim now is to extend the vector field $F_{*}(X)$ to the whole $\SS^{n}$, so that $F(N)$ is an $r$-normally hyperbolic invariant manifold:
\begin{definition}[Normally hyperbolic invariant manifold, \cite{HSS} Section 1]
Let $Y$ be a smooth vector field on a manifold $M$. We will say that a submanifold $V \subset M$ is an $r$-normally hyperbolic invariant manifold of $Y$ if $Y$ is tangent to $V$ and, moreover, there is a continuous splitting
\[
TM|_{V}=E^{u} \oplus TV \oplus E^{s}
\]
and constants $c>0$, $0\leq \mu< \lambda$ such that, for any $x \in V$ we have:
\begin{enumerate}
\item For any $t\in \mathbb{R}$, $d_x\phi_{Y}^{t}(E^{u}_{x})=E^{u}_{\phi^{t}_{Y}(x)} $, and analogously for $E^{s}$.
\item For any $v \in E^{s}_{x}$ and $t\geq 0$, $\|d_x\phi_{Y}^{t}(v)\| \leq c e^{-\lambda t}\|v\|$.
\item For any $v \in E^{u}_{x}$ and $t\geq 0$, $\|d_x\phi_{Y}^{-t}(v)\| \leq c e^{-\lambda t}\|v\|$.
\item For any $v \in T_xV$ and $t\in \mathbb{R}$, $\|d_x\phi_{Y}^{t}(v)\| \leq c e^{\frac{\mu}{r}|t|}\|v\|$.
\end{enumerate}
\end{definition}
Let $(x, z_1,...,z_k)$, $x \in N$, $(z_1,...,z_k) \in N_{F(x)} F(N)$, $k=n-\text{ dim }N$, be local coordinates parametrizing a tubular neighbourhood $V$ of $F(N)\subset \SS^n$. Define a vector field $Z$ that is given in the local coordinates by
\[
Z(x, z_1,...,z_k)=X(x)-C(z_1 \frac{\partial}{\partial z_1}+...+z_k \frac{\partial}{\partial z_k})\,.
\]
Because $N$ is compact, if we take the constant $C$ large enough the field $Z$ is $r$-normally hyperbolic on $F(N)$ for any a priori chosen $r$. We then extend it smoothly in an arbitrary way to the rest of $\SS^n$, still denoting this extension by $Z$.
A crucial property of $r$-normally hyperbolic flows is their structural stability (Theorem 4.1 in \cite{HSS}), that is, any other vector field $Z'$ close enough to $Z$ in the $C^r$ norm, $r\geq 1$, has the following property: there is an embedding $F': N \rightarrow \SS^n$, close to $F$ in the $C^r$ norm, such that $Z'$ is tangent to $F'(N)$. This also implies, in particular, that the vector field $dF'^{-1}( Z'|_{F'(N)})$ on $N$ is close to $X$ in the $C^{r-1}$ norm.
We now take $Z'$ to be a polynomic vector field on $\SS^n$, approximating $Z$ to any desired degree of accuracy (more precisely, we approximate by polynomials the components of the vector field $Z$ with respect to the vector fields $h_{\mu}$ defined in Section \ref{s11}). Theorem \ref{main} follows by applying Theorem \ref{main2} to $Z'$.
\subsection{The dimension of $M$}\label{dimension}
The dimension of the Riemannian manifold $M$ in Theorem \ref{main} will depend only on the dimension $n$ of the sphere in which we embed $N$, and on the degree of the polynomic vector field $Z'$ approximating the extension of $X$. More precisely, $M=SO(d)\times \mathbb{T}^d$, where $d$ is the dimension of the space of spherical harmonics in $\SS^n$ whose degrees are less than that of $Z'$ (see Subsection \ref{dimension2}). The degree of $Z'$ will in turn depend on the acceptable error $\epsilon$ in the approximation of $Z$ by $Z'$, which in applications will be determined by the robustness of the dynamical feature of $X$ we are interested in.
To get a quantitative bound on the degree, and hence on the dimension, as a function of $\epsilon$, we can use a multidimensional Jackson-type theorem. For example, by virtue of Theorem 2 in \cite{Bagby}, we have that, for any integer $k\geq 0$, there are polynomials $Z'$ of any degree satisfying:
\[
||Z-Z'||_{C^{m}(V)} \leq \frac{c}{(\text{ degree }Z')^{k}} ||Z||_{C^{m+k}(V)}
\]
where $c$ is a constant depending on the dimension $n$, on the desired norm of approximation $m$, and on the tubular neighbourhood $V$ of $F(N)$. Observe that, by how the extension $Z$ was constructed, the $C^{m+k}$ norms of $Z$ can be bounded by those of $X$ in $N$, modulo a constant depending on the Lyapunov exponents of $X$ and on the derivatives of the embedding of $N$.
Thus we conclude
\[
\text{ degree } Z'\leq \bigg(\frac{C'}{\epsilon}||X||_{C^{m+k}(N)}\bigg)^{\frac{1}{k}}
\]
so that the parameter $d$ in $SO(d)\times \mathbb{T}^d$ can be bounded by the dimension of the space of harmonic polynomials of degrees up to the degree $D$ of $Z'$
\[
d < \binom{D+n}{n}\frac{D(2Dn+n^2+1)}{n(n+1)(D+n)} \sim D^{n+1}
\]
and finally
\[
\text{ dim } M < \bigg(\frac{C'}{\epsilon} ||X||_{C^{m+k}(N)}\bigg)^{2\frac{n+1}{k}}\,.
\]
\section{Proof of Theorem \ref{chaos}}\label{s4}
The family of vector fields on $\mathbb{T}^3$ defined as
\[
u_{ABC}(x_1, x_2, x_3) = (A \sin x_3 + C \cos x_2) \frac{\partial}{\partial x_1} + (B \sin x_1 + A \cos x_3) \frac{\partial}{\partial x_2} +
\]
\[
+(C \sin x_2 + B \cos x_1) \frac{\partial}{\partial x_3}
\]
for parameters $A, B, C \in \mathbb{R}$, are called the ABC (Arnold-Beltrami-Childress) flows. They have been extensively studied (see e.g \cite{AKh} and references therein), and are known to have chaotic invariant sets for certain values of the parameters $A, B, C$ \cite{Ch, Zi}.
An ABC flow is given by sines and cosines with integer frequencies in the unit sphere. Thus, arguing as in the proof of Theorem \ref{main2} for the torus, we see that ABC vector fields are embedded in the Euler dynamics on $M=SO(6) \times \mathbb{T}^{6}$ (there are seven integer points in the ball of radius $1$, but we can exclude the zero vector by virtue of Remark \ref{harmonics}). Theorem \ref{chaos} follows.
\section{Additional comments}\label{sf}
Here we give some additional constructions of interesting dynamics inside the Euler system which do not follow immediately from Theorems \ref{main} and \ref{main2}, but rather need the concert of other results:
\subsection{The Lorenz attractor in the Euler equations}
The Lorenz attractor is a paradigmatic example of attractor and a popular emblem of chaos. It arises in an ODE in $\mathbb{R}^3$ that is obtained from a Galerkin truncation of the Boussinesq equation (itself a PDE approximating the Navier-Stokes equations).
The goal of this subsection is to embed 3-dimensional geometric Lorenz flows (vector fields introduced in \cite{W} that have the same qualitative dynamics as the Lorenz system) into the Euler dynamical system on the manifold $SO(d)\times \mathbb{T}^{d}$, so that the Euler equations reduce to the Lorenz dynamics in a finite dimensional subset of the phase space. In other words, in that manifold, the Lorenz dynamics are not a toy model of the Navier-Stokes equations, but an exact description of ideal fluid motion.
Lorenz attractors are not stable under perturbation of the flow \cite{W}, which precludes the direct application of Theorem \ref{main}.
Nevertheless, by a theorem of Guckenheimer and Williams (see the main Theorem in \cite{GW}), the set of vector fields in $\mathbb{R}^3$ having a geometric Lorenz attractor contains an open set in the $C^{0}$ topology. Since geometric Lorenz attractors are contained in a bounded set, this fact carries over to any other $3$-dimensional compact manifold; in particular, to $\SS^3$. Thus there are polynomial vector fields in $\SS^3$ having a geometric Lorenz attractor, because they are dense for the $C^{0}$ topology. Applying Theorem \ref{main2} to one of these vector fields, we obtain a finite-dimensional family of solutions to the Euler equation on some Riemannian manifold $M$ that converge to a geometric Lorenz attractor.
\subsection{The universal template}
There is a Riemannian manifold $M$, and a 3-dimensional family of divergence-free vector fields $\Sigma \subset \mathfrak{X}^{\infty}(M)$, with the following properties:
\begin{enumerate}
\item $\Sigma$ is diffeomorphic to a 3-sphere, and invariant under the Euler dynamical system.
\item The Euler dynamics on $\Sigma$ contains sets of periodic orbits representing every isotopy class of knots.
\end{enumerate}
Indeed, there is a vector field in $\SS^{3}$ containing periodic orbits of every isotopy class of knots and links, which are moreover stable under sufficiently small $C^{m}$ perturbations (Corollary 3.2.19 in \cite{GS}). Applying Theorem \ref{main}, the claim follows.
\subsection{Euler trajectories and translation surfaces of triangular billiards}\label{sbilliards}
Let $P\subset \mathbb{R}^2$ be a triangle. We will henceforth assume that $P$ comprises the interior and the sides of the triangle, but not the vertices.
The billiard on $P$ is the dynamical system defined thus: a particle at point $q \in P$, with initial velocity $v$ of modulus $1$, moves with constant velocity while in the interior of $P$, and is reflected every time the trajectory hits a side of the triangle. If the trajectory hits a vertex, the particle stops. Any billiard trajectory is thus completely determined by the initial position $q \in P$ and the angle $\theta \in \mathbb{R} / 2 \pi \mathbb{Z}$ of the initial velocity.
By means of the unfolding construction of Katok-Zemljakov \cite{KZ}, we can associate to any triangle $P$ an open, smooth Riemann surface $S_{P}$ with a flat metric, so that the geodesic flow on the unit tangent bundle of $S_{P}$ is equivalent to the billiard on $P$.
The surface $S_{P}$ can be endowed with an atlas whose transition maps are euclidean translations, and so it is called the translation surface of $P$. In the coordinates given by this atlas, the geodesics on $S_{P}$ are straight lines, and their slope is globally well-defined, because the transition functions are translations. Thus for any given angle $\theta$, we can define a foliation on $S_{P}$ whose leaves are the geodesics of slope $\tan(\theta)$.
\begin{corollary}\label{billiards}
For any triangle $P$, there is a metric on $M=SO(30)\times \mathbb{T}^{30}$ so that the Euler equation on $M$ has the following property: for any angle $\theta$, there is an compact surface $\Sigma_{P, \theta} \subset \mathfrak{X}^{\infty}(M)$, invariant under the Euler evolution, and such that
\begin{enumerate}
\item The surface $\Sigma_{P, \theta}$ minus a finite number of points is diffeomorphic to the translation surface $S_{P}$. These finite points in $\Sigma_{P, \theta}$ are stationary solutions of the Euler equation.
\item The one-dimensional foliation on $\Sigma_{P, \theta}$ whose leaves are given by the Euler trajectories is diffeomorphic to the foliation on $S_{P}$ given by geodesics of slope $\tan(\theta)$.
\end{enumerate}
\end{corollary}
\begin{proof}
They key ingredient in the proof is the dictionary between polygonal billiards and homogeneous foliations in $\mathbb{C}^2$ due to F. Valdez \cite{V}. Let $\lambda_{1}, \lambda_2, \lambda_3$ be the angles of the triangle $P$. In $\mathbb{C}^2$ we define the following homogeneous holomorphic vector field:
\[
X=z_1(\lambda_3 z_2+\lambda_2(z_2-z_1)) \partial_{z_1}+z_2(\lambda_3 z_1+\lambda_1(z_1-z_2))
\]
The integral curves of $\Re (X)$ and $\Im (X)$ generate a $2$-dimensional homogeneous foliation $\mathcal{F}$ in $\mathbb{R}^4$.
By virtue of Theorem 1.1 in \cite{V}, for any angle $\theta$ we can find a leaf $L$ of $\mathcal{F}$ that is diffeomorphic to $S_P$ through a diffeomorphism that maps the integral curves of $\Re(X)$ in $L$ to the leaves of the foliation in $S_{P}$ given by the geodesics of slope $\tan(\theta)$.
As argued in Section 1.1 of \cite{V}, the homogeneity of the construction above allows us to deduce the analogous result in $\mathbb{RP}^{3}$ and $\SS^3$. In particular, the foliation in $\SS^3$ defined by the integral curves of the vector fields
\[
U=\Re(X)-(\Re(X)\cdot \partial_{r}) \partial_{r}
\]
\[
V=\Im(X)-(\Im(X)\cdot \partial_{r}) \partial_{r}
\]
(where $\partial_r$ denotes the radial unit vector field) has a leaf which, minus the zeros of $U$, is diffeomorphic to $S_P$, and on which $U$ defines a foliation diffeomorphic to the geodesic foliation of any given slope.
The vector fields $U$ and $V$ in $\SS^3$ are the restrictions of polynomial vector fields of degree $4$ on $\mathbb{R}^4$. Arguing as in the proof of Theorem \ref{main2} in the case of the $\SS^3$, we see that we can embed the vector field in $SO(30)\times \mathbb{T}^{30}$ ($30$ being the dimension of the linear space of spherical harmonics in $\SS^3$ of degree up to $3$). Corollary \ref{billiards} follows.
\end{proof}
\begin{remark}
Corollary \ref{billiards} can be generalized to more general polygonal billiards using the construction in Section 5 of \cite{V}.
\end{remark}
\section{Acknowledgements}
This work owes a great deal to Daniel Peralta-Salas, who shared with the author many crucial insights. The author also wants to thank Theodore Drivas, Boris Khesin and \'Angel David Mart\'inez for useful conversations and for helping to improve the manuscript with their suggestions. Finally, we acknowledge the excellent working conditions and financial support provided by the Max Planck Institute for Mathematics, the University of Toronto, and the Fields Institute.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,167 |
<?php
namespace Flower\View\Pane\Factory;
use Flower\View\Pane\PaneClass\PaneInterface;
/**
* Description of ListPaneFactory
*
* @author Tomoaki Kosugi <kosugi at kips.gr.jp>
*/
class ListPaneFactory extends PaneFactory
{
protected static $paneClass = 'Flower\View\Pane\PaneClass\ListPane';
public static function parseBeginEnd(PaneInterface $pane, array $config)
{
if (isset($config['begin'])) {
$pane->setBegin((string) $config['begin']);
} elseif(!isset($pane->tag) || empty($pane->tag)) {
$pane->setBegin('<!-- start pane -->');
} else {
$attributeString = self::parseAttributes($pane);
if (strlen($attributeString)) {
$pane->setBegin(sprintf('<%s%s>', $pane->tag, $attributeString));
}
else {
$pane->setBegin(sprintf('<%s>', $pane->tag));
}
}
if (isset($config['end'])) {
$pane->setEnd((string) $config['end']);
} elseif(! strlen($pane->tag)) {
$pane->setEnd('<!-- end pane -->');
} else {
$pane->setEnd('</' . $pane->tag . '>');
}
}
public static function parseWrapBeginEnd(PaneInterface $pane, array $config)
{
if (isset($config['wrapBegin'])) {
$pane->setWrapBegin((string) $config['wrapBegin']);
} elseif(!isset($pane->wrapTag) || empty($pane->wrapTag)) {
$pane->setWrapBegin('<!-- start wrap pane -->');
} else {
$attributes = $pane->getOption('wrap_attributes');
if (is_array($attributes)) {
$attributeString = self::attributesToAttributeString($attributes, $pane->getOption('wrap_attr_options'));
$pane->setWrapBegin(sprintf('<%s%s>', $pane->wrapTag, $attributeString));
} else {
$pane->setWrapBegin(sprintf('<%s>', $pane->wrapTag));
}
}
if (isset ($config['wrapEnd'])) {
$pane->setWrapEnd((string) $config['wrapEnd']);
} elseif (! strlen($pane->wrapTag)) {
$pane->setWrapEnd('<!-- end wrap pane -->');
} else {
$pane->setWrapEnd('</' . $pane->wrapTag . '>');
}
}
public static function parseContainerBeginEnd(PaneInterface $pane, array $config)
{
if (isset($config['containerBegin'])) {
$pane->setContainerBegin((string) $config['containerBegin']);
} elseif(!isset($pane->containerTag) || empty($pane->containerTag)) {
$pane->setContainerBegin('<!-- start container pane -->');
} else {
$attributes = $pane->getOption('container_attributes');
if (is_array($attributes)) {
$attributeString = self::attributesToAttributeString($attributes, $pane->getOption('container_attr_options'));
$pane->setContainerBegin(sprintf('<%s%s>', $pane->containerTag, $attributeString));
} else {
$pane->setContainerBegin(sprintf('<%s>', $pane->containerTag));
}
}
if (isset($config['containerEnd'])) {
$pane->setContainerEnd((string) $config['containerEnd']);
} elseif(!isset($pane->containerTag) || empty($pane->containerTag)) {
$pane->setContainerEnd('<!-- end container pane -->');
} else {
$pane->setContainerEnd('</' . $pane->containerTag . '>');
}
}
public static function treatment(PaneInterface $pane)
{
if (isset($pane->var) && ($pane->var !== array($pane, 'render'))) {
$pane->_var = $pane->var;
$pane->var = array($pane, 'render');
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,512 |
Historic environment advice assistant
Providing technical, research and logistical support to Historic Environment professionals working with heritage assets
Occupation summary
This occupation is found in the Cultural Heritage, Historic Environment, Engineering, Construction, Design, Planning, Local Government, Education and Tourism sectors. The broad purpose of the occupation is to provide technical, research and logistical support to Historic Environment professionals working with heritage assets in the planning and development process, and on the legal and policy frameworks for their protection.
Examples of heritage assets include historic buildings, places, landscapes, townscapes, monuments, the historic marine environment, archaeological sites and deposits of heritage interest, registered parks, gardens and battlefields. Legislation or policy frameworks may be international, national or local, and include (but are not limited to) the Planning (Listed Buildings and Conservation Areas) Act 1990, Ancient Monuments and Archaeological Areas Act 1979, National Planning Policy Framework, Protection of Wrecks Act 1973, Town and Country Planning (Environmental Impact Assessment) Regulations 2017, Ecclesiastical Exemption Order 2010. In their daily work, an employee in this occupation interacts with a range of clients, colleagues and stakeholders in the public, private and third sectors, within
organisations of any size. The Historic Environment Advice Assistant is typically office based but duties also include site visits, outdoor working and attendance at meetings with stakeholders, clients or colleagues. An employee in this occupation will be responsible for researching, investigating, analysing and reporting upon aspects of the historic environment in order to inform advice and recommendations on change, protection, maintenance, interpretation, conservation or restoration. Their work supports the evaluation of the significance of the historic environment, assessment of its condition, and its sensitivity to change. Where necessary, they are required to present arguments clearly and effectively based on their analysis. They also assist with the design and help monitor the implementation and compliance of programmes of work relating to heritage assets.
The Historic Environment Advice Assistant will typically be expected to work independently conducting research and preparing documentation and may be responsible for providing advice and recommendations to clients or the public without the supervision of senior colleagues, where appropriate to individual cases.
Typical job titles
Typical job titles include Heritage at Risk Projects Officer; Built Heritage Assistant; Assistant Conservation Officer; Assistant Heritage Consultant; Assistant Historic Environment Officer; Assistant Historic Buildings Officer; Associate Heritage Consultant; Assistant Archaeological Advisor; Assistant Historic Environment Record Officer; Heritage Assistant; Assistant Heritage Officer; Assistant Heritage Consultant; Assistant Conservation & Design Officer, Listing Officer, Designation Officer, Casework Officer.
While any entry requirements will be a matter for individual employers, typically an apprentice might be expected to have already achieved five 9-4 (previously A* to C) GCSEs on entry.
Occupation duties
Criteria for measuring performance
KSBs
Duty 1 Research and compile information on the historic environment through desk based or site-based investigations,
using site inspections, online resources and archive research,
e.g. using National Monuments Record, National Heritage List, Historic Environment Records etc.
Work must be accurate and meet industry standards and relevant professional body guidelines and codes of practice (those issued by the Chartered Institute of
Building (CIOB), Institute of Historic Building Conservation (IHBC), Chartered Institute for Archaeologists (CIfA)).
K1, K2, K3, K4, K5, K6, K7, K8, K9, K11, K12, K14
S3, S4, S5, S9, S10, S11, S15, S16
B1, B2, B3, B4, B5, B6, B7
Duty 2 Apply the relevant legislation and policy to historic environment projects, casework or applications, within broad but generally well-defined parameters, e.g. BS7913 (Guide to Conservation of Historic Buildings), Planning (Listed Buildings and Conservation Areas) Act 1990, Ancient Monuments and Archaeological Areas Act 1979, National Planning Policy Framework (NPPF), Historic England: Conservation Principles, Policies and Guidance for the sustainable management of the Historic Environment, DCMS: Principles of selection for listing buildings)
Work must be accurate, comply with relevant legislation and statutory principles, and meet industry standards and professional body guidelines and codes of practice (CIOB, IHBC, CIfA).
K1, K2, K3, K4, K5, K6, K7, K8, K9, K10, K11, K12, K13
S2, S3, S4, S6, S9, S14
Duty 3 Assess the condition of the historic environment within broad but generally well-defined parameters, based on understanding of principles of conservation in the historic environment. For example assessing the extent and rate of deterioration of part of the fabric of a listed building or an archaeological site.
Work must be accurate, comply with relevant legislation and statutory principles, and meet industry standards and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA).
K1, K2, K3, K4, K5, K6, K7, K8, K9, K10, K11, K12, K13, K14
S4, S5, S7, S8, S15, S16
Duty 4 Design, develop or determine the impact of applications and proposals for change to the historic environment e.g. alterations or repairs to a listed building or its demolition, or development-led changes to an archaeological site or Conservation Area.
S2, S4, S5, S6, S8, S9, S12
Duty 5 Identify risks to the historic environment, constraints and opportunities (such as regeneration, adaptive reuse).
For example the impact of development, risks of adverse weather conditions to historic fabric, or deterioration of historic sites through a lack of intervention/neglect.
S4, S5, S6, S7, S8, S9, S15, S16
Duty 6 Produce and evaluate documentation supporting the management of change of the historic environment. For example this could involve producing, assessing or updating conservation management plans, statements of significance, identifying where specialist skills may be required and commissioning specialist services. Work must be accurate, comply with relevant legislation and statutory principles, and meet industry standards and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA.
S2, S3, S4, S5, S6, S7, S8, S9, S10, S11
Duty 7 Interpret design documentation that relates to change in the historic environment, this may involve reviewing graphical material including building and engineering drawings. Work must be accurate, comply with relevant legislation and statutory principles, and meet industry standards and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA).
Duty 8 Provide technical advice, recommendations, or consultation responses related to the historic environment to stakeholders, within broad but generally well-defined parameters, such as providing technical advice on the protection, maintenance, interpretation, conservation or restoration of an archaeological site, listed building, or registered park or garden.
Work must be accurate, comply with relevant legislation and statutory principles, and meet industry standards and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA). Work must also meet client expectations and criteria includes customer feedback.
Duty 9 Monitor the implementation of investigation or work programmes and statutory compliance concerning the historic environment in well- defined circumstances, e.g. checking that conservation work has been carried out to an acceptable standard, monitoring planning conditions where they relate to Written Schemes of Investigation, monitoring grant awards, or checking listed building consent conditions have been met. Work must be accurate, comply with relevant legislation and statutory principles, and meet industry standards and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA). Work must also meet client expectations and criteria includes customer feedback.
S1, S2, S4, S5, S6, S7, S9, S12, S14, S15, S16
Duty 10 Demonstrate public benefit of the historic environment, presenting arguments clearly and effectively and conducting the public dissemination of information about the historic environment. For example demonstrating how a historic site or conservation area can contribute to a sense of civic pride and improve well- being, improve local employment and economic growth. Work must be accurate, comply with relevant legislation and statutory principles, and meet industry standards and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA). Work must also meet client expectations and criteria includes customer feedback.
K1, K2, K3, K4, K5, K6, K7, K8, K9, K12
S1, S4, S10, S13
Duty 11 Raise awareness, appreciation and understanding of the historic environment, and promote good conservation practice, through an understanding of historic environment principles, policies and guidance Work must be accurate and meet industry standards and statutory principles, and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA). Work must also meet client expectations and criteria includes customer feedback.
S1, S4, S13
Duty 12 Assist with the design of technical briefs and specifications for carrying out or procuring a range of historic environment products and services Work must be accurate and meet industry standards and statutory principles, and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA). Work must also meet client expectations and criteria includes customer feedback.
Duty 13 Manage and catalogue data recovered from research or investigation on the historic environment, this includes use of GIS and map-based database systems, such as the National Heritage List, Historic Environment Records, National Monuments Record etc. Work must be accurate and meet industry standards and statutory principles, and relevant professional body guidelines and codes of practice (CIOB, IHBC, CIfA).
K1, K2, K3, K4, K7, K8, K9, K10, K11, K12
S3, S4, S5, S14
K1: How to respond to client or public requests and organisational requirements e.g. requests for advice from owners of Listing Buildings, or requests for record information about Listed Buildings from colleagues
K2: Knowledge of stakeholder communication methods and strategies and how to maintain honest and constructive relationships.
K3: The requirements for projects, including timescales, deadlines, cost implications, and identifying milestones/targets.
K4: Heritage policies, frameworks, strategies, and best practice Standards for historic building conservation and archaeological work e.g. National Planning Policy Framework (NPPF), Historic England: Conservation Principles, Policies and Guidance for the sustainable management of the Historic Environment, DCMS: Principles of selection for listing buildings
K5: Where and how to find the relevant statutory legislation and other guidance concerning change in the historic environment, e.g. Planning (Listed Buildings and Conservation Areas) Act 1990, Ancient Monuments and Archaeological Areas Act 1979, National Planning Policy Framework, Protection of Wrecks Act 1973, Town and Country Planning (Environmental Impact Assessment) Regulations 2017, Ecclesiastical Exemption Order 2010
K6: The Historic Environment conservation principles (BS7913) and other published criteria for assessment and design.
K7: The fundamentals of archaeological, landscape or architectural history and building practice and chronology, including a broad understanding and recognition of archaeological site or building types, styles, technologies, materials and periods.
K8: The wider context of the historic environment, and the roles and responsibilities of statutory authorities, heritage organisations of all types and specialists, e.g. public, private and third sector organisations, subject specialist networks
K9: Documentation used in the identification, management, design or recording of the Historic Environment, e.g. assessments of significance, Local Development Plans, listing and designation statements, desk-based assessments, Historic Environment Records.
K10: Compliance processes for the historic environment, including heritage at risk and enforcement.
K11: Their employer's health and safety policy and procedures and operational procedures and how those relate to industry standards, and the fundamentals of relevant Health and Safety legislation and construction site Health & Safety. How to recognise and report risks.
K12: Their role in the context of the project which they are working, what is required of them, and the implications of the project on the wider context of the historic environment. The limits of their own understanding, abilities and responsibilities, and how to practice within them. The ethical requirements of the relevant professional body.
K13: Knowledge of learning and self-development opportunities within the sector and how to develop a personal action plan.
K14: How to identify archaeological sites or building types, styles, technologies, materials and periods in practice.
S1: Work as part of a team and communicate effectively with colleagues, clients and stakeholders
S2: Coordinate and administer casework and projects, manage projects for clients or organisations including using systems and processes specific to the historic environment sector.
S3: Demonstrate time management skills, prioritise workload, work under pressure and be able to meet critical deadlines.
S4: Write reports, specifications and briefs for projects that are complex and non-routine but well defined, e.g. Written Schemes of Investigation, Conservation Management Plans, listing and designation advice, desk-based assessments.
S5: Interpret technical information and documents on the historic environment, e.g architectural plans, design and access statements, listed building consent orders.
S6: Link, contextualise and apply legislation, policy, standards and guidance to projects, e.g. Planning (Listed Buildings and Conservation Areas) Act 1990, Ancient Monuments and Archaeological Areas Act 1979, National Planning Policy Framework, Protection of Wrecks Act 1973, Town and Country Planning (Environmental Impact Assessment) Regulations 2017, Ecclesiastical Exemption Order 2010
S7: Assess and describe the condition of the historic environment, e.g conducting evaluation visits, desk-based assessment, assessments of significance
S8: Problem solve and evaluate proposals for change and gauge appropriateness of proposed actions
S9: Apply relevant historic environment sector standards to check and review work
S10: Identify, compile and process data on the historic environment
S11: Review and maintain records on the historic environment in accordance with relevant standards
S12: Exercise appropriate judgement and decision making, escalating to/involving others when dealing with complex queries or sensitive cases
S13: Recognise the potential for work in the historic environment to deliver public benefit, identify opportunities for research and to deliver new knowledge for society
S14: Be responsible for mapping and working to data standards, carrying out information reviews or technical investigation on the historic environment.
S15: Work in a variety of outdoor and indoor site types safely, recognise and report risks in order to reduce the risk of incidents
S16: Identify and use of range of methods and techniques to identify archaeological sites or building types, styles, technologies, materials and periods in practice.
B1: Take reasonable care for the health and safety of themselves and of others who may be affected by their acts or omissions at work
B2: focus and pay attention to detailB3: Ability to problem-solve and negotiate
B3: Ability to problem-solve and negotiate
B4: Commit to quality and their continuous professional development
B5: Work effectively individually and as part of a team
B6: Be approachable and able to communicate with all levels of their own and other organisations, as well as the general public, in workplace settings, as well as during site visits and stakeholder meetings.
B7: Be sensitive to and aware of the significance of the historic environment, and the needs of its stakeholders, being conscious of integrity, honesty and professional ethical requirements
English and Maths qualifications
Apprentices without level 2 English and maths will need to achieve this level prior to taking the End-Point Assessment. For those with an education, health and care plan or a legacy statement, the apprenticeship's English and maths minimum requirement is Entry Level 3. A British Sign Language (BSL) qualification is an alternative to the English qualification for those whose primary language is BSL.
Chartered Institute for Archaeologists / Practitioner
Chartered Institute for Building / Registered
Institute of Historic Building Conservation / Affiliate
Occupational Level: 4
Duration (months): 24
This standard will be reviewed after three years.
Approved for delivery: 11 April 2019
Route: Creative and design
Trailblazer contact(s): philip.Pollard@HistoricEngland.org.uk
Employers involved in creating the standard: Historic England, English Heritage, National Trust, Mott MacDonald, Atkins, Chris Blandord Associates, Warwickshire County Council, National Parks (Northumberland), National Parks (Peak), Centre for Applied Archaeology University of Salford, Lichfields, Savills, Cullen Conservation, Alan Baxter, Arup
Historic environment advice assistant assessment plan
1 11/04/2019 Assessment plan and funding band first published | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,450 |
\section{Introduction}
Confining energetic particles, especially alpha particles born in nuclear fusion reactions, is of key importance for magnetic confinement fusion reactors. In configurations where axisymmetry is not present, either tokamaks with non-axisymmetric perturbations, or stellarators, some particle orbits will have a non-zero bounce-averaged radial drift causing them to leave the confined region, sometimes very quickly. These promptly lost particles can cause significant damage to plasma facing surfaces and reduce the lifetime of the plasma wall \cite{mau2008divertor}. It will never be possible to confine all alpha particles in a fusion device, but reducing the losses, especially prompt losses, is crucial for the longevity of the device. This paper will show collisional energetic particle transport results from various stellarator configurations at the reactor scale with the goal of identifying the properties of configurations with good confinement.
Several metrics have been developed for neo-classical confinement in 3D systems \cite{grieger1992physics}. Some configurations possess a symmetry in the magnetic field strength, $|B|$, and therefore are isomorphic to axisymmetric systems. These configurations are called quasi-symmetric, because they possess a symmetry in $|B|$ similar to an axisymmetric system \cite{reiman1999physics}. This paper includes configurations of two quasi-symmetric types, quasi-axisymmetric (QA) where $|B|$ contours connect toroidally, and quasi-helical (QH) where $|B|$ contours connect helically \cite{Nuhrenberg_PLA_1988}. In all the configurations exact quasisymmetry is not present, but rather an approximate symmetry exists. The deviation from a strict symmetry is referred to in this paper as quasisymmetric deviation. Mathematically this is obtained by transforming coordinates into the Boozer coordinate system, and then calculating the energy in the non-symmetric modes.
It has been shown experimentally that quasi-helical configurations improve neoclassical confinement \cite{canik2007experimental}. Additionally, many numerical explorations of quasisymmetric configurations of all types exist as well. Specifically, recent results indicate that low quasi-symmetric deviation can help alpha confinement in both quasiaxisymmetric \cite{henneberg2019improving} and quasi-helically symmetric configurations \cite{bader2019stellarator}.
There are alternative methods for improving neoclassical confinement in the absence of quasisymmetry exist. A broader class of configurations are omnigenous, in which the second adiabatic invariant, $J_\parallel = \oint v_\parallel dl$, is constant on a flux surface. A consequence of this optimization is that all the maxima and minima of a field line on a flux surface are the same. Configurations that approximate omnigeneity are called quasi-omnigenous (QO). If, in addition, flux surfaces close poloidally, the configuration is quasi-isodynamic. A consequence of this optimization is that drifts are purely poloidal, and bootstrap and Pfirsch-Schl\"uter currents vanish. This optimization was used to produce W7-X \cite{nuhrenberg2010development}. Confinement of energetic particles in quasi-isodynamic configurations is expected to improve at high pressure when the alignment between $J_\parallel$ and flux surfaces improves \cite{lotz1992collisionless}.
An even less restrictive optimization for improved confinement is described as $\sigma$-optimization \cite{mynick1983improved}. In LHD it is possible to achieve an equilibria where $\sigma = 1$, where for a given flux surface the minima of each field line are equivalent, but the maxima do not. This is achieved in LHD by shifting the axis inward, creating the "inward shifted" configuration \cite{murakami2002neoclassical}. In this optimization, collisionless drift orbits are not fully confined, but the coefficient of neoclassical transport drops significantly.
Various metrics have been used to quantify the degree of neoclassical transport. One metric, $\epsilon_\mathrm{eff}$, is the coefficient of the neoclassical diffusion in the low-collisionality ($\sim 1/\nu$) regime has been regularly used for stellarator optimization \cite{nemov1999evaluation}. However, recent results indicate that there is little correlation between $\epsilon_\mathrm{eff}$ and good energetic particle confinement \cite{bader2019stellarator}. However, a different metric $\Gamma_c$ \cite{Nemov_PoP_2005, Nemov_PoP_2008}, that seeks explicitly to align contours of $J_\parallel$ with flux surfaces similar to the omnigeneity constraint, has been shown to correlate better with energetic particle confinement. This metric has been used to optimize quasi-helically symmetric configurations with good energetic particle confinement \cite{bader2019stellarator, bader2020advancing}.
In previous publications stellarators were compared only between variations of similar classes (QA, QH, QO). Comparisons between stellarators of different classes, have often been hampered by different choices of magnetic field strength, size, and profiles of density and temperature. Some calculations include collisions where others do not. Comparisons between published results is therefore very difficult. This paper attempts to rectify the situation by providing consistent scalings across a broad class of configurations and then comparing the energetic particle confinements both collisionlessly and with collisions.
The layout of this paper is as follows. In Section \ref{sec:configs}, we will briefly describe the configurations used in this paper. Section \ref{sec:scaling} will explain how the reactor scale configurations were constructed. Section \ref{sec:coll} will show results from both collisional and collisionless calculations of alpha particles. Section \ref{sec:metrics} compares the alpha particle losses for the metrics of interest in each configuration. Section \ref{sec:disc} will discuss the results and describe the limitations of the current work. Section \ref{sec:conc} will conclude the paper and provide areas for future research.
\section{Configurations}
\label{sec:configs}
This paper considers three quasi-helically symmetric configurations, three quasi-axisymmetric configurations, a W7-X like configuration, and two LHD-like configurations. An ITER configuration is included for comparison. A table of all the configurations and their relevant properties has been included in table \ref{tab:configs}.
\begin{table}
\caption{\label{jlab1}A list of configurations along with relevant properties}
\footnotesize
\begin{tabular}{@{}lllll}
\br
Name&Type&Periods&Aspect ratio&$\beta$\\
\mr
Wistell-A & QH & 4 & 6.7 & Vacuum\\
Wistell-B & QH & 5 & 6.6 & Vacuum\\
Ku5 & QH & 4 & 10.0 & 10.0\%\\
ARIES-CS & QA & 3 & 4.5 & 4.0\%\\
NCSX & QA & 3 & 4.4 & 4.3\%\\
Simsopt & QA & 2 & 6.0 & Vacuum\\
W7-X & QI & 5 & 10.5 & 4.4\%\\
LHD st & Torsotron & 10 & 6.5 & Vacuum\\
LHD in. & Torsotron & 10 & 6.2 &Vacuum\\
ITER&Tokamak&N/A&2.5&2.2\%\\
\br
\label{tab:configs}
\end{tabular}\\
\end{table}
\normalsize
The quasi-helically symmetric configurations are: the "Wistell-A" configuration which has been described in a previous publication \cite{bader2020advancing}; the "Wistell-B" configuration, a five-field period vacuum configuration optimized with the \texttt{ROSE} code \cite{Drevlak_NF_2018} explicitly for quasisymmetry and $\Gamma_c$; "Ku4" a four field period configuration from \cite{ku2010new} that was optimized for quasisymmetry at high normalized pressure, $\beta$. The three quasi-axisymmetric configurations are comprised of the NCSX (specifically "li383") \cite{koniges2003magnetic, mynick2002exploration} and ARIES-CS (specifically: "n3are") \cite{ku2008physics, mynick2006improving} configurations. Also included amongst quasiaxisymmetric configurations is a more recent vacuum configuration. called Simsopt, which was optimized solely for quasisymmetry at the $s=0.5$ surface using the \texttt{SIMSOPT} optimizer \cite{simsoptcode}. Here and throughout the paper, $s$ represents the normalized toroidal flux, $\psi/\psi_{\mathrm{edge}}$. The W7-X like configuration is a high-mirror configuration designed for improved energetic particle confinement \cite{grieger1991physics}, with coefficients given by Table IV in \cite{nuhrenberg1996global}. The two LHD \cite{murakami2002neoclassical} configurations are vacuum configurations, with one in the standard (outward) configuration and the other in an inward shifted configuration which is known to have improved confinement properties \cite{mynick1983improved}. Finally the ITER configuration is a near-axisymmetric configuration, although this equilibrium includes coil ripple, blanket modules and ferritic inserts \cite{tobita2003reduction}.
\section{Scaling to Reactor size}
\label{sec:scaling}
In order to properly scale configurations to each other it is necessary to adjust to a benchmark size. The ARIES-CS parameters are used for the scaling, representing a fairly compact reactor size. There are two possible ways to scale the configurations. One option is to scale the configurations to have the same volume (444 m$^3$) the other is to scale the minor radii to the same value (1.7m). For this paper we only show results using the volume scaling, although the main conclusions do not change when the configurations are scaled to have equivalent minor radii.
All configurations are represented by \texttt{VMEC} equilibria \cite{Hirshman_POF_1983}, and the size scaling is accomplished by adjusting the boundary coefficients such that all configurations have the same volume. The magnetic field strengths are made equivalent by ensuring the volume averaged magnetic field is equivalent across all configurations (5.86 T). For non-vacuum configurations, the rotational transform are adjusted for each configuration by using $I \propto RB$, where $I$ is the total current in the plasma. The normalized pressure, $\beta$ is similarly kept constant through the scaling procedure. The boundary flux surfaces for the configurations at the $\phi$=0 plane, often referred to as the "bean" or "crescent", are plotted in figure \ref{fig:fs}
\begin{figure}
\centering
\begin{subfigure}{0.25\textwidth}
\centering
\includegraphics[width=\linewidth]{fs_iaea1.png}
\end{subfigure}%
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=\linewidth]{fs_iaea2.png}
\end{subfigure}%
\begin{subfigure}{0.43\textwidth}
\centering
\includegraphics[width=\linewidth]{fs_iaea3.png}
\end{subfigure}%
\caption{Boundary flux surfaces for the scaled configurations are shown. Left: Quasihelically-symmetric configurations. Center: Quasiaxisymmetric configurations (including ITER) Right: non-quasisymmetric configurations}
\label{fig:fs}
\end{figure}
For all configurations, with three exceptions, the scaling is accomplished by starting from an idealized fixed boundary equilibrium and scaling the coefficients. These fixed boundary equilibria are usually generated through optimization and do not include effects from finite coils. One exception is the ITER equilibrium which includes coil ripple and the effect from ferritic inserts and blanket modules. The scaled ITER equilibrium is a direct replica of the unscaled ITER equilibrium, but with slightly larger volume and higher field. There was no attempt to recalculate the effects of blanket modules and ferritic inserts on the larger size. The other exceptions are the two LHD equilibria which represent two configurations very similar to those generated in the actual LHD device. In these cases, the coils used to generate the equilibria were adjusted and new free-boundary equilibria were generated from the enlarged coils. Results from these free-boundary equilibria are presented in this paper. A second set of LHD calculations were undertaken with a scaled fixed-boundary equilibrium and the differences were not noticeable, and are not included here.
In order to perform the collisional calculations, it is necessary to define the temperature and density profiles. These profiles determine the initial launch points for collisional calculations as well as the slowing down behavior of the alpha particles. The density profile chosen for these simulations is mostly flat with $n = n_0 \left(1-s^5\right)$, the temperature profile is more peaked with $T = T_0\left(1-s\right)$. These profiles are roughly consistent with those chosen for the ARIES-CS studies \cite{ku2008physics}. In the previous equations, $s$ represents the normalized toroidal flux, $\psi/\psi_{edge}$. The values of the core temperatures, $T_0$ and $n_0$ are approximately equivalent to those of ARIES-CS: $n_{e,0} = 4.8 \times 10^{20}$ and $n_{D,0} = n_{T,0} = 2.25 \times 10^{20}$, with $T_{e,0} = T_{i,0} = 11.5$ keV. The difference between $n_D + n_T$ and $n_e$ arises from a flat profile of $Z_\mathrm{eff} = 1.13$ as in the ARIES-CS equilibrium. However, collisions with impurity ions are not included in these calculations. Once the temperature profiles are chosen, the reaction profile is determined. The temperature, density and reaction profiles are shown in figure \ref{fig:profiles}. The same reaction profile is used for each equilibrium, and is estimated as
{\setlength{\mathindent}{1cm}
\begin{equation}
R \frac{dV}{ds} = n_D n_T \langle \sigma v \rangle \frac{dV}{ds};\; \langle \sigma v \rangle = 3.6\times10^{-18} T^{-2/3} \exp{\left(-19.94 * T^{-1/3}\right)} \mathrm{m}^3/\mathrm{sec}
\end{equation}}
where $n_D$ and $n_T$ are the deuterium and tritium concentrations, $T$ is the temperature in keV and $dV/ds$ is the derivative of the volume with respect to normalized toroidal flux $s$ for the ARIES-CS equilibrium. Even though the reaction profile varies slightly from equilibrium to equilibrium due to variations in $dV/ds$, the same fusion reaction profile is maintained in order to keep consistent particle launch profiles across equilibria.
\begin{figure}
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{ntprofile_iaea.png}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{reactivity_iaea.png}
\end{subfigure}%
\caption{Left: Temperature (red) and density (blue) profiles as a function of normalized toroidal flux $s$. Right: The derived reaction rate given as a function of temperature and density for the ARIES-CS equilibrium.}
\label{fig:profiles}
\end{figure}
Since the configurations vary in pressure from vacuum configurations to normalized pressures of 10\%, it is impossible to choose profiles consistent across configurations that also match the pressure profiles from each configuration. The main goal is to determine what magnetic configuration properties affect alpha particle confinement rather than to do self-consistent studies of each of the configurations. Therefore, the same temperature and density profiles are used in all configurations for alpha particle confinement calculations even though there is no self-consistency with the plasma pressure used in the equilibrium.
\section{Alpha Particle Losses}
\label{sec:coll}
Particles are sourced by first choosing a radial location such that the distribution matches the reaction profile given in figure \ref{fig:profiles}. Next a random location on the surface and the velocity pitch angle is chosen in the same manner as described in \cite{bader2019stellarator}. The guiding centers of the particles are followed using an Adams-Bashford integration scheme and can under go both slowing down and pitch angle scattering. The \texttt{ANTS} code is used for all particle following calculations \cite{Drevlak_NF_2014}. If a particle passes beyond the penultimate flux surface at any point in time it is considered lost. If the particle's energy is the same as the background thermal particles it is considered confined and is no longer followed.
The results from the collisional calculation are shown in Figure \ref{fig:eloss}. The line style indicates the configuration type, with solid lines indicating QH, dotted lines QA, dashed lines for the LHD-like configurations and both ITER and W7-X use dashed-dotted lines. To help the reader, throughout the paper consistent colors and linestyles (where possible) for each configuration are used. Among the quasisymmetric configuration, the QHs strongly outperform the QAs with the exception of the Simsopt configuration which performs as well as the best QHs. The three best performing configurations shown (outside of ITER) are the Ku5 configuration (QH), the Wistell-B configuration (QH) and the Simsopt configuration (QA). W7-X performs about equivalently to both the WISTELL-A configuration and the inward shifted LHD configuration. This behavior will be examined in depth later. Note that due to differences in machine size, magnetic field, and particle sourcing, the results shown here may differ from previously published results on energetic particle confinement.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{loss_energy_iaea.png}
\caption{Energy loss from alpha particles as a function of time for all configurations. }
\label{fig:eloss}
\end{figure}
In addition to the collisional calculation, calculations without collisions are also presented in Figure \ref{fig:ploss}. For these calculations particles were started on a specific flux surface, in this case $s$=0.3, representing a surface just outside the midradius is chosen. Particles are launched on this surface and followed until they are lost or 200 ms have elapsed, corresponding to several ($\sim3$) slowing down times. Figure \ref{fig:ploss} shows the particle loss versus time rather than the energy loss, but because no collisions are included, all lost particles have the full energy. Collisionless calculations were previously used to distinguish between configurations \cite{bader2019stellarator}, and they are very useful to highlight the specific loss behaviors of the configurations, which will be examined below. Note that the addition of collisions tend to enhance energetic particle losses due to pitch angle scattering onto lost orbits.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{loss_s3_iaea.png}
\caption{Particle loss as a function of time for all configurations for particles born on the $s$=0.3 surface.}
\label{fig:ploss}
\end{figure}
Before a more detailed look at the configurations, it will be useful to distinguish between the different types of losses seen. Some particles are born on lost orbits and leave the confined region at almost the full energy values even in the collisional calculation. Particles born on the outer regions of the plasma are likely to be lost in this manner. In fact all the losses from ITER are particles born near the edge that are promptly lost. We will refer to these particles as ``prompt" losses.
There is a second class of lost particles that undergo many orbits before being lost. Sometimes this is the result from diffusive properties, especially pitch-angle scattering which becomes increasingly important at low energies. However, these slow losses can also occur in collisionless calculations discussed below. As such we will refer to all losses on extended time scales as "stochastic" losses, following the convention in \cite{albert2020accelerated}.
The exact boundary between prompt and stochastic losses is not clear in all configurations, but it is often easy to see the distinction in some of the configurations. The W7-X collisionless losses at, say, $s = 0.3$ are particularly clear. The W7-X configuration loses about 2\% of launched particles born at $s=0.3$ before 0.2 ms. There are almost no additional losses until about 1 ms when additional stochastic losses begin accumulating again. Many of the particles are lost stochastically, however the precise behavior is important. Slow stochastic losses are less problematic because particles will be able to deposit most of their energy. The same distinction between prompt and stochastic losses in W7-X exists with collisionless losses on other flux surfaces (not shown). It also is visible in the collisional losses, however, with collisional losses, diffusive behavior causes there to be some particles losses between 0.2 and 1 ms. A more detailed discussion about prompt versus stochastic losses is in sections \ref{ss:lhd} and \ref{ss:qh}.
\section{Alpha Particle Loss Metrics}\
\label{sec:metrics}
We consider two metrics for alpha particle losses, quasisymmetry and $\Gamma_c$. A given configuration is quasisymmetric if the variation of $|B|$ along a field line is the same for all field lines on a flux surface \cite{helander2014theory}. Quasisymmetry can be determined by Fourier decomposing $|B|$ on a flux surface in the straight field line coordinate system known as Boozer coordinates \cite{Boozer_PoF_1982}. If the only modes present are ones where the ratio of the toroidal mode $n$ to the poloidal mode $m$ is constant, the configuration is quasisymmetric. For quasiaxisymmetric equilibria, $n/m = 0$ and a perfectly quasiaxisymmetric equilibrium will only have modes with $n=0$. For quasihelically-symmetric equilibria, the ratio $n/m$ is usually equal to the number of field periods (modulo a sign). So for a perfectly quasihelically-symmetric equilibrium with four periods, the only modes present are ones where $n/m$ = 4. A third symmetry, quasipoloidal symmetry, where only modes with $m=0$ are present, is not considered in this paper.
Excepting precisely axisymmetric configurations, it is conjectured that perfect quasisymmetry can only be achieved on a single flux surface \cite{garren1991existence}, and a metric is needed to describe the deviation from perfect quasisymmetry. The metric used in this paper is calculated by first Fourier decomposing the two dimensional flux surface in Boozer coordinates, and then summing the magnetic energy in all non-symmetric modes normalized to the $m=0, n=0$ mode, which is representative of the background field strength. That is,
\begin{equation}
Q_{qs}(s) = \frac{1}{B_{0,0}(s)}\left(\sum_{m/n \neq C_{qs}} B_{m,n}^2(s) \right)^{1/2}
\end{equation}
where $C_{qs}$ represents the target for quasisymmetry, 0 for quasiaxisymmetry and the number of field periods for quasihelical symmetry. Lower values of $Q_{qs}$ indicate better quasisymmetry.
Figure \ref{fig:qsqx} shows the results of $Q_{qs}$ for quasiaxisymmetric configurations (left) and quasihelically symmetric configurations (right) as a function of flux surface. There is clear separation among the quasiaxisymmetric configurations. At all $s$ values, the ARIES-CS configuration is the least quasisymmetric and the Simsopt configuration is the most quasisymmetric. The story is less clear for the quasihelical configurations. The Ku5 configuration is the most quasisymmetric in the core and the least quasisymmetric in the edge. Overall the Wistell-B configuration has the best average quasisymmetry.
\begin{figure}
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{qsqa_iaea.png}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{qsqh_iaea.png}
\end{subfigure}%
\caption{The deviation from quasisymmetry as a function of normalized toroidal flux, $s$ for quasiaxisymmetric configurations (left) and quasihelically symmetric configurations (right)}
\label{fig:qsqx}
\end{figure}
The second metric, $\Gamma_c$ was introduced by Nemov \cite{Nemov_PoP_2008} (eqs 61, 50, and 36) as a measure of the energetic ion confinement properties and is given by,
\begin{equation}
\Gamma_c = \frac{\pi}{\sqrt{8}} \lim_{L_s \rightarrow \infty} \left( \int_0^{L_s} \frac{ds}{B}\right)^{-1}
\int_1^{B_\mathrm{max}/B_\mathrm{min}} db' \sum_\mathrm{well_j} \gamma_c^2 \frac{v \tau_{b,j}}{4 B_\mathrm{min}
b'^{2}};\;\gamma_c = \frac{2}{\pi} \mathrm{arctan}\frac{v_r}{v_\theta}
\end{equation}
Here, $v_r$ and $v_\theta$ are the bounce average radial and poloidal drifts respectively; $v$ is the particle velocity; $\tau_b$ is the bounce time, $B_\mathrm{max}$ and $B_\mathrm{min}$ are the maximum and minimum field strength on a flux surface or suitably long field line; $b'$ represents a normalized field strength, here equivalent to $|B|/B_\mathrm{min}$ and $L_s$ is the length along a field line. The summation is over every well along a field line, where the boundaries of the wells are themselves a function of the integrating variable, $b'$. When $\Gamma_c$ is small, contours of $J_\parallel$ align with flux surfaces, and the bounce average radial drift goes to zero. More information about $\Gamma_c$ and its use for stellarator optimization can be found in \cite{bader2019stellarator}. Unlike quasisymmetry, the $\Gamma_c$ metric can be calculated for all stellarator configurations. All calculations for both $\Gamma_c$ and quasisymmetry were carried out using the \texttt{ROSE} code. Due to an unresolved difficulty with handling single field period equilibria, $\Gamma_c$ for the ITER calculation is unavailable for this paper.
Nine configurations are represented in figure \ref{fig:gcvss} in the two plots showing $\Gamma_c$ as a function of normalized toroidal flux $s$. The six quasisymmetric configurations are plotted on the left, and the three non-quasisymmetric configurations are plotted on the right. Looking at the non-quasisymmetric configurations first, there is a clear distinction between the optimized configuration W7-X and the two LHD configurations. There is also a clear improvement between the LHD inward shifted configuration compared to the outward shifted configuration. However, the LHD inward shifted configuration has roughly the same magnitude of $\Gamma_c$ as the worst of the quasisymmetric configurations NCSX (note the difference in y-axis scale).
\begin{figure}
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{gcqs_iaea.png}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{gcnonqs_iaea.png}
\end{subfigure}%
\caption{$\Gamma_c$ as a function of normalized toroidal flux, $s$ for quasisymmetric configurations (left) and non-quasisymmetric configurations (right)}
\label{fig:gcvss}
\end{figure}
The quasisymmetric configurations also show considerable spread in the $\Gamma_c$ metric. Once again there is clear separation among the three quasiaxisymmetric configurations. The best performing case is the Simsopt equilibrium. Contrary to the quasisymmetry result, ARIES-CS outperforms NCSX with regard to the $\Gamma_c$ metric. This behavior is not surprising since the ARIES-CS optimization explicitly degraded quasisymmetry in order to improve energetic particle confinement \cite{mynick2006improving}. The particle loss results shown in figures \ref{fig:eloss} and \ref{fig:ploss} indicate that this optimization was successful.
For the three quasihelically symmetric configurations the Wistell-A and Wistell-B configurations have almost identical values of $\Gamma_c$ (these are similar in scale to the W7-X value). The Ku5 configuration has a larger value in the core, but a lower value in the outer half of the plasma. Since the Ku5 and Simsopt configurations represent the best performing configurations, it appears that the edge values in the outer half may be more important. The importance of the quasisymmetric values on the outer half of the plasma has already been discussed with respect to optimizations of quasisymmetry \cite{henneberg2019improving} and the results presented here indicate that the values of $\Gamma_c$ in the outer half of the plasma may be more closely related to energetic particle confinement as well.
Figure \ref{fig:scatter} shows the total energy loss for each configuration plotted against the value of a parameter of interest evaluated at $s$=0.6. The deviation from quasisymmetry (for the QS configurations) is plotted on the left hand plot and $\Gamma_c$ is in the right hand plot. A correlation between alpha energy confinement and $\Gamma_c$ is clear from the right hand plot. Although a perfect correlation does not exist, it is clear that the best/worst performing configurations also have the best/worst performance with regard to this metric. For quasisymmetry the correlation is weaker. While the Simsopt and Wistell-B configuration both perform very well on this metric, the Ku5 configuration performs worse despite having the best overall energetic particle confinement.
\begin{figure}
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{scatter_qs_iaea.png}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{scatter_gc_iaea.png}
\end{subfigure}%
\caption{Values of total collisional energy lost as a function of quasisymmetry on the $s$=0.6 surface (left) and $\Gamma_c$ on the $s$=0.6 surface (right)}
\label{fig:scatter}
\end{figure}
Among the QA configurations the best performing configuration is the Simsopt QA, which also performs the best on both metrics, despite only optimizing for quasisymmetry. As noted above, ARIES-CS performs worse in quasisymmetry but better in both $\Gamma_c$ and energetic particle confinement.
The QH configurations include two stellarator configurations with excellent confinement, Ku5 and Wistell-B. For the QH configurations, Wistell-A performs worse in the quasisymmetry metric but approximately as well in $\Gamma_c$. Both Wistell-A and Wistell-B were optimized including both quasisymmetry and $\Gamma_c$ in the target function. The Ku5 configuration only optimized for quasisymmetry, but despite this has the lowest $\Gamma_c$. There is a caveat to the performances of the two high performing QH configurations. In both the Wistell-B and Ku4 configurations, strong indentations in the plasma boundary make designing coils extremely challenging (see for example figure 13 in \cite{ku2010new}). However, coils that reproduce the energetic particle properties have already been designed for the Wistell-A configuration \cite{bader2020advancing}.
The W7-X configuration was specifically designed for good energetic particle transport \cite{grieger1991physics} and indeed it outperforms ARIES-CS and is on par with the Wistell-A configuration. Interestingly, the W7-X and Wistell-A configuration also have almost the same minimum value for $\Gamma_c$ so the points are very close in Figure \ref{fig:scatter}b.
The inward shifted LHD-like configuration has properties that deserve some attention. The overall losses for this configuration are on par with both W7-X and Wistell-A. Even more interesting is that this configuration does exceedingly well at confining prompt losses. This good performance in the collisional results appears despite not performing particularly well on the $\Gamma_c$ metric. Figure \ref{fig:elosshist} illustrates this by plotting a histogram of the number of lost particles in the collisional calculation against the energy at which they are lost. Prompt losses are on the far right of the graph. These prompt losses are lowest for the LHD inward shifted configuration compared to both Wistell-A and W7-X. The LHD losses reach their maximum between 2.5 and 3.0 MeV after which they fall to a very low level. The Wistell-A and W7-X, in contrast have fewer particles lost between 2.5 and 3.0 MeV but considerably more particles lost at energies under 1 MeV. These configurations will be examined closer in sections \ref{ss:lhd} and \ref{ss:qh}.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{eloss_histogram_iaea.png}
\caption{Histogram of number of particles lost (per 10k) as a function of the loss energy for Wistell-A (black), W7-X (magenta dashed) and LHD-inward shifted (green dashed). Prompt losses at 3.5 MeV are at the far right side of the graph.}
\label{fig:elosshist}
\end{figure}
\section{Discussion}
\label{sec:disc}
\subsection{Optimization}
One salient feature of the configuration scan is that two of the configurations with lowest achieved values of $\Gamma_c$ were not actually optimized for $\Gamma_c$ but rather for quasisymmetry only. These are the Ku5 and Simsopt configurations. Since perfect quasisymmetry will confine all particles, and a perfectly quasisymmetric configuration will have $\Gamma_c$ = 0, an optimization for quasisymmetry will almost always improve $\Gamma_c$ as well. The cases where this does not occur, such as one of the configurations presented in \cite{bader2019stellarator} are fairly uncommon. In fact, in several steps of the Wistell-B optimization it was found that the best improvement on both metrics, quasisymmetry and $\Gamma_c$ was obtained when the $\Gamma_c$ optimization was turned off in the optimizer.
The reverse is not true. Optimization for $\Gamma_c$ alone will almost never improve quasisymmetry. There are many pathways to improving $\Gamma_c$ that do not include quasisymmetry, for example, the optimizations that lead to W7-X.
Another important point of consideration is that many of the configurations were designed with additional metrics included. Both the NCSX and ARIES-CS equilibria placed strong emphasis on stability properties at finite $\beta$. Among other things, this generates a strong crescent shape at the $\phi = 0$ plane. In contrast the Simsopt QA is a vacuum configuration without a vacuum magnetic well and no attempt to provide stability at high pressure. The Simsopt configuration should be viewed as what is possible if you attempt to make the most quasiaxisymmetric configuration possible perhaps in opposition to other desired or even necessary properties. As always, significant effort is needed to weigh different optimizations considerations together to produce the ideal configuration for an experiment or reactor.
A similar story exists in the QH configurations. While Wistell-A and Wistell-B are both vacuum configurations, Wistell-A has a vacuum magnetic well and Wistell-B does not. Furthermore, attempts to generate coils to reproduce the Wistell-A configuration were successful, while attempts to produce coils for Wistell-B or Ku5 have not been successful to date. Of course this does not mean that it is impossible to find coils for these configurations, just that it is comparatively easier to design coils for Wistell-A. Despite having entirely different optimization schemes, Wistell-B and Ku5 have similar features. Specifically, both have a strong indentation in the teardrop shape that is very difficult to reproduce with coils. Future work that incorporates coil buildability should examine whether this feature is necessary for good confinement or not.
Finally we note that several of the configurations, namely the LHD configurations are actually built machines. It is much easier to design a configuration with good parameters than to actually build one. It is possible that the performance of the other configurations would degrade due to accumulated errors in the construction process. Efforts to optimize taking into account manufacturing errors are being undertaken by others \cite{lobsien2020physics} and will not be discussed further here.
\subsection{LHD Inward Shifted}
\label{ss:lhd}
A surprising result from the configuration scan was the performance achieved by the LHD configuration. While it has been theorized and experimentally verified that the confinement improves in LHD with inward-shifted configurations, the actual performance deserves some additional discussion here.
The specific optimization in question aligns the minimal values of the magnetic field on the surface, referred to as $\sigma = 1$ optimization. This can be seen in figure \ref{fig:lhdmodb} where a field line from both the outward and inward shifted cases are plotted as a function of toroidal angle. The minima align for the inward shifted case (green) but do not align for the outward shifted case (yellow). Another feature of these LHD-like equilibria is because the field is generated with helical coils, the field strength is smoothly varying with no local minima above the global minimum value. These local minima are problematic for particle confinement and can lead to promptly lost particles, similar to ripple trapped particles in a tokamak.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{lhd_modb_iaea.png}
\caption{Magnitude of the magnetic field, $|B|$ along a field line for the LHD inward shifted (green) and outward shifted (yellow) configurations}
\label{fig:lhdmodb}
\end{figure}
Since the maxima along the field line do not align, particles in the the LHD inward shifted configuration have a finite radial drift. In fact, as visible in figure \ref{fig:ploss} even in the absence in collisions, all trapped particles in both LHD configurations are eventually stochastically lost. The end result is a configuration which has no prompt losses due to the $\sigma=1$ optimization, but eventually loses all the particles. The parameters used for this calculation use the ARIES-CS parameters which have high plasma density and low temperature giving a slowing down time of $\approx$50 ms in the core, and considerably lower in the edge. When examining the collisional results under these conditions, the LHD inward shifted configuration compares favorably to configurations such as Wistell-A even though stochastic losses are very low in Wistell-A.
\subsection{QH: Collisionless vs Collisional losses}
\label{ss:qh}
The performance of Wistell-A is worth looking at closer. In the collisionless losses (figure \ref{fig:ploss}) the total losses are low, below all other configurations except for Wistell-B and ITER. Furthermore, almost all the losses that do exist are prompt, occurring well before 1 ms. Yet, when collisions are added, Wistell-A performs significantly poorer to Wistell-B and Ku5 and instead performs equally well to W7-X which has more prompt losses and significantly more collisionless stochastic losses. The performance of Wistell-A is actually slightly worse than LHD, which has fewer prompt losses, but very large values of collisionless stochastic losses.
To understand this behavior it is necessary to not only distinguish between prompt and stochastic losses, but between the pitch angle of promptly lost particles. Particles can diffuse through phase space by pitch-angle scattering. Although pitch-angle scattering is small for 3.5 MeV alpha particles compared to momentum loss (by roughly a factor of 20), it still exists. If a particle diffuses into a region of phase space which is promptly lost, it will likely be lost before it can diffuse out. The distribution of these loss regions in phase space is important. If there is one major region of losses, such as all deeply trapped particles, the only particles that will be lost are those born in the region or close to it. However, if the prompt-loss regions are scattered around phase space, even if the total volume is lower, the amount of particles that may drift through a prompt-loss may be higher. Although verification will require statistical analysis tools beyond the scope of this paper, some basic analysis can be done by examining the pitches of promptly lost particles. Figure \ref{fig:lvp} shows a histogram of prompt (within 1 ms) collisionless particle losses for W7-X and Wistell-A as a function of pitch. The pitch parameter is given as $E/\mu$ where E is the particle energy and $\mu$ is the first adiabatic invariant. This ratio is the maximum field a particle can reach before reflecting. The trapped-passing boundary is slightly different for the configurations and is shown with vertical dashed lines. All the lost particles are trapped. Most of the losses from W7-X are from deeply trapped particles. All the losses for Wistell-A are near the trapped-passing boundary. While there are some losses near the trapped passing boundary for W7-X, these losses are significantly less in number than for Wistell-A.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{lvp_iaea.png}
\caption{Histogram of collisionless prompt losses (less than 0.1 s) as a function of starting pitch for Wistell-A (black) and W7-X (magenta). The vertical dotted lines represent the minimum possible value of $E/\mu$ and the vertical dashed lines represent the trapped-passing boundaries for each configuration. The large population for W7-X corresponds to deeply trapped particles, while the other peak, larger on Wistell-A is near the trapped-passing boundary.}
\label{fig:lvp}
\end{figure}
One explanation for the relatively poor performance of Wistell-A compared to the expectations from collisionless losses is as follows. The LHD-inward shifted has no prompt losses and the collisional results appear similar to the collisionless results. Most particles are lost, but they are lost slowly. W7-X does have prompt losses, but these occur mostly in the deeply trapped particles. Only particles that are close to the deeply trapped region can diffuse onto lost orbits and be lost. In contrast, the losses from Wistell-A occur mostly near the trapped-passing boundary. The phase space volume near the trapped-passing boundary is significantly larger than the deeply trapped volume. For this reason, it is easier to diffuse into loss regions near the trapped-passing boundary and the losses are enhanced for Wistell-A when collisions are included. One result from this analysis is that if a configuration is to have prompt losses, it is far better to have them in deeply trapped regions.
\subsection{Limitations and Caveats}
The analysis presented in this paper is useful particularly for comparing different configurations, but to actually calculate losses in reactor configurations additional steps need to be taken. This section outlines some of the limitations of the calculations.
As noted above, the same profiles were used for every configuration despite significant differences in $\beta$. Furthermore, including a realistic profile for each configuration would obscure some of the differences between the configurations, which is the primary purpose of the results presented here.
Another limitation is the particle following algorithm is a guiding center algorithm and does not include finite gyro-orbits. Finite orbits for alpha particles can be large and a full-orbit analysis between configurations would help determine whether the alpha loss estimates are accurate. At increased machine size, the effects of finite orbits are smaller and the guiding center approximation gets increasingly better. Many other effects are also not included, including any transport from Alfven Eigenmodes.
The configurations presented here all rely on \texttt{VMEC} equilibria. \texttt{VMEC} describes the equilibria as having nested toroidal flux surfaces without magnetic islands or stochastic field regions. This limitation is mitigated somewhat because the large orbits of alpha particles may average over small regions of stochasticity. Calculations with more realistic field, which also includes the effects from fields generated from coils are left for future work.
Particles are considered lost if they pass beyond the penultimate surface in the \texttt{VMEC} equilibrium. In reality, particles may leave the confined plasma and reenter. This effect may be strongest in QA configurations which have the longest connection lengths and thus the largets banana widths.
\section{Conclusions and Outlook}
\label{sec:conc}
The analysis presented here shows that it is possible to optimize for stellarators to have good energetic particle confinement, often by ensuring very high quasisymmetry, as was done in the Ku5 and Simsopt equilibria. Post-hoc analysis of these two configurations indicate that it may be possible to achieve acceptable levels of energetic particle loss by optimizing for $\Gamma_c$ instead. Since $\Gamma_c$ is less restrictive than quasisymmetry, a large configuration space is available, and it may be possible to find a configuration that satisfies various other needs as well. Indeed, the Wistell-A configuration was optimized with $\Gamma_c$ and has both a vacuum magnetic well and a buildable coil set.
Unfortunately, none of the configurations presented here satisfy all of our needs for a stellarator reactor, which requires not only energetic particle confinement, but performance at high pressure, a buildable coil set, as well as other properties that are more difficult to quantify, like a viable divertor solution and reduced turbulent transport. As optimization algorithms and the physics metrics that feed into them improve, it is more likely that configurations which perform satisfactorily on all required axes will be found.
\section*{Acknowledgments}
The authors would like to acknowledge Mike Zarnstorff and Sam Lazerson for providing the NCSX and ARIES-CS equilibria, Don Spong for providing the ITER equilibrium, and Joachim Geiger and Carolin N\"uhrenberg for providing the W7-X equilibrium. Work for this paper was supported by DE-FG02-93ER54222, DE-FG02-00ER54546 and UW 2020 135AAD3116. Matt Landreman was supported by Simons Foundation (560651, ML)
\section*{References}
\bibliographystyle{iopart-num}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 346 |
package base
import (
"time"
sgbucket "github.com/couchbase/sg-bucket"
)
// A wrapper around a Bucket that transparently adds logging of all the API calls.
type LoggingBucket struct {
bucket Bucket
}
func (b *LoggingBucket) GetName() string {
//LogTo("Bucket", "GetName()")
return b.bucket.GetName()
}
func (b *LoggingBucket) Get(k string, rv interface{}) (uint64, error) {
start := time.Now()
defer func() { LogTo("Bucket", "Get(%q) [%v]", k, time.Since(start)) }()
return b.bucket.Get(k, rv)
}
func (b *LoggingBucket) GetRaw(k string) (v []byte, cas uint64, err error) {
start := time.Now()
defer func() { LogTo("Bucket", "GetRaw(%q) [%v]", k, time.Since(start)) }()
return b.bucket.GetRaw(k)
}
func (b *LoggingBucket) GetAndTouchRaw(k string, exp int) (v []byte, cas uint64, err error) {
start := time.Now()
defer func() { LogTo("Bucket", "GetAndTouchRaw(%q) [%v]", k, time.Since(start)) }()
return b.bucket.GetAndTouchRaw(k, exp)
}
func (b *LoggingBucket) GetBulkRaw(keys []string) (map[string][]byte, error) {
start := time.Now()
defer func() { LogTo("Bucket", "GetBulkRaw(%q) [%v]", keys, time.Since(start)) }()
return b.bucket.GetBulkRaw(keys)
}
func (b *LoggingBucket) Add(k string, exp int, v interface{}) (added bool, err error) {
start := time.Now()
defer func() { LogTo("Bucket", "Add(%q, %d, ...) [%v]", k, exp, time.Since(start)) }()
return b.bucket.Add(k, exp, v)
}
func (b *LoggingBucket) AddRaw(k string, exp int, v []byte) (added bool, err error) {
start := time.Now()
defer func() { LogTo("Bucket", "AddRaw(%q, %d, ...) [%v]", k, exp, time.Since(start)) }()
return b.bucket.AddRaw(k, exp, v)
}
func (b *LoggingBucket) Append(k string, data []byte) error {
start := time.Now()
defer func() { LogTo("Bucket", "Append(%q, ...) [%v]", k, time.Since(start)) }()
return b.bucket.Append(k, data)
}
func (b *LoggingBucket) Set(k string, exp int, v interface{}) error {
start := time.Now()
defer func() { LogTo("Bucket", "Set(%q, %d, ...) [%v]", k, exp, time.Since(start)) }()
return b.bucket.Set(k, exp, v)
}
func (b *LoggingBucket) SetRaw(k string, exp int, v []byte) error {
start := time.Now()
defer func() { LogTo("Bucket", "SetRaw(%q, %d, ...) [%v]", k, exp, time.Since(start)) }()
return b.bucket.SetRaw(k, exp, v)
}
func (b *LoggingBucket) Delete(k string) error {
start := time.Now()
defer func() { LogTo("Bucket", "Delete(%q) [%v]", k, time.Since(start)) }()
return b.bucket.Delete(k)
}
func (b *LoggingBucket) Remove(k string, cas uint64) (casOut uint64, err error) {
start := time.Now()
defer func() { LogTo("Bucket", "Remove(%q) [%v]", k, time.Since(start)) }()
return b.bucket.Remove(k, cas)
}
func (b *LoggingBucket) Write(k string, flags int, exp int, v interface{}, opt sgbucket.WriteOptions) error {
start := time.Now()
defer func() { LogTo("Bucket", "Write(%q, 0x%x, %d, ..., 0x%x) [%v]", k, flags, exp, opt, time.Since(start)) }()
return b.bucket.Write(k, flags, exp, v, opt)
}
func (b *LoggingBucket) WriteCas(k string, flags int, exp int, cas uint64, v interface{}, opt sgbucket.WriteOptions) (uint64, error) {
start := time.Now()
defer func() {
LogTo("Bucket", "WriteCas(%q, 0x%x, %d, %d, ..., 0x%x) [%v]", k, flags, exp, cas, opt, time.Since(start))
}()
return b.bucket.WriteCas(k, flags, exp, cas, v, opt)
}
func (b *LoggingBucket) Update(k string, exp int, callback sgbucket.UpdateFunc) (err error) {
start := time.Now()
defer func() { LogTo("Bucket", "Update(%q, %d, ...) --> %v [%v]", k, exp, err, time.Since(start)) }()
return b.bucket.Update(k, exp, callback)
}
func (b *LoggingBucket) WriteUpdate(k string, exp int, callback sgbucket.WriteUpdateFunc) (err error) {
start := time.Now()
defer func() { LogTo("Bucket", "WriteUpdate(%q, %d, ...) --> %v [%v]", k, exp, err, time.Since(start)) }()
return b.bucket.WriteUpdate(k, exp, callback)
}
func (b *LoggingBucket) Incr(k string, amt, def uint64, exp int) (uint64, error) {
start := time.Now()
defer func() { LogTo("Bucket", "Incr(%q, %d, %d, %d) [%v]", k, amt, def, exp, time.Since(start)) }()
return b.bucket.Incr(k, amt, def, exp)
}
func (b *LoggingBucket) WriteCasWithXattr(k string, xattr string, exp int, cas uint64, v interface{}, xv interface{}) (casOut uint64, err error) {
start := time.Now()
defer func() { LogTo("Bucket", "WriteCasWithXattr(%q, ...) [%v]", k, time.Since(start)) }()
return b.bucket.WriteCasWithXattr(k, xattr, exp, cas, v, xv)
}
func (b *LoggingBucket) WriteUpdateWithXattr(k string, xattr string, exp int, callback sgbucket.WriteUpdateWithXattrFunc) (casOut uint64, err error) {
start := time.Now()
defer func() {
LogTo("Bucket", "WriteUpdateWithXattr(%q, %d, ...) --> %v [%v]", k, exp, err, time.Since(start))
}()
return b.bucket.WriteUpdateWithXattr(k, xattr, exp, callback)
}
func (b *LoggingBucket) GetWithXattr(k string, xattr string, rv interface{}, xv interface{}) (cas uint64, err error) {
start := time.Now()
defer func() { LogTo("Bucket", "GetWithXattr(%q, ...) [%v]", k, time.Since(start)) }()
return b.bucket.GetWithXattr(k, xattr, rv, xv)
}
func (b *LoggingBucket) DeleteWithXattr(k string, xattr string) error {
start := time.Now()
defer func() { LogTo("Bucket", "DeleteWithXattr(%q, ...) [%v]", k, time.Since(start)) }()
return b.bucket.DeleteWithXattr(k, xattr)
}
func (b *LoggingBucket) GetDDoc(docname string, value interface{}) error {
start := time.Now()
defer func() { LogTo("Bucket", "GetDDoc(%q, ...) [%v]", docname, time.Since(start)) }()
return b.bucket.GetDDoc(docname, value)
}
func (b *LoggingBucket) PutDDoc(docname string, value interface{}) error {
start := time.Now()
defer func() { LogTo("Bucket", "PutDDoc(%q, ...) [%v]", docname, time.Since(start)) }()
return b.bucket.PutDDoc(docname, value)
}
func (b *LoggingBucket) DeleteDDoc(docname string) error {
start := time.Now()
defer func() { LogTo("Bucket", "DeleteDDoc(%q, ...) [%v]", docname, time.Since(start)) }()
return b.bucket.DeleteDDoc(docname)
}
func (b *LoggingBucket) View(ddoc, name string, params map[string]interface{}) (sgbucket.ViewResult, error) {
start := time.Now()
defer func() { LogTo("Bucket", "View(%q, %q, ...) [%v]", ddoc, name, time.Since(start)) }()
return b.bucket.View(ddoc, name, params)
}
func (b *LoggingBucket) ViewCustom(ddoc, name string, params map[string]interface{}, vres interface{}) error {
start := time.Now()
defer func() { LogTo("Bucket", "ViewCustom(%q, %q, ...) [%v]", ddoc, name, time.Since(start)) }()
return b.bucket.ViewCustom(ddoc, name, params, vres)
}
func (b *LoggingBucket) SetBulk(entries []*sgbucket.BulkSetEntry) (err error) {
start := time.Now()
defer func() { LogTo("Bucket", "SetBulk(%q, ...) --> %v [%v]", entries, err, time.Since(start)) }()
return b.bucket.SetBulk(entries)
}
func (b *LoggingBucket) Refresh() error {
start := time.Now()
defer func() { LogTo("Bucket", "Refresh() [%v]", time.Since(start)) }()
return b.bucket.Refresh()
}
func (b *LoggingBucket) StartTapFeed(args sgbucket.TapArguments) (sgbucket.TapFeed, error) {
start := time.Now()
defer func() { LogTo("Bucket", "StartTapFeed(...) [%v]", time.Since(start)) }()
return b.bucket.StartTapFeed(args)
}
func (b *LoggingBucket) Close() {
start := time.Now()
defer func() { LogTo("Bucket", "Close() [%v]", time.Since(start)) }()
b.bucket.Close()
}
func (b *LoggingBucket) Dump() {
LogTo("Bucket", "Dump()")
b.bucket.Dump()
}
func (b *LoggingBucket) VBHash(docID string) uint32 {
LogTo("Bucket", "VBHash()")
return b.bucket.VBHash(docID)
}
func (b *LoggingBucket) GetMaxVbno() (uint16, error) {
return b.bucket.GetMaxVbno()
}
func (b *LoggingBucket) CouchbaseServerVersion() (major uint64, minor uint64, micro string, err error) {
return b.bucket.CouchbaseServerVersion()
}
func (b *LoggingBucket) UUID() (string, error) {
return b.bucket.UUID()
}
func (b *LoggingBucket) GetStatsVbSeqno(maxVbno uint16, useAbsHighSeqNo bool) (uuids map[uint16]uint64, highSeqnos map[uint16]uint64, seqErr error) {
return b.GetStatsVbSeqno(maxVbno, useAbsHighSeqNo)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,215 |
Teargas, clashes at funeral for Bahraini protester
Bahraini Shiite Muslims hold national flags on the top of a van during the funeral of Habib Ebrahim Abdullah on January 13, 2013.(AFP Photo / Mohammed Al-Shaikh) © AFP
Bahraini government forces clashed with protesters on Sunday at the funeral of a demonstrator killed by teargas at a previous rally. Violent crackdowns continue as the demonstrators protest against the royal family, ruling for more than 40 years.
Scores of Bahrainis attended the funeral of the protestor Haj Habib Ibrahim Abdullah, which resulted in another demonstration against the ruling Al-Khalifa family's regime.Protesters shouted slogans against the regime and called for release of jailed activists. Government forces fired teargas and birdshot to disperse the crowd. A demonstration on Sunday was held at the funeral of Abdullah, who died after inhaling poisonous teargas. He and his grandson were exposed to lethal gas fired by government forces during a peaceful anti-regime demonstration last Monday. The nine-year-old grandson is now suffering from side effects and is being treated abroad, Ahlul Bayt News Agency reports. Abdullah and his grandson were exposed to gas during a previous peaceful demonstration which was held in Malkiya village last Monday aftertop Bahraini court overturned an appeal by 13 anti-regime activists on their sentences for protesting in 2011. Seven of the activists received life sentences, the other sentences ranged from five to 15 years in prison.The 13 whose appeals were overturned were originally part of a group of 20 activists convicted by a military tribunal of conspiring against Bahrain's Sunni royal family and "setting up terror groups to topple the regime." They then lost an appeal in civilian court in September 2011. Seven chose not to appeal again.Asma Darwish from the European-Bahraini Organization for Human Rights told RT that Bahraini authorities are regularly using excessive volumes of "toxic gases" against citizens even inside peoples homes."Bahrain has been using teargas canisters or as I like to call it toxic gases against citizens even in their houses not even protesting in the streets," she said, sharing her own experience of being attacked with tear gas in her home while being eight months pregnant. "I was nearly 8-months pregnant as our house was targeted by teargas canisters and the smell of teargas stormed into the house."She also claimed that the house of an 88-year-old man was targeted on several occasions, leading to his death."We have been pointing to the amount of teargas being used in residential areas against peaceful protesters and we are very disappointed at the international community's reaction, especially the US and the UK's position towards the Bahraini revolution, being an ally and applying double standards when it comes to dealing with the situation in Bahrain," she added.
Thousands of Shiites protested against the jailing of 13 activists on charges of plotting to overthrow Bahrain's monarchy on Saturday near Manama, witnesses said, AFP reports. The demonstrators shouted "we will not resign ourselves to it" and "we will not forget the prisoners", while some carried photos of the convicted. The protest was held under close supervision of police forces. Anti-government protesters have been holding peaceful demonstrations across Bahrain since February 2011, calling for an end to the Al-Khalifa family rule which continues in the country since 1974. The protesters also stand for an end to the discrimination of the state's Shiite majority by the predominant Sunni government.Eighty people have been killed and thousands arrested since the unrest began. Many opposition figures have been arrested on the allegation of planning to topple the government.According to a 2011 reports by Human Rights Watch, the government regularly abuses its citizens. The abusesinclude denying defendants the right to counsel and to present a defense, denial of medical access to protesters injured by security forces, torture and ill-treatment during interrogation.
Trends:Arab world protests
Undefeated protests: Bahraini court jails young woman as rallies continue
Bahrain police fire tear gas, grenades on protesters during 'Martyrs Day' rally | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,426 |
package org.wso2.siddhi.query.api.definition.partition;
import org.wso2.siddhi.query.api.ExecutionPlan;
import org.wso2.siddhi.query.api.condition.Condition;
import org.wso2.siddhi.query.api.expression.Variable;
import java.util.ArrayList;
import java.util.List;
/**
* {@linkplain PartitionDefinition} class is used to represent the definition of
* a partition for a Siddhi instance. The partition definition consists of a
* partition ID and a list of partition types that belong to a given definition
* instance.
*/
public class PartitionDefinition implements ExecutionPlan {
private String partitionId;
private List<PartitionType> partitionTypeList = new ArrayList<PartitionType>();
public PartitionDefinition name(String name) {
this.partitionId = name;
return this;
}
public PartitionDefinition partitionBy(Variable variable) {
this.partitionTypeList.add(new VariablePartitionType(variable));
return this;
}
public PartitionDefinition partitionBy(Condition condition, String label) {
this.partitionTypeList.add(new RangePartitionType(condition, label));
return this;
}
public String getPartitionId() {
return partitionId;
}
public void addPartitionType(PartitionType partitionType) {
this.partitionTypeList.add(partitionType);
}
public List<PartitionType> getPartitionTypeList() {
return partitionTypeList;
}
@Override
public String toString() {
return "PartitionDefinition{" +
"partitionId='" + partitionId + '\'' +
", partitionTypeList=" + partitionTypeList +
'}';
}
@Override
public boolean equals(Object o) {
if (this == o) {
return true;
}
if (o == null || getClass() != o.getClass()) {
return false;
}
PartitionDefinition that = (PartitionDefinition) o;
if (partitionId != null ? !partitionId.equals(that.partitionId) : that.partitionId != null) {
return false;
}
if (partitionTypeList != null ? !partitionTypeList.equals(that.partitionTypeList) : that.partitionTypeList != null) {
return false;
}
return true;
}
@Override
public int hashCode() {
int result = partitionId != null ? partitionId.hashCode() : 0;
result = 31 * result + (partitionTypeList != null ? partitionTypeList.hashCode() : 0);
return result;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,843 |
Participants of capacity need assessment in Birendranagar.
Capacity need assessment training in Nepalgunj.
Capacity need assessment training in Birendranagar.
The objective of the project is that Urban centres of Nepal become people-centred and resilient to climate extremes and disasters.
The project has four main objectives, namely, i) building climate adaptation and resilience into vulnerable urban centres, ii) developing adaptation-friendly policy, practice and business, iii) equipping vulnerable urban dwellers with information, resources and appropriate technology to respond to climate extremes and disasters, and iv) knowledge documentation and dissemination.
20% increase in project districts and municipalities' budgets for Climate Change Adaptation (CCA) and Disaster Risk Reduction (DRR) activities.
5 private sector actors incorporating climate adaptive practices into their business plans.
60% (7063) of target groups demonstrate skills and showcase knowledge to minimise exposure and vulnerability and adapt to climate extremes and disaster.
5 briefing papers and policy briefs used as tool for influencing national policies and programmes.
5 multi-stakeholder dialogues organised to disseminate project learning and best practices.
The project will be implemented in two municipalities in the Mid-western region of Nepal, Birendranagar in Surkhet district and Nepalgunj in Banke district. These districts have suffered from increased frequency of extreme weather events such as landslides, floods and droughts resulting to the loss of human lives as well as high social and economic costs. The project aims to develop adaptation plans in the two municipalities and leverage resources from private sectors in financing adaptation actions. Approximately 235,418 populations residing in two cities will benefit from the project. A detailed climate vulnerability assessment will be conducted; local adaptation plans will be formulated, integrated and implemented in development plans of municipalities. A guideline will be developed to integrate climate adaptation in urban plans.
Skill development trainings in coordination with Council for Technical Education and Vocational Training (CTEVT) and Federation of Nepalese Chambers of Commerce and Industry (FNCCI) would be conducted for the vulnerable communities. The project will support daily wage earners by initiating innovative schemes and securities plans for ensuring their employment. They will be linked with micro credit facility for enterprise development after skill development activities for sustainability. The project will also work with private sectors to retrofit their business and prepare business continuity plans to make their businesses resilient to climate extremes and disasters. The project has identified the private sector (e.g. hotel owners, local retailers) as a key actor for building urban resilience and seeks to harness private sector growth as a positive force for change in urban areas and develop new partnerships and alliances to influence the path of private sector development so that it contributes to positive social, economic and environmental gains. This includes existing water service providers who need to retrofit their system to respond to needs of growing population and current availability of water.
As Banke and Surkhet districts are highly vulnerable to climatic extremes like drought, flood, inundation and landslide, the project will play a key role in bringing awareness to people about climate change and its impact in building climate resilience. One of the major components of this project is to develop and implement local adaptation plans in an urban context based on adopted Local Adaptation Plans of Action (LAPA), Local Disaster Risk Management Planning (LDRMP) and Environment Friendly Local Governance (EFLG) guidelines and frameworks and integrate the plans in local development and sectorial plans and programme as envisioned in national plans and policy. This project aims to disseminate the knowledge of climate change and adaptation to the private sectors so that they can build an adaptation-friendly business plan, which ultimately helps to improve the livelihood. The private sector will be influenced to adopt climate-adaptive strategies into existing processing and distribution systems. The project will support women's groups in setting up climate-adaptive livelihoods to build their resilience.
The project will build resilience of the two municipalities to climate extremes and disasters by working with all the relevant stakeholders including different tiers of the government, private sector, chambers of commerce, technical training institutes, and local communities. The project aims to mainstream the climate change and DRR frameworks within an urban context for the first time in Nepal. The tri-partite relationship being set up between the private sector, technical training institutes and vulnerable communities is a unique feature of this initiative. A resilience fund, which is an investment in a range of business enterprises and services, will help in reducing exposure to climate extremes and disasters for most of the vulnerable populations of urban centres. The business partnerships between Nordic enterprise providing appropriate technologies and local enterprises with capacity to gain new business will be promoted. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,741 |
\section{Introduction}
The last 25 years have seen the detection of more than 4,000 exoplanets \citep{Schneider2011}. Despite the indirect nature of most detections, existing observations already provide us with a wealth of information on the properties of exoplanetary systems: their mass, size, and orbital elements. Yet direct detection of a planet's reflected, or radiated light, for its direct spectral analysis for a large sample of targets, remains an exciting prospect that will contribute to further characterize individual planets, in particular the properties of their atmospheres \citep{Marois2008, Zurlo2015}.\par
The use of coronagraphic instruments is now leading to the detection of young giant planets in wide orbits around nearby stars \citep{Macintosh2015, Chauvin2017, Mesa2019}. This success is carried by the continued improvements of extreme adaptive optics systems \citep{Sauvage2016, Lozi2018, Boccaletti2020}. For smaller separations approaching the diffraction limit (below $\sim3\lambda/D$), small residual wavefront errors still dominate the error budget and coronagraphic solutions become less favorable.\par
\cite{Lacour2019} have demonstrated the advantages brought by long baseline interferometry for the characterization of extrasolar planets. This observing mode takes advantage of the spatial filtering provided by the resolving power of each of the 8 meter telescopes of the VLTI, coupled into single mode fibers to reach the required contrast. Interferometric nullers \citep{Bracewell1978, Colavita2009, Serabyn2019, Hoffmann2014, Defrere2015, Norris2019} offer the possibility to explore smaller angular separations through the use of fragmented apertures and long-baseline interferometry. Some combining solutions have been found that optimize the rejection of resolved stars \citep{Angel1997, Guyon2013}. The exploitation of these instruments is still limited by their vulnerability to optical path differences (OPD) errors, and requires sophisticated statistical analysis, like those proposed by \cite{Hanot2011}, \cite{Defrere2016} and more recently used by \cite{Norris2019} to disentangle the off-axis astrophysical signal from the effects of unwanted OPD.\par
Classical long-baseline and Fizeau interferometry make extensive use of the production of robust observables, like closure phases \citep{Jennison1958} and their generalized form, kernel phases \citep{Martinache2010}, to sidestep the limitations brought by the OPD residuals. This approach has provided reliable performance at very small separations, down to one resolution element and below.\par
Bringing together the robustness of interferometric observables and the photon-noise suppression of nulling, is an exciting perspective as it opens a novel high-contrast, high-precision regime. The double Bracewell architecture \citep{Angel1997} was remarked to offer such robustness \citep{Velusamy2003} when implemented with the adequate phase shift between the two stages (called "sin-chop"). A different approach was later proposed by \cite{Lacour2014}, exploiting the measurement of fringes in the leakage light.\par
\cite{Martinache2018} introduced an alternative, more efficient solution for a four-telescope beam-combiner architecture that produces six nulled outputs. By analyzing the response of these outputs to parasitic OPDs (instrumentally or atmospherically induced phase error), the authors identify linear combinations of outputs that are robust to these aberrations to second order. The solution they propose concerns a four-input nuller that provides three nulled kernel observables.\par
In this paper, we look for the properties ensuring that a combiner will produce kernel nulls. They help outline a general strategy for the design of kernel nullers for an arbitrary number of apertures.\par
\section{Analysis of the existing four-input kernel-nulling architecture}
\subsection{The kernel-nulling approach} \label{sec_kernel_principle}
Using the nuller architecture laid out by \cite{Martinache2018} for a four-beam interferometer as a starting point, and reexamining its properties, we look into ways of generalizing this special case to a wider range of configurations, involving different numbers of apertures.\par
The inner structure of a homodyne interferometric combiner (nulling or not) is conveniently represented by a combiner matrix $\mathbf{M}$ that acts on a vector $\mathbf{z}$ of input electric fields and leads to the production of an output electric field vector $\mathbf{x}$.
\begin{equation}\label{eq_xMz}
\mathbf{x} = \mathbf{M} \cdot \mathbf{z}.
\end{equation}
Eventually, a detector records the intensity associated to the square norm of this output electric field.\par
The fact that only the square norm of the field is recorded has two consequences. The first is that the response of the combiner is insensitive to the absolute phase of the input electric field: one of the sub-apertures can therefore be arbitrarily picked as a reference, and the phases of the different electric fields sampled by the other sub-apertures are measured relative to that reference. The second is that the output intensity is equally insensitive to any global phase shift $\phi$ applied to any row $\mathbf{m}$ of the matrix $\mathbf{M}$ describing the combiner. This is of consequence when identifying distinct combinations (or rows).\par
Assuming that the recombiner is fed by a balanced array of identical sub-apertures, the complex amplitude of the input electric field can be described by a vector of phasors. We will further assume that the combiner benefits from a fringe tracker that, although not perfect, brings the system close to its nominal state. The fringe tracking residuals are assumed to be small and all phasors $e^{-j\varphi_k}$ are here approximated using the following expansion:
\begin{equation}\label{eq_simplified_inputs}
z_{k} = e^{-j \varphi_k} \approx 1 -j \varphi_k,
\end{equation}
where $j$ is the imaginary unit.\par
Since only intensities are measured, the overall response of the system is a quadratic function of the perturbation phase vector. \cite{Martinache2018} thus use this description to look at the properties of the second order derivative of the intensity relative to the phase. One of the $n_a$ sub-apertures being used as a phase reference, there are $n_a - 1$ degrees of freedom, and $n_d = n_a * (n_a - 1) / 2$ such derivatives. One can store this response into a $n_o \times n_d$ matrix $\mathbf{A}$ called the matrix of second order derivatives, where $n_o$ is the number of relevant outputs. Linear combinations of rows of $\mathbf{A}$ that equal 0 cancel out the second order intensity deviations caused by small input phase errors. The same linear combination applied to the intensity measured after the recombiner will be equally insensitive to small input phase errors. We refer to these linear combinations as kernel outputs or kernel nulls when applied to a collection of nulled outputs.\par
The rank of $\mathbf{A}$ and the possibility of forming such robust observables rely entirely on the properties of the matrix $\mathbf{M}$, and therefore does not depend on the geometry of the input array. However, the question of whether a kernel null carries astrophysically relevant information also depends on the configuration of the array. Throughout this work, phase and amplitude contributions are considered independently, but their coupled contribution is neglected. For now, we will further examine the properties of the combiner and introduce a convenient visual representation of the structure of $\mathbf{M}$.
\subsection{Visualization of complex combiner matrices}\label{sec_complex_matrix_plots}
The effect of the matrix $\mathbf{M}$ on the complex amplitude of the input electric field can be conveniently visualized by a series of plots of the complex plane. For a given combiner, each input is represented by a colored arrow which, in the absence of environmental perturbation, is aligned with the real axis. Each plot illustrates the effect of a row of $\mathbf{M}$ on such inputs: the resulting electric field is the sum of all colored arrows present in the plot. A nuller is characterized by several outputs for which the sum of the arrows, associated to the electric fields, sum up to zero. These complex matrix plots (CMP) will be used throughout this work to describe several nuller designs of varying complexity.\par
\subsection{From real to complex nulls} \label{sec_real_to_complex}
The architecture of the kernel nuller described in \cite{Martinache2018} builds from an initial all-in-one four-beam nuller whose overall effect can be described by the following matrix:
\begin{equation}\label{eq_N}
\mathbf{N}_4 =
\frac{1}{\sqrt{4}}\left[\begin{matrix}
1 & 1 & 1 & 1\\
1 & 1 & -1 & -1\\
1 & -1 & 1 & -1\\
1 & -1 & -1 & 1\\
\end{matrix}\right]
\end{equation}
This matrix is real. Each nulled row of $\mathbf{N_4}$ recombines distinct arrangements of the four input electric fields such that the coefficients on the corresponding rows sum up to zero, as represented in Fig. \ref{fig_paper_plot_N}, with arrows aligned with the real axis: two positive (or not phase-shifted), and two negative (or phase-shifted by $\pi$). As discussed in this reference, this nuller does not allow the formation of kernels: the output intensities it produces are a degenerate function of the target information and input phase perturbations. The outputs of this nuller can however be fed to a second stage, described by the following matrix:
\begin{equation}\label{eq_S}
\mathbf{S}_4 =
\frac{1}{\sqrt{4}}\left[\begin{matrix}
2 & 0 & 0 & 0 \\
0 & 1 & e^{j\frac{\pi}{2}} & 0\\
0 & e^{j\frac{\pi}{2}} & 1 & 0\\
0 & 1 & 0 & e^{j\frac{\pi}{2}}\\
0 & e^{j\frac{\pi}{2}} & 0 & 1\\
0 & 0 & 1 & e^{j\frac{\pi}{2}}\\
0 & 0 & e^{j\frac{\pi}{2}} & 1\\
\end{matrix}\right],
\end{equation}
which leaves the bright output untouched but further splits the nulled ones, and selectively introduces $\pi/2$ phase shifts. The overall effect of the combiner will be described by a now complex combiner matrix, result of the product $\mathbf{M}_4 = \mathbf{S}_4 \cdot \mathbf{N}_4$. The CMPs of this modified combiner, shown in Fig. \ref{fig_paper_plots}, offer a more easily readable description of its effect, with components of the output electric field no longer simply aligned with the real axis, but spanning the complex plane.\par
The new complex configuration enables the larger diversity that is required to disentangle the otherwise degenerate effect that environmental perturbations have on the input electric fields. The modified nuller indeed features more outputs than inputs, and a close examination of the CMPs of Fig. \ref{fig_paper_plots} shows that all six combinations offer a distinct arrangement of the four input fields. The construction of a larger number of distinct nulls is one of the requirements for the existence of a non-empty left null space for $\mathbf{A}$ described in Sec. \ref{sec_kernel_principle}. In effect, pairs of outputs produce the same response to environmental effects, while still producing different response to off-axis light.\par
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{viking_nulls_simplified.pdf
\caption{CMP of the matrix $\mathbf{N}_4$ of Eq. (\ref{eq_N}) representing a four-input nulling combiner. The first row constitutes the bright channel, with all inputs combined constructively. Note how each output is a contribution of all the inputs, and not just a pair of them, which prevents the direct interpretation through the uv plane.}
\label{fig_paper_plot_N}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{viking_nulls_piover2_plot.pdf
\caption{CMP of the $\mathbf{S}_4\cdot\mathbf{N}_4$ combination. The first output is the bright channel for which all the inputs add-up constructively. The vectors are staggered for readability. Pairs of nulled rows represented side-by-side are mirror images of each-other (enantiomorph). Note that the amplitude of the phasors is reduced compared to Fig. \ref{fig_paper_plot_N} due to additional splitting.}
\label{fig_paper_plots}
\end{figure}
\section{Properties of conjugate pairs of nulls}\label{sec_properties}
\subsection{Kernel outputs}
Identifying the kernel-forming combinations of outputs no-longer requires building the second order derivative matrix $\mathbf{A}$ but can be achieved by examination of the CMP representation of the nuller. Figure \ref{fig_paper_plots} lays out, side by side, the two outputs leading to one kernel-null observable. We call these outputs enantiomorph: close examination of any such pair of outputs reveals that the electric field combination patterns are the mirror image of one-another.\par
Given that the measured intensity associated to any output is insensitive to a global phase shift, one can always apply such a shift so as to align the arrow corresponding to the phase reference input with the real axis, and point it towards the positive direction. After such a rotation is applied, enantiomorph outputs simply become complex conjugate. This makes it possible to write simple equations that describe the two key properties of kernel nulls: their robustness to small phase perturbation, and the antisymmetric nature of the signal they provide.
\subsection{Robustness} \label{sec_robustness}
Considering $\mathbf{m}_1$ and $\mathbf{m}_2$, a conjugate pair of null rows of $\mathbf{M}$:
\begin{equation}\label{eq_m}
\mathbf{m}_2 = \mathbf{m}_1^*.
\end{equation}
A corresponding kernel null $\kappa(\mathbf{z})$ writes as the difference of the two measured intensities:
\begin{equation}\label{eq_km}
\kappa(\mathbf{z}) =
|\mathbf{m}_1 \mathbf{z}|^2 -
|\mathbf{m}_2 \mathbf{z}|^2 =
\mathbf{m}_1 \mathbf{z} (\mathbf{m}_1 \mathbf{z})^* -
\mathbf{m}_2 \mathbf{z} (\mathbf{m}_2 \mathbf{z})^*.
\end{equation}
Using (\ref{eq_m}) and (\ref{eq_km}) gives:
\begin{equation}\label{eq_mstar}
\kappa(\mathbf{z}) =
\mathbf{m}_1 \mathbf{z} (\mathbf{m}_1 \mathbf{z})^* -
\mathbf{m}_1^* \mathbf{z} (\mathbf{m}_1^* \mathbf{z})^*.
\end{equation}
In the case of the approximation mentioned in Eq. (\ref{eq_simplified_inputs}):
\begin{multline}
\kappa(\mathbf{z}) =
\mathbf{m}_1 (\mathbf{a} + j\boldsymbol{\varphi}) (\mathbf{m}_1 (\mathbf{a} + j\boldsymbol{\varphi}))^*\\
- \mathbf{m}_1^* (\mathbf{a} + j\boldsymbol{\varphi}) (\mathbf{m}_1^* (\mathbf{a} + j\boldsymbol{\varphi}))^*,
\end{multline}
where $\mathbf{a}$ is a vector of ones. Developing this expression, since $\mathbf{m}_1\mathbf{a} = 0$ and $\mathbf{m}_1^*\mathbf{a} = 0$, the only terms left are the ones containing only the imaginary perturbation term $j\boldsymbol{\varphi}$:
\begin{equation}
\kappa(\mathbf{z}) =
\mathbf{m}_1 j\boldsymbol{\varphi} (\mathbf{m}_1 j\boldsymbol{\varphi})^* -
\mathbf{m}_1^* j\boldsymbol{\varphi} (\mathbf{m}_1^* j\boldsymbol{\varphi})^*.
\end{equation}
Distributing the conjugate operator gives:
\begin{equation}
\kappa(\mathbf{z}) =
- \mathbf{m}_1 j\boldsymbol{\varphi} \mathbf{m}_1^* j\boldsymbol{\varphi} +
\mathbf{m}_1^* j\boldsymbol{\varphi} \mathbf{m}_1 j\boldsymbol{\varphi},
\end{equation}
and therefore $\kappa(\mathbf{z}) = 0$ due to the commutativity. This shows that the subtraction of intensity of complex conjugate pairs of nulled outputs always produces a kernel null that is robust to arbitrary imaginary phasors, to which the small input phase aberrations are approximated.\par
This property also applies to arbitrary purely real input electric fields that would correspond to pure photometric error generated by fluctuations of the coupling efficiencies. Considering a purely real input vector $\mathbf{a}$:
\begin{equation}
\kappa(\mathbf{z}) =
\mathbf{m}_1 \mathbf{a} (\mathbf{m}_1 \mathbf{a})^* -
\mathbf{m}_1^* \mathbf{a} (\mathbf{m}_1^* \mathbf{a})^*.
\end{equation}
Distributing the conjugate operator gives:
\begin{equation}\label{eq_robust_last}
\kappa(\mathbf{z}) =
\mathbf{m}_1 \mathbf{a} \mathbf{m}_1^* \mathbf{a} -
\mathbf{m}_1^* \mathbf{a} \mathbf{m}_1 \mathbf{a},
\end{equation}
and therefore $\kappa(\mathbf{z}) = 0$ due to the commutativity.\par
At any instant, the subtraction of the signals recorded by conjugate (or more generally enantiomorph) outputs forms a kernel null.
Conjugate pairs of nulls allow the formation of kernel nulls. This property generalizes to enantiomorph pairs of nulls through the rotation by a single common phasor. A complementary approach for the identification of robust combinations of outputs is the use of the singular value decomposition (SVD) of the second order derivative matrix $\mathbf{A}$, as mentioned by \cite{Martinache2018}, which ensures the identification of all the robust combinations of outputs.\par
This behavior can be illustrated by adding different phased contributions to the inputs, and plotting the resulting electric field on top of the original perfectly cophased CMP (kept in dashed lines). The first panel of Fig. \ref{fig_didactic_plots} uses this representation of the combined light to illustrate how small input phase aberrations affect the amplitude (and therefore the intensity) of the combiner's outputs. In particular it shows how, for small phase errors, conjugate pairs of nulls suffer the same leakage light intensity.\par
\begin{figure*}
\centering
\begin{tabular}{c|c|c}
Small error & Partially resolved & Null peak\\
\includegraphics[width=0.31\textwidth]{didactic_binary_4T_step_0.pdf}&
\includegraphics[width=0.31\textwidth]{didactic_binary_4T_step_1.pdf}&
\includegraphics[width=0.31\textwidth]{didactic_binary_4T_step_2.pdf}\\
\end{tabular}
\caption{CMP for a four-input combiner, representing in dashed lines the coefficients of the combiner matrix $\mathbf{M}_4'$, functionally equivalent to $\mathbf{M}_4$, and in solid arrows the contributions of the complex amplitude of an example input electric field to the output electric field represented in black. Most dashed lines are hidden under the arrows. A black circle of radius equal to the modulus of this output is plotted for visual cue, its area being proportional to the corresponding intensity. Enantiomorph pairs that generate kernel combinations by subtraction are represented side-by-side. Like \cite{Martinache2018}, we use the example of the VLTI UT configuration observing at zenith at a wavelength of $3.6\mu m$. On the left-hand panel, a source located at 0.2 mas from the optical axis and for which the corresponding input phase shifts are within the small phase approximation. As a result, the output intensities within each pair are fully correlated and result in no kernel signal. On the central panel, the source is located 1.1 mas off-axis which generates larger phase shifts. As a result, the null intensities from the enantiomorph pairs begin to decorrelate and generate kernel-null signal. On the right-hand panel, the source is located 4.3 mas from the optical axis, in the position where the first nulled output peaks. At this position, the second output gets to zero.}
\label{fig_didactic_plots}
\end{figure*}
\subsection{Symmetry of the response}\label{sec_odd_response}
The second and third panels of Fig. \ref{fig_didactic_plots} show how input light coming from a significantly off-axis source (input phases $\phi \geq 1$ radian) propagates to the nulled outputs. They highlight how this off-axis light produces different intensities at the outputs of the conjugate pairs, translating into kernel-null signal.\par
For a combiner that is fed by an array of apertures collecting the light from the sky, the value of this response as a function of the incidence of the light is the response map of the interferometer, and depends on the position of each of the apertures. \cite{Martinache2018} note how this map is antisymmetric, therefore providing a rejection of the photosphere of stars and symmetric circumstellar disks that could hide a planetary companion, and provide the astrometry of such companions without ambiguity. Using our formalism, we can demonstrate this antisymmetric property for any aperture configuration. If $\mathbf{z}$ and $\mathbf{z}'$ are two input electric field vectors coming from sources located at symmetric positions in the field of view, then:
\begin{equation}\label{eq_z}
\mathbf{z}' = \mathbf{z}^*.
\end{equation}
Considering again a conjugate pair of null rows $\mathbf{m}_1$ and $\mathbf{m}_2$, and by substitution of (\ref{eq_z}) into (\ref{eq_km}) we get:
\begin{equation}
\kappa(\mathbf{z}) =
\mathbf{m}_1 \mathbf{z}'^* \mathbf{m}_1^* \mathbf{z}' -
\mathbf{m}_2 \mathbf{z}'^* \mathbf{m}_2^* \mathbf{z}'.
\end{equation}
After substitution of (\ref{eq_m}), this becomes:
\begin{equation}
\kappa(\mathbf{z}) =
\mathbf{m}_2^* \mathbf{z}'^* \mathbf{m}_2 \mathbf{z}' -
\mathbf{m}_1^* \mathbf{z}'^* \mathbf{m}_1 \mathbf{z}'.
\end{equation}
This leads to the conclusion that the response is antisymmetric:
\begin{equation}\label{eq_response_last}
\kappa(\mathbf{z}) = - \kappa(\mathbf{z}').
\end{equation}
\par
Conversely, one may also extract from this pair of nulls the complementary observable:
\begin{equation}\label{eq_tau}
\tau(\mathbf{z}) =
\mathbf{m}_1 \mathbf{z} (\mathbf{m}_1 \mathbf{z})^* +
\mathbf{m}_2 \mathbf{z} (\mathbf{m}_2 \mathbf{z})^*
\end{equation}
whose response is symmetric. The observables $\kappa$ and $\tau$ therefore carry complementary information on the target field, much like the amplitude and phase of complex visibility in classical interferometry.\par
Although $\tau$ does not have the same robustness to aberrations, there may be ways to use it with the processing methods employed by \cite{Hanot2011} and \cite{Norris2019} so as to provide additional information on the target in different science cases. $\kappa$ is best suited for the study of high-contrast non-symmetrical features such as planetary companions, while $\tau$ may be used to study brighter symmetrical features such as debris disks or stellar envelopes. Combining both types of observables could enable image reconstruction.\par
The $\tau$ observables carry some information about the input phase errors. One can use their values over the course of a scan or modulation of the OPDs to locate the setpoint of the kernel-nuller, for which they will reach a minimum.
\section{Construction of new nullers}\label{sec_construction}
\subsection{Blueprints of kernel-nulling matrices} \label{sec_generative}
The properties used in Sects. \ref{sec_robustness} and \ref{sec_odd_response} to demonstrate the robustness of kernel nullers to small phase perturbations may be used as constraints to guide the design of an arbitrary kernel nuller matrix. For the output of any row $l$ to provide an on-axis null, the matrix coefficients must satisfy:
\begin{equation}\label{eq_null_condition}
\sum_{k=0}^{n_a-1}{M_{k,l}} = 0.
\end{equation}
Output intensities are unchanged when the coefficients of a row are all multiplied by a common phasor. We therefore apply one such phasor so as to get $\mathrm{Arg}(M_{0,l}) = 0$. We also set output \#0 by combining all the inputs with zero phase offset: $\mathrm{Arg}(M_{k,0}) = 0$.\par
Simple solutions to Eq. (\ref{eq_null_condition}) for a balanced array can be found by picking arrangements of uniformly spaced phase values in the $[0,2\pi]$ interval as can be seen of Figs. \ref{fig_paper_plots}, \ref{fig_3x3} and \ref{fig_6T_combiner}. The phase of each coefficient is therefore a multiple of $\Phi_0 = 2\pi/n_a$. On the CMPs seen thus far, this would result in the rotation of all of the arrows on the nulled outputs until the one associated with input \#0 is aligned with the real axis in the positive direction. With these constraints in place, outputs will only differ in the order in which the remaining $n_a - 1$ phase offsets are associated with the inputs. The maximum number of distinct nulled outputs is therefore:
\begin{equation}\label{eq_nmax}
n_{max} = (n_a-1)! .
\end{equation}
\par
The phase term $\phi_{k,l}$ writes:
\begin{equation}
\phi_{k,l} = c_{k,l} \Phi_0,
\end{equation}
where $c_{k,l}a$ is the $k$-th term of the $l$-th possible combination on the circle. In general, a complex coefficient of $\mathbf{M}$ will therefore write:
\begin{equation}\label{eq_generative}
M_{k,l} = a_l \cdot e^{j \phi_{k,l}},
\end{equation}{}
where $a_l$ is a real coefficient, normalizing the matrix, so that $\mathbf{M}$ represents a lossless beam-combiner for which each column vector is of unit norm. As mentioned in appendix \ref{sec_normalization}, this condition on the norm is necessary (but not sufficient) to ensure that the matrix represents a lossless beam combiner, and one solution for it is to have:
\begin{equation}
\begin{cases}\label{eq_normalization}
a_l = \frac{1}{\sqrt{n_a}} & \text{for the bright output}\\
a_l = \frac{1}{\sqrt{n_a}} \sqrt{\frac{n_a-1}{n_{null}}} & \text{for the nulled outputs}
\end{cases}
\end{equation}{}
where $n_{null}$ is the number of nulled outputs. Normalization is not mandatory to study the qualitative properties of the combiner, but it is necessary to study their throughput in a quantitative manner and their practical implementation.\par
The matrix $\mathbf{M}$ obtained with Eq. (\ref{eq_generative}), represents a combiner for which pairs of complex conjugate nulls can be subtracted to build the kernel nulls that are the focus of this work.
\subsection{Information redundancy}\label{sec_info_redundancy}
As shown by Eq. (\ref{eq_nmax}) and Table \ref{tab_nuller_growth}, the number of nulled outputs that would result from a strict application of these blueprint rules rapidly grows as the factorial of the number of inputs. However, for numbers of apertures larger than four, although all the nulls produced with the presented scheme are distinct, some of them do not carry new information on the target, as their response function to off-axis signal is a linear combination of the response function of other nulls.\par
Here, we analyze this property empirically by examining the response maps (analogous to Fig. 5 and 7 of \cite{Martinache2018}) and assembling them as vectors of a set of nulled outputs and a set of kernel outputs. The ranks of these sets provide the number of independent observables produced by the combiner. Although we were not able to link this property to particular traits of the combinations, the largest number of independent kernel nulls obtainable by a given non-redundant array of apertures was always the same as the number of independent closure-phases, which is in agreement with the expectations set by \cite{Martinache2018}. For any non-redundant array of apertures, this number is:
\begin{equation}\label{eq_na-2}
n_{kn}(n_a) = \binom{n_a}{2} - (n_a - 1) =\frac{(n_a-1)(n_a-2)}{2}.
\end{equation}
The underlying relationship between the baselines and our new observables is non-trivial but will be assumed to hold for any non-redundant array. For redundant arrays, this number decreases. We will call complete a nuller that provides the aforementioned maximum number $n_{kn}$ of independent observables. The number of independent nulls in the full set is $n_{indep.} = 2\times n_{kn}$. These results obtained empirically for up to seven inputs are shown in Table \ref{tab_nuller_growth}, along with their expected progression for larger numbers of inputs.\par
\begin{table}[]
\centering
\begin{tabular}{c|c|c|c}
Inputs & Distinct nulls & Indep. nulls & Kernel nulls\\
$n_a$ & $n_{max}$ & $n_{indep.}$ & $n_{kn}$ \\
\hline
3 & 2 & 2 & 1 \\
4 & 6 & 6 & 3 \\
5 & 24 & 12 & 6 \\
6 & 120 & 20 & 10 \\
7 & 720 & 30 & 15 \\
\end{tabular}
\caption{Growth of kernel-nuller combiners with the number of apertures.}
\label{tab_nuller_growth}
\end{table}{}
As seen in Eq. (\ref{eq_normalization}), an increase in the number of nulled rows decreases the normalization coefficients $a_i$, as in practice fewer splittings of the input light are necessary to obtain the fewer combinations. Our goal may therefore be to construct complete combiners using the minimum number of nulled combinations from the full matrix $\mathbf{M}$, with the intent of increasing its throughput. Expecting this number to be twice the number of kernel nulls (if we consider only pairwise kernel nulls), this produces a very large number $n_{crops}(n_a)$ of possible combinations:
\begin{equation}\label{eq_big_permutation}
n_{crops}(n_a) = \binom{(n_a-1)!}{(n_a-1)(n_a-2)}
\end{equation}
Only for the cases of three and four inputs is the solution unique ($n_{crops}(n_a)=1$), and all null rows must be kept. For more inputs, this number grows rapidly. Although a large fraction of them are complete, fewer satisfy the conditions detailed in appendix \ref{sec_normalization} for conservation of energy.\par
The following characteristics are shared by the three- and four-input combiners, as well as all the lossless realizations of the cropped five-input combiner:
\begin{itemize}
\item All nulls appear in conjugate (or enantiomorph) pairs, which implies that robust observables can be constructed by subtraction.
\item Each phasor appears in each column the same number of times (except for the one of phase zero which serves as the reference). Equation (\ref{eq_na-2}) implies that for a combiner producing $n_o = 2n_{kn}$ nulls, each phasor is used $(n_a-2)$ times for each input.
\end{itemize}
Enforcing these characteristics as rules for reducing the number of outputs has helped us to identify lossless realizations of the cropped six-input combiner outlined in Sect. \ref{sec_6T_combiner} by reducing the parameter space. The process leading to a valid solution remains one of trial and error: chosen randomly among all the possible arrangements that respect the aforementioned characteristics, a solution is eventually only accepted when the corresponding combiner is both lossless (cf. appendix \ref{sec_normalization}) and produces a complete set of kernel nulls.\par
\section{Examples of combiners}
\subsection{Three-input kernel nuller}\label{sec_3Tcombiner}
The simplest practical example of this architecture appears for the combination of three inputs. Here, the algorithm results in the formation of two enantiomorph nulled outputs. Those two outputs will, by subtraction, produce one robust observable. The resulting combiner matrix writes:
\begin{equation}\label{eq_3T_matrix}
\mathbf{M}_{3} =
\frac{1}{\sqrt{3}}\left[\begin{matrix}1 & 1 & 1 &\\
1 &
e^{\frac{2 j \pi}{3}} & e^{\frac{4 j \pi}{3}}\\
1 & e^{\frac{4 j \pi}{3}} & e^{\frac{2 j \pi}{3}}\end{matrix}\right].
\end{equation}
The combinations offered by this matrix are illustrated in Fig. \ref{fig_3x3}.\par
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{3_generalized_nulls.pdf}
\caption{CMP for a three-input kernel nuller of Eq. (\ref{eq_3T_matrix}). The first row corresponds to the bright channel with the overlapping phasors staggered for readability. The two nulled outputs are complex conjugates of one-another and will form a kernel null.}
\label{fig_3x3}
\end{figure}
As an example, we built a response map of the robust observable produced by this combiner fed by three of the VLTI \citep{vonderLuhe1997} unit telescopes (UTs) observing at zenith. Figure \ref{fig_3T_maps} shows the values of the kernel-null observable represented as a two-dimensional function of the relative position of a source normalized by the flux of one aperture. While simpler than the one provided in Fig. 7 of \cite{Martinache2018} for the four-input combiner, this pattern retains the same antisymmetric property.\par
\begin{figure}[H]
\centering
\includegraphics[width=0.45\textwidth]{3_generalized_pupils.pdf}\par
\includegraphics[width=0.40\textwidth]{3T_kernel_response.pdf}
\caption{Top: the three telescope configuration picked for the example and corresponding to the position of three of the VLTI UTs. Bottom: the value of the kernel null as a function of the relative position of a source of unit contrast at $3.6\mu m$, normalized to the throughput of one aperture. The map is relevant for a target observed at Zenith and would evolve with the projected aperture map. Note the antisymmertric nature of the response, as demonstrated in Sect. \ref{sec_odd_response}}
\label{fig_3T_maps}
\end{figure}
Assuming the practical implementation of the combiner itself can be manufactured either with bulk or integrated optics, this configuration would allow the production of robust high-contrast observables with the least amount of infrastructure. Drawing a parallel between this type of combinations and closure triangles used for closure phases is tempting but misleading. Here, as the combination must be done optically rather than in post-processing, kernel nulling does not scale in the same way. The advantages and drawbacks of using these simple combiners as building blocks is briefly discussed in Sect. \ref{sec_modular}.\par
\subsection{Six-input kernel nuller}\label{sec_6T_combiner}
We also outline a solution for a kernel-nulling recombiner for six telescopes that could, for example, be used at the focus of the CHARA array. The initial algorithm produces a combiner matrix $\mathbf{M}_6$ with $121$ rows with redundancy in the off-axis response. It is cropped to $\mathbf{M}_6'$ using the guidelines offered in Sect. \ref{sec_info_redundancy} to reduce it to the minimum of $21$ rows while making sure the number of independent kernel nulls $n_{kn}$ is preserved. Furthermore, by enforcing the properties outlined in Appendix \ref{sec_normalization}, we make sure that $\mathbf{M}_6'$ remains the matrix of a lossless beam combiner.\par
The matrix describing this 6-input combiner writes:
\begin{equation}\label{eq_6T_matrix}
\begin{split}
\mathbf{M}_6' =
\frac{1}{\sqrt{6}}\frac{1}{\sqrt{4}}\left[\begin{matrix}2 & 2 & 2 & 2 & 2 & 2\\1 & e^{\frac{5 j \pi}{3}} & e^{\frac{4 j \pi}{3}} & -1 & e^{\frac{j \pi}{3}} & e^{\frac{2 j \pi}{3}}\\1 & e^{\frac{j \pi}{3}} & e^{\frac{2 j \pi}{3}} & -1 & e^{\frac{5 j \pi}{3}} & e^{\frac{4 j \pi}{3}}\\1 & -1 & e^{\frac{j \pi}{3}} & e^{\frac{2 j \pi}{3}} & e^{\frac{5 j \pi}{3}} & e^{\frac{4 j \pi}{3}}\\1 & -1 & e^{\frac{5 j \pi}{3}} & e^{\frac{4 j \pi}{3}} & e^{\frac{j \pi}{3}} & e^{\frac{2 j \pi}{3}}\\1 & e^{\frac{4 j \pi}{3}} & e^{\frac{5 j \pi}{3}} & e^{\frac{2 j \pi}{3}} & e^{\frac{j \pi}{3}} & -1\\1 & e^{\frac{2 j \pi}{3}} & e^{\frac{j \pi}{3}} & e^{\frac{4 j \pi}{3}} & e^{\frac{5 j \pi}{3}} & -1\\1 & -1 & e^{\frac{2 j \pi}{3}} & e^{\frac{5 j \pi}{3}} & e^{\frac{4 j \pi}{3}} & e^{\frac{j \pi}{3}}\\1 & -1 & e^{\frac{4 j \pi}{3}} & e^{\frac{j \pi}{3}} & e^{\frac{2 j \pi}{3}} & e^{\frac{5 j \pi}{3}}\\1 & e^{\frac{4 j \pi}{3}} & e^{\frac{2 j \pi}{3}} & e^{\frac{j \pi}{3}} & e^{\frac{5 j \pi}{3}} & -1\\1 & e^{\frac{2 j \pi}{3}} & e^{\frac{4 j \pi}{3}} & e^{\frac{5 j \pi}{3}} & e^{\frac{j \pi}{3}} & -1\\1 & e^{\frac{j \pi}{3}} & e^{\frac{2 j \pi}{3}} & e^{\frac{4 j \pi}{3}} & -1 & e^{\frac{5 j \pi}{3}}\\1 & e^{\frac{5 j \pi}{3}} & e^{\frac{4 j \pi}{3}} & e^{\frac{2 j \pi}{3}} & -1 & e^{\frac{j \pi}{3}}\\1 & e^{\frac{4 j \pi}{3}} & e^{\frac{j \pi}{3}} & -1 & e^{\frac{2 j \pi}{3}} & e^{\frac{5 j \pi}{3}}\\1 & e^{\frac{2 j \pi}{3}} & e^{\frac{5 j \pi}{3}} & -1 & e^{\frac{4 j \pi}{3}} & e^{\frac{j \pi}{3}}\\1 & e^{\frac{2 j \pi}{3}} & e^{\frac{5 j \pi}{3}} & e^{\frac{j \pi}{3}} & -1 & e^{\frac{4 j \pi}{3}}\\1 & e^{\frac{4 j \pi}{3}} & e^{\frac{j \pi}{3}} & e^{\frac{5 j \pi}{3}} & -1 & e^{\frac{2 j \pi}{3}}\\1 & e^{\frac{5 j \pi}{3}} & -1 & e^{\frac{j \pi}{3}} & e^{\frac{4 j \pi}{3}} & e^{\frac{2 j \pi}{3}}\\1 & e^{\frac{j \pi}{3}} & -1 & e^{\frac{5 j \pi}{3}} & e^{\frac{2 j \pi}{3}} & e^{\frac{4 j \pi}{3}}\\1 & e^{\frac{5 j \pi}{3}} & -1 & e^{\frac{4 j \pi}{3}} & e^{\frac{2 j \pi}{3}} & e^{\frac{j \pi}{3}}\\1 & e^{\frac{j \pi}{3}} & -1 & e^{\frac{2 j \pi}{3}} & e^{\frac{4 j \pi}{3}} & e^{\frac{5 j \pi}{3}}\end{matrix}\right] .
\end{split}
\end{equation
This combiner offers a total of 20 independent nulls, and 10 independent kernel nulls. The corresponding CMP is showed in Fig. \ref{fig_6T_combiner} and highlights how each aperture contributes to all of the outputs. \par
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{6T_recipe_0.pdf}
\caption{Representation of the 20 nulled outputs of a six input beam combiner proposed in Eq. (\ref{eq_6T_matrix}). The conjugate pairs that form the 10 kernel nulls are represented side-by-side.}
\label{fig_6T_combiner}
\end{figure}
To illustrate the astrophysical information gathered by the larger number of kernel nulls, we construct response maps of the kernel-null observables. These plots, shown in Fig. \ref{fig_6T_maps} display the response of each of the observables for the combiner being fed by the CHARA array observing a target at zenith in the $3.6\mu m$ wavelength. The patterns reflects the richness of the uv coverage provided by an array like CHARA, and the fact that each output uses information collected by every telescope. As a consequence, each map covers the field of view differently, and brings new constraint on the properties of the astrophysical scene observed.\par
\begin{figure}
\centering
\includegraphics[width=0.3\textwidth]{6T_chara_pupils.pdf}\par
\includegraphics[width=0.45\textwidth]{6T_recipe0_kernel_response.pdf}
\caption{Top: the six telescope configuration for the CHARA array used as example. Bottom: the value of all 10 kernel nulls as a function of the relative position of a source at the wavelength $3.6\mu m$ observed at zenith. The transmission is normalized by the flux of a single aperture. Again, each map remains antisymmetric.}
\label{fig_6T_maps}
\end{figure}
\section{Discussion}
No active long-baseline optical interferometer currently provides more than six sub-apertures. However, the masking of monolithic apertures to produce interferometric arrays is an established practice \citep{Tuthill2010a, Jovanovic2012} that may be used in conjunction to nulling interferometry \citep{Norris2019}. Therefore, the use of even larger combiners may prove to be a viable alternatives to small inner working angle coronagraphs \citep{Guyon2006a}. Their robustness to small aberrations might provide unprecedented contrast performance in the $1-3 \lambda /D$ regime in the near infrared.\par
\subsection{Multiplexing nullers} \label{sec_modular}
Instead of building an all-in-one combiner, which may be difficult to construct for a large number of apertures, an alternative approach would be to multiplex several independent nullers, that each recombines a smaller number of apertures. For example, instead of a six-input nuller producing 20 nulled outputs, one conservative option would be to use two three-input kernel nullers, identical to the one presented in Sect. \ref{sec_3Tcombiner}, side by side producing four nulled outputs.\par
While the latter of these two options results in a reassuring higher throughput per output, it can only produce distinct robust observables, where the $\mathbf{M}_6'$ combiners offers 10. Moreover, this multiplexed option also results in two bright channels, where some of the off-axis light is also lost, further reducing the overall efficiency of the combiner. In between these two extreme scenarios, intermediate solutions can be imagined to alleviate some of their risks and deficiencies, with a modular design multiplexing the nullers much like a number of ABCD combiners are multiplexed inside the beam combiner of VLTI/GRAVITY.
It was already argued by \cite{Guyon2013} that efficient nulling solutions concentrate the most starlight into the smallest number of outputs, which favors the all-in-one combiner over the multiplexed versions. If manufacturability or operational constraints were to prevent the deployment of an all-in-one combiner at the focus of a specific observatory, one way to alleviate this inefficiency could be to recombine the light from the multiple bright outputs, into an additional nulling stage so as to extract additional useful observables. This type of architecture, in part inspired by the hierarchical fringe tracker idea of \cite{Petrov2014}, might prove a necessary compromise to the implementation of a kernel nuller at the focus of a very long baseline observing facility such as the envisioned Planet Formation Imager \citep{Monnier2016}, for which a distributed hierarchical recombination mode seems particularly apt.
\subsection{Evolution of robustness}
In addition to trade-off considerations between the total number of observables and the throughput efficiency of the available options, one must also consider whether the number of inputs has an impact on the phase-noise rejection performance of a kernel nuller.\par
To evaluate this risk, we trace the evolution of the noise affecting the outputs and their kernels as a function of the amount of phase noise affecting the inputs. We do this for the 3T, 4T and the 6T designs described in the previous sections. For simplicity, this study assumes that the phase noise affecting all inputs is Gaussian, non-correlated, and characterized by a single rms value equally affecting all inputs. Following a Monte Carlo approach, random realizations of input piston errors are drawn, propagated through the different combiner matrices, and the standard deviation of the output intensities are evaluated. Figure \ref{fig_plot_comparison} thus shows the evolution of the standard deviation of the output intensities of the nulls and of the kernel nulls of the different architecture normalized by the peak null intensity $I_{peak}$ from their response map. This therefore constitutes a noise-to-signal ratio of sorts. The simple Bracewell nuller is also added to this study for comparison, as modelled by the combiner matrix:
\begin{equation}
\mathbf{M}_B = \left[\begin{matrix}1 & 1\\
- j & j\end{matrix}\right] .
\end{equation}
\par
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{perf_comparison.pdf}
\caption{Propagation of phase noise from the input to the nulled intensities for different kernel-nulling architectures. Values are normalized by the peak transmission of an off-axis signal. The raw nulls (dashed lines) are compared with their corresponding kernel-null observable (solid lines), showing the suppression of 2nd order phase noise (by a few orders of magnitude) for small input phase error. This effect decreases as the input phase errors depart from the small phase approximation. The behavior of the Bracewell nuller is shown in dashed blue for reference.}
\label{fig_plot_comparison}
\end{figure}{}
This plot shows that the larger kernel-nulling combiners provide a rejection of the phase noise that is very similar to the smaller ones, if not slightly better. The improvement on the raw observables may be credited to a manifestation of the central limit theorem affecting the distribution of the sum of a larger number of complex intensities. Further interpretation of this plot must be undertaken with caution. Indeed, while the distribution of kernel nulls under such conditions is close to Gaussian \citep{Martinache2018}, the distribution of null intensities is not \citep{Hanot2011} and is therefore poorly described by its standard deviation. While a full performance comparison of the different designs lies outside the scope of the present paper and would include the coupled effects of phase and amplitude fluctuations \citep{Lay2004}, these elements already indicate that kernel nullers recombining a large number of sub-apertures are intrinsically at least as robust to phase noise as their smaller, simpler counterparts. This is an encouraging prospect for single telescope applications of the kernel nuller for which a potentially large number of sub-apertures can be used.
\section{Conclusions}
In this work, we offer a new description of the kernel-nuller design introduced by \cite{Martinache2018}. This is done by introducing a new graphical representation of the complex matrix that models the nuller and the transformations it operates on the input electric field. Combined with an analytical description of the outputs used to form a kernel, these representations explain the origin of a kernel nuller's main properties: their intrinsic robustness to small input piston and amplitude fluctuations, and their sensitivity to asymmetric features of the observed scene. We incidentally show that the same outputs can also be summed so as to fall back on the original outputs of an all-in-one $\mathbf{N}_4$ nuller stage, while not robust to perturbations, can nevertheless provide further astrophysical information.\par
Our graphical and analytical representations help devise a systematic way to build a kernel nuller as a combiner featuring pairs of channels that are enantiomorph in the complex plane. It is this feature that makes two channels equally sensitive to perturbations although they respond differently to the presence of an off-axis structure. This approach allow us to design kernel nullers for an arbitrary number of apertures, which we here apply to three- and six-aperture arrays.\par
We discuss the possibility of simplifying kernel nullers that grow in complexity when they recombine a larger number of input beams, for instance, using distinct nullers operating in parallel over a subset of input beams. For a given total number of inputs, a global architecture, giving access to a larger number of high-contrast observables is more efficient and offers the means to explore and characterize complex astrophysical scenes. For the same number of inputs, we can also note that the total number of outputs for a kernel nuller (exactly twice the number of theoretical closure-phases) is in fact less than that of non-nulling combiners designed to measure the complex visibility of all baselines. Integrated optical circuits in particular already enable the implementation of such complicated designs in small and stable packages, and are a very promising avenue for the construction of these larger kernel-nulling combiners.\par
While no existing long baseline optical interferometric facility currently offers the simultaneous combination of more than six apertures, a kernel nuller sampling the pupil of a single telescope could prove to be a valuable complement to a coronagraph, producing high contrast observations near one resolution element that would be insensitive to the small but ever present adaptive optics residuals. The evaluation of performance in practical implementations including the contribution of coupled phase an amplitude contributions and the consideration of relevant science cases will be the topic of future theoretical and experimental work.
\begin{acknowledgements}
We thank Alban Ceau and Coline Lopez for their suggestions to improve the manuscript.
KERNEL has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement CoG - 683029).
\end{acknowledgements}
\bibliographystyle{aa}
| {
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} | 4,694 |
Lawsuit Filed to Stop "Barbaric" Wild Horse Sterilization Surgeries
WASHINGTON, DC (December 11, 2020) —The American Wild Horse Campaign (AWHC) and Utah citizen Robert Hammer filed suit Thursday in U.S. District Court for the District of Columbia against the U.S. Department of the Interior to stop the Bureau of Land Management (BLM) from performing risky and inhumane surgical sterilization surgeries on federally-protected wild mares (female horses) recently rounded up in Utah. The lawsuit was filed on behalf of AWHC by Eubanks and Associates, one of the nation's top public interest environmental law firms.
At issue is the BLM's decision to proceed with the invasive procedure despite significant public and congressional outcry, veterinary opposition, and a National Academies of Sciences recommendation against using the surgery for the management of wild horses.
"We've successfully sued previously to stop the BLM from galloping ahead with its irresponsible plan to conduct risky, painful, and invasive surgeries on wild horses and we intend to win again," said Brieanah Schwartz, policy counsel for the AWHC. "The BLM's decision to subject wild mares to this inhumane procedure violates four federal laws and is particularly egregious in light of humane, scientifically-recommended and cost-effective management alternatives such as the PZP birth control vaccine."
Over the last four years, AWHC and its coalition partners filed successful litigation to block three previous attempts by the BLM to perform the controversial procedure on wild horses in Oregon. The agency has now turned its sights on Utah and a group of wild mares who were captured in a roundup that ended Wednesday in the Confusion Herd Management Area (HMA).
The procedure in question - "ovariectomy via colpotomy" - is widely regarded to be an outdated and inhumane surgical procedure. It involves a veterinarian inserting their arm into a mares' abdominal cavity through an incision in the vaginal wall, manually locating the ovaries, then twisting, severing, and removing them using a rod-and-chain tool called an ecraseur.
"I find it perplexing that the BLM continues to push their population management philosophy in such a barbaric direction, particularly when effective, cost-efficient and, above all, humane alternatives are available," said Robert Hammer, author and photographer, native Utahan, and plaintiff in the suit said in a statement. "The agency seems to give little or no consideration to the welfare of the mares they would subject to this cruel experimental procedure, nor the bands that rely upon intact adults of both genders for their ongoing health and survival."
Video of the ovariectomy procedure can be viewed here. It was taken by Simone Netherlands, president of the Salt River Wild Horse Management Group, who attended a demonstration workshop that resulted in the deaths of a horse and a burro and severe infections in three additional burros. Netherlands concludes that ovariectomies are "100% not a viable or humane option for wild horse management." She wrote an affidavit describing her observations.
When used infrequently for domestic mares, it requires adequate anesthesia and rigorous post-operative care -- including pain relief, antibiotics, and stall confinement -- that cannot be provided to wild, untamed horses. The main reason the procedure is performed in domestic horses is to alter their behaviors, something that is harmful to wild horses whose natural free-roaming behaviors allow them to survive in the wild.
The American Wild Horse Campaign (AWHC) is the nation's leading wild horse protection organization, with more than 700,000 supporters and followers nationwide. AWHC is dedicated to preserving the American wild horse in viable, free-roaming herds for generations to come, as part of our national heritage.
Robert Hammer is a native Utahan, photographer, and author of the book, Salt Desert Mustangs: Discovering wild horses and historic trails in Tooele County, Utah. He runs Wild Horse Tourist, an informational website on viewing America's wild horses and burros. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,758 |
**_Hell's
_****_Cartographers_**
_Edited by_
Brian W. Aldiss
_and_
Harry Harrison
Dover Publications, Inc.
Mineola, New York
_Bibliographical Note_
This Dover edition, first published in 2018, is an unabridged republication of the work originally published by Weidenfeld & Nicolson, Great Britain, in 1975.
_International Standard Book Number_
_ISBN-13: 978-0-486-82448-2_
_ISBN-10: 0-486-82448-9_
Manufactured in the United States by LSC Communications
82448901 2018
www.doverpublications.com
**Contents**
Introduction
[Robert Silverberg:
Sounding Brass, Tinkling Cymbal](06_Robert_Silverberg__Sounding_Bras.html#page_7)
[Alfred Bester:
My Affair With Science Fiction](07_Alfred_Bester__My_Affair_With_Sc.html#page_46)
[Harry Harrison:
The Beginning of the Affair](08_Harry_Harrison__The_Beginning_of.html#page_76)
[Damon Knight:
Knight Piece](09_Damon_Knight__Knight_Piece.html#page_96)
[Frederik Pohl:
Ragged Claws](10_Frederik_Pohl__Ragged_Claws.html#page_144)
[Brian Aldiss:
Magic and Bare Boards](11_Brian_Aldiss__Magic_and_Bare_Boa.html#page_173)
Appendices:
How We Work
Selected Bibliographies
About the editors:
Harry Harrison was born in Stamford, Connecticut, in 1925 and died in southern England in 2012. Apart from enjoying an enviable reputation as one of the best writers on the science fiction scene, he was also an experienced editor, anthologist, translator and cartoonist. Harrison's style leaps from the humour of his _Stainless Steel Rat_ to the splendid combination of fantasy and sf found in _Captive Universe_ and the _Deathworld_ trilogy.
Brian Aldiss was born in Dereham, a town in Norfolk, England, in 1925 and lived with his family in Oxford, England, until his death in 2017. Voted Great Britain's most popular science fiction writer by the British Science Fiction Association and awarded the Ditmar for being the Best Contemporary Writer of sf, he also won a Hugo Award for _Hothouse_ and a Nebula Award for _The Saliva Tree._
**Introduction**
A few years ago, there was a man living down in Galveston or one of those ports on the Gulf of Mexico who helped make history. He did not enjoy that honour – a feeling shared by many who find themselves in that position.
His name was Claude Eatherly, and at one time he was something of a legend. For all I know, he still lives down in Galveston, for all I know he still feels himself to be one of the scapegoats of history. For Major Claude Eatherly, back in 1945, piloted the weather plane which flew over Hiroshima and reported that cloud conditions were suitable for the dropping of the first A-bomb.
The responsibility for the deaths which followed rode hard on Eatherly's shoulders, although nobody until then had mistaken him for a thinking man. He liked drink, gambling, women, and horseplay, and read nothing more profound than comic books. Then he got mixed up with lethal technology.
After the war, Eatherly became a misfit and eventually a jailbird, before being turned into a myth-figure by some of the dark father-figures of our time – politicians, psychiatrists, philosophers, preachers, and publicists. Even Bertrand Russell weighed in.
It was not the importance of Eatherly's life as such. The extraordinary crucifying incident in which he was involved was what gave him significance.
This volume contains brief autobiographies by six eminent science fiction writers. With only one exception, we are all within a few years of the same age. We were all old enough to appreciate the spectacle of the mushroom clouds rising over Hiroshima and Nagasaki. There are few things more vexing than modesty exercised upon someone else's behalf, so I trust my contributors will forgive me if I say that some of our interest is, like Eatherly's, extrinsic.
For the atomic bomb meant something particular to science fiction writers and readers. Despite our differences, we held at least two items of faith unshakably. One of them was – and it is curious to look back to the forties and see how absolutely bizarre, lunatic even, was this faith then – that space travel waited just round the corner (well, so it did, and we were right, although on many of the details we were instructively wrong).
The other item of faith concerned science fiction itself, by which was meant at that time magazine science fiction, disreputable stuff which has only recently been graced by such sociologists' terms as Alternative Literature. We believed that sf was of genuine merit. More, we saw those merits as being unique.
Whatever else the A-bomb meant to Eatherly and all the rest of mankind, to a small handful of us it meant vindication. We who had been regarded as mad were proved dangerously sane. The Future had happened, and blown the lid off the Old Order.
From then on, we wrote sf with greater confidence, treated it more seriously, and were ourselves treated by our critics with slightly less scorn and by our readers with positive veneration. Never have critics and readers in any field been more divided than they are over sf.
Science fiction, to my mind, is not a matter of prediction, and never has been, although prediction is one of the ingredients which makes it fun. Rather, it mirrors the present in such a way as to dispense with inessentials and dramatize new trends. In my own fiction, each decade would typically present some central image which differed from the one before: in the forties and early fifties, a bleak landscape cleared of people by some almost-forgotten catastrophe; in the fifties, men imprisoned in huge spaceships and technologies; in the sixties, men's minds altered by drugs or engines; and now – well, maybe a great windjammer, fully automated and computerized, bearing the goods that formerly went by air, its complex rig of sails operated without the need of human crew. . . . There's always a new scenario round the corner.
My thought was to invite the men who have been most successful in inventing such fictional scenarios to write a brief memoir of themselves. They were asked to be as frank as possible about their lives and to discuss their involvement in the world of science fiction.
The result is a book of unique significance. We have been the weather men flying above alien cities, and we have not delivered our reports before. When we began to write, it seemed as if we were doomed by our beliefs to work in obscurity. Yet it turned out that there was something prodromic in our approach to life; what we had to say proved to be on a subject with which millions of people of our generation were concerned; and, as a result, our books have been reprinted and translated all round the world (not least in Eatherly's old target, Japan, one of sf's global capitals). We are an entirely new sort of popular writer, the poor man's highbrows.
We wrote against the grain and were accepted against it. We wrote for kicks and ha'pence. There is a certain emphasis on finance in our memoirs, and with reason; for the smaller the payment, the larger it looms. We had faith in what we were doing; individualists though we were, it transpired that the faith virtually created a movement. A lot of people needed to re-dream our nightmares.
What we see today is the too-easy acceptance of sf. The sharp idiom we created has blurred to become one of the bland flavourings of mass media; the unembarrassed muse we espoused is one of the jades of television. And the younger writers now writing have an entirely different approach to their art. They have found how easy it is to rely on formula, or how simply success can come through self-advertisement.
For us it was different.
Well, that's a good motto for this volume. Where I think the difference showed in our work was that, for all our sleight-of-hand with the wonders of space-and-time, our fiction gained its power by having as unspoken topic one of the great issues of the day: the sense that the individual's role in society is eroded as society itself becomes wealthier and more powerful. This is certainly so with novels as unalike as Pohl and Kornbluth's _The Space Merchants,_ Silverberg's _The Time Hoppers,_ and Knight's A _for Anything._ Harrison and Bester, in their most characteristic fiction, allow the individual much more latitude; their heroes can save worlds or defeat the solar system; but nobody who ever meets them is likely to forget the oppressions of the decadent society in _Tiger! Tiger!_ or of the hunger-line crowds of _Make Room! Make Room!_
I chose the men I did because they were friends of mine, though not always particularly close friends, since the Atlantic separates us much of the time; although it is true to say that I have danced the samba with Damon Knight's wife, Kate Wilhelm – and (darn it) Damon at the same time, because it was one of those mad nights in Rio de Janeiro; while I have been reasonably stoned with Fred Pohl and his wife Carol in the Tokyo hotel room of our Russian pal Julius Kagarlitski; and so on.
They were also chosen because I admired their innovations in sf. Knight published _In Search of Wonder,_ the first book of critical reviews of sf, and it would be hard to overestimate the influence of his cool appraisals in a field over-fond of puffery. Bester was quite simply the popular writer who showed greatest verve and swagger in short stories and novels; although he does not realize it, he is something of a cult figure in England. One man in his time plays many parts; Pohl has played all the parts in the sf world, fan, editor, writer, adviser, ambassador. Among his other lesser virtues, he was the first guy in the States to buy one of my stories. Silverberg made a great deal of money from sf; in his exemplary piece, he relates how sf made him a millionaire – and it is a story which gives me a great deal of pleasure. He provides a deep insight into what it means to be a popular writer.
Harrison had to be in here, simply because we co-edit books. We co-edit books because we work well together and get pleasure from so doing. I cannot recall all the odd places in which Harrison and I have found ourselves together. My life would be poorer in many ways without his friendship.
The sixth writer is me because I could not bear to be left out. I originally approached seven writers. The seventh was Michael Moorcock. Because there has always been a bond between Moorcock and me; because I am one of the handful of people who know just how much lifeblood Moorcock gave to his sf magazine. Moorcock was the only guy who said he could not talk about himself. True modesty.
The rest of us, happily, have no such qualms. I hope at a future date to produce a second – who knows, even a third and fourth – volume of Hell's Cartographers, since I am convinced that this is the way in which I can most easily earn posterity's gratitude. And the book, in a sense, follows naturally after my history of science fiction.
The overall interest in this volume resides, I think, in the fact that vital parameters of our six lives lie between the Bomb and the Apollo. These two events mark out our sceptical approach to life from both those writers of a generation before us and those of a later generation. We have all been keeping the reading public reading for two or more decades; this is how and why we did it.
My particular thanks go to Miss Nancy Neiman of Weidenfeld & Nicolson for all her understanding and assistance.
Brian W. Aldiss
Heath House
Southmoor
February 1974
**Robert Silverberg:**
**sounding brass, tinkling cymbal**
' . . . _and even Silverberg, who sometimes, with all his skill and knowledge and sophistication, does tend to the androidal._...'
John Clute in _New Worlds 5_
_Though I speak with the tongues of men and of angels, and have not charity, I am become as sounding brass, or a tinkling cymbal._
_And though I have the gift of prophecy, and understand all mysteries, and all knowledge; and though I have all faith, so that I could remove mountains, and have not charity, I am nothing._
I Corinthians, 13
At last to speak of one's self. An odd temptation, which mostly I have resisted, in the past, maintaining that I'm not yet ready to undertake a summing up, or that I'm in the midst of some intricate new transition still not fully understood, or that I'm bored with myself and talking about myself. Yet I have granted all sorts of interviews, and spoken quite explicitly, all the while protesting my love of privacy; the one thing I've never attempted is explicit written autobiography. I manage to hold all poses at once, modest and exhibitionistic, esthete and man of commerce, puritan and libertine: probably the truth is that I have no consistent positions at all. We'll see.
Autobiography. Apparently one should not name the names of those one has been to bed with, or give explicit figures on the amount of money one has earned, those being the two data most eagerly sought by readers; all the rest is legitimate to reveal. Very well. The essential starting point, for me, is the confession (and boast) that I am a man who is living his own adolescent fantasies. When I was sixteen or so I yearned to win fame as a writer of science fiction, to become wealthy enough to indulge in whatever amusements I chose, to know the love of fair women, to travel widely, to live free from the pressures and perils of ordinary life. All these things have come to me, and more; I have fewer complaints to make about the hand destiny has dealt me than anyone I know. Here at what I assume is my midpoint I feel a certain inner security, a self-satisfaction, which I suppose borders occasionally on smugness. (But not on complacency. The past is unchangeable and the present delightful, yet the future still must be regarded warily. I live in California, a land where the earth might literally open beneath my feet this afternoon; and I've already once had, in my pre-California incarnation, the experience of awakening before dawn to find my world in flames.)
Because my life has been so generally satisfactory, and because I'm a literary enough man to know the dangers of _hubris,_ I sometimes affect a kind of self-deprecatory shyness, a who- _me?_ kind of attitude, whenever I am singled out for special attention. This pose gets more and more difficult to maintain as the years go on and the accomplishments and money and awards pile up; by now certain objective measures of achievement exist, for me, and there's an element of hypocrisy in trying to deny them purely for the sake of trying to avoid the fate that chops down the boastful. Ten years ago, or even five, I probably would have refused the opportunity to contribute to this book, claiming that I was unworthy (and privately fearing that others would say so if I did not). To hell with that now.
I am the youngest of the six contributors here: the youngest by nearly a decade, I suspect, since as I write this I'm still more than a year short of my fortieth birthday, and my companions, I know, all cluster around the half-century mark. A familiar feeling, that one. I was always the youngest in any group, owlishly precocious, a nastily bright little boy who was reading at three, writing little stories at six, spouting learned stuff about European dynasties and the sexual habits of plants at seven or eight, publishing illegible magazines at thirteen, and selling novels at eighteen. I was too unruly and too clever to remain in the same class at school with my contemporaries, so I grew up two years younger than all my friends, thinking of myself as small and weak and incomplete. Eventually, by surviving, I caught up with everyone. I am the oldest in my immediate circle of friends, with a beard alas now tinged with grey, and I am as tall as most and taller than many, and within the tiny world of science fiction I have become something of an elder statesman, and the wounds I received by being fourteen years old in a universe of sixteen-year-olds are so well sheathed in scar-tissue now that I might as well consider them healed. And yet it still is strange to be included as an equal in this particular group of writers, since three of them–Alfred Bester, Damon Knight, Frederik Pohl – were among my own literary idols when I was indulging in those adolescent fantasies of a writer's career twenty-odd years ago. A fourth, Harry Harrison, had not yet begun writing seriously then himself, but he was the editor who first paid me for writing anything, in 1953; and only Brian Aldiss, the originator of this project, played no part in shaping me in my teens, for I had never heard his name until I myself was an established writer. Yet I make no apologies for being here among my elders. Here we all are: professional writers, diligent craftsmen, successful creators – artists, if you will. And good friends as well.
I am an only child, born halfway through the Great Depression. (There would have been a sibling, I think, when I was about seven, but it miscarried; I often wonder what pattern my life would have taken had I not grown up alone, pampered, self-indulgent.) My ancestors were Jews from Eastern Europe, and my grandparents, three of whom survived well into my adulthood, were reared in Poland or Russia in villages beyond my easy comprehension. My father was born in London in the first year of this century, and came to the United States a few years thereafter. My mother was born in Brooklyn, New York, and so was I.
I have no very fond recollections of my childhood. I was puny, sickly, plagued with allergies and freckles, and (I thought) quite ugly. I was too clever by at least half, which made for troubles with my playmates. My parents were remote figures; my father was a certified public accountant, spending his days and many of his evenings adding up endless columns of red figures on long yellow sheets, and my mother taught school, so that I was raised mainly by Lottie, our mulatto housekeeper, and by my loving and amiable maternal grandmother. It was a painful time, lonely and embittering; I did make friends but, growing up in isolation and learning none of the social graces, I usually managed to alienate them quickly, striking at them with my sharp tongue if not my feeble fists. On the other hand, there were compensations: intelligence is prized in Jewish households, and my parents saw to it that mine was permitted to develop freely. I was taken to museums, given all the books I wanted, and allowed money for my hobbies. I took refuge from loneliness in these things; I collected stamps and coins, harpooned hapless butterflies and grasshoppers, raided the neighbours' gardens for specimens of leaves and flowers, stayed up late secretly reading, hammered out crude stories on an ancient typewriter, all with my father's strong encouragement and frequent enthusiastic participation, and it mattered less and less that I was a troubled misfit in the classroom if I could come home to my large private room in the afternoon and, quickly zipping through the too-easy homework, get down to the serious business of the current obsessional hobby.
Children who find the world about them distasteful turn readily to the distant and the alien. The lure of the exotic seized me early. These were the years of World War II and real travel was impossible, but in 1943 a friend of my father's gave me a subscription to the _National Geographic Magazine,_ and I was off to Zanzibar and Surinam and Jamaica in my imagination decades before I ever reached those places in actuality. (Typically, I began buying old _National Geographics_ with lunatic persistence, and didn't rest until I had them all, from the 1880's on. I still have them.) Then, an hour's journey from home on the subway, there was the American Museum of Natural History, with its mummies and arrowheads, its mastodons and glyptodons, above all its brontosaurs and tyrannosaurs; Sunday after Sunday my father and I made the pilgrimage, and I revelled in the wonders of prehistory, soberly lecturing him on the relative chronological positions of Neanderthal and Peking and Piltdown Man. (Yes, Piltdown, this was 1944, remember.) From dinosaurs and other such fantastic fossils to science fiction was but a short journey: the romantic, exotic distant past is closely tied to the romantic, exotic distant future in my imagination.
So there was Jules Verne when I was nine – I must have taken that voyage with Captain Nemo a hundred times – and H. G. Wells when I was ten, most notably _The Time Machine_ (which promised to show me all the incredible eons I would never live to know) but also _The Island of Dr Moreau_ and _War of the Worlds,_ the myriad short stories, and even an obscure satire called _Mr Blettsworthy on Rampole Island,_ to which I often returned because Mr Blettsworthy encountered living ground-sloths. There was Twain's _Connecticut Yankee in King Arthur's Court,_ which also I read repeatedly. (How early my fascination with time travel emerged!) I dabbled in comic books, too, and I have gaudy memories of Buck Rogers and _Planet Comics._ But somehow I missed Edgar Rice Burroughs altogether; and it was not until early 1948, when I was already a veteran of scores of hardbound science fiction books, that I even knew such things as science fiction magazines existed.
The magazines mostly repelled me by their covers and their titles. I did buy _Weird Tales_ – my first one had an Edmond Hamilton novelette about the Norse gods, which delighted me since I had gone through whole libraries of Norse mythology in early boyhood. I bought _Amazing Stories,_ then the sleaziest representative of the genre, because it happened to publish an uncharacteristically respectable-looking issue about then. I bought John Campbell's dignified little _Astounding Science Fiction,_ but found the stories opaque and unrewarding to my thirteen-year-old mind. Because I was rather a snob, I would not even open magazines with names like _Thrilling Wonder Stories_ and _Famous Fantastic Mysteries_ and _Startling Stories,_ especially since their covers were bright with paintings of hideous monsters and scantily clad damsels. (Sex was very frightening to me just then, and I had sworn never to have anything to do with women.) More than a year passed before I approached those magazines in what was by then an unquenchable thirst for science fiction, and discovered they were publishing some of the best material of the day.
Then there were the books: the wondrous Healy-McComas _Adventures in Time and Space,_ the big Groff Conklin titles, Wollheim's _Pocket Book of Science Fiction,_ and the other pioneering anthologies. My father was more than a little baffled by my increasing obsession with all this trash, when previously I occupied myself with decent books on botany and geology and astronomy, but he saw to it that I bought whatever I wanted. One collection in particular had enormous impact on me: Wollheim's _Portable Novels of Science,_ published in 1945 and discovered by me three years later. It contained Wells' _First Men in the Moon,_ which amused me; Taine's _Before the Dawn,_ which fed my always passionate interest in dinosaurs; Lovecraft's _Shadow Out of Time,_ which gave me that peep into unattainable futures that originally led me to science fiction; and above all Stapledon's _Odd John,_ which spoke personally to me as I suppose it must to any child who is too bright for his own good. I was up almost till dawn reading that book, and those novels marked me.
I was at that time still talking of some sort of career in the sciences, perhaps in botany, perhaps in paleontology, perhaps astronomy. But some flaws in my intelligence were making themselves apparent, to me and to my teachers if not to my parents: I had a superb memory and a quick wit, but I lacked depth, originality, and consistency; my mind was like a hummingbird, darting erratically over surfaces. I wanted to encompass too much, and mastered nothing, and though I always got high marks in any subject that caught my interest, I noticed, by the time I was thirteen, that some of my classmates were better than I at grasping fundamental principles and drawing new conclusions from them. I doubt that I would have been of much value as a scientist. But already I was writing, and writing with precocious skill – for school newspapers and magazines, for my own abominably mimeographed magazine, and, without success, for professional science fiction magazines. Off went stories, double-spaced and bearing accurate word-counts (612, 1814, 2705). They were dreadful, naturally, and they came back, usually with printed rejection slips but sometimes – when the editors realized they were dealing with a bright child of thirteen or fourteen and not with a demented adult – with gentle letters suggesting ways I might improve my style or my sense of plot. And I spoke openly of a career in writing, perhaps earning my living as a journalist while writing science fiction as a sideline.
Why science fiction? Because it was science fiction that I preferred to read, though I had been through Cervantes and Shakespeare and that crowd too. And because writing science fiction allowed me to give free play to those fantasies of space and time and dinosaurs and supermen that were so gratifying to me. And because I had stumbled into the world of science fiction fandom, a world much more comfortable than the real world of bullies and athletes and sex, and I knew that my name on the contents page of _Astounding_ or _Startling_ would win me much prestige in fandom, prestige that I could hardly hope to gain among my classmates.
So, then, the stories went forth, awkward imitations on a miniature scale of my favourite moments out of Lovecraft or Stapledon or Taine or Wells, and the stories came back, and I read textbooks on the narrative art and learned a good deal, and began also to read the stories in the science fiction magazines with a close analytical eye, measuring the ratio of dialogue to exposition, the length of paragraphs, and other technical matters that, I suppose, few fifteen-year-olds study as carefully as I did. Nothing got published, or even came close, but I was growing in skill.
I was growing in other ways, too. When I was about fourteen I went off, for the first time, to summer camp, where I lived among boys (and girls) of my own age and no longer had to contend with being the youngest and puniest in my peer-group. I had always been known as 'Robert', but at camp I was speedily dubbed 'Bob', and it seemed to me that I was taking on a new identity. _Robert_ was that spindly misfit, that maladjusted, isolated little boy; _Bob_ was a healthy, outgoing, normal young man. To this day I wince when some stranger presumes on my public persona and addresses me as Robert – it sends me rocketing backward in time to the horrors of being ten again. Although I sign my stories _Robert_ for reasons of formality, my friends know me as _Bob,_ and my parents managed the transition fairly gracefully at my request (though my father sometimes slips, a quarter of a century after the change), and when I occasionally encounter some childhood friend I let him know, rapidly, the name I prefer and the reason I prefer it.
This new Bob was able to cope. He grew to a reasonable height, halting just a bit short of six feet; he became a passable athlete; he discovered how to sustain friendships and how to manage conversations. For a few years I led a split life, introverted and lonely and secretive at home, open and lighthearted and confident during the summers; and by the time I was about seventeen, some integration of the two lives had begun. I had finished high school (where I had become editor of the high-school newspaper and was respected for my skill as a writer) and, by way of surrendering some of my precocity, had declined to go immediately into college. Instead I spent a few months reading and writing, and a few months working in a furniture warehouse on the Brooklyn waterfront, among rough, tough illiterates who found my cultivated manner a charming novelty rather than a threatening intrusion, and then I went off to the summer camp, not as a camper but as an employee. In the autumn I entered Columbia University with old slates wiped clean: I was no longer morbidly too young, I was free of the local playmates who could never forget the maladjustments of my childhood, I was able to begin in the _Bob_ persona, without hauling the burden of my past problems.
I lived away from home, in a little apartment of my own. I manifested previously unknown skills for drinking and carousing. I discovered that women were not really very frightening after all. I plunged myself into new worlds of the mind: into Aquinas and Plato, into Bartok and Schoenberg, into Kafka, Joyce, Mann, Faulkner, Sartre. I continued to read science fiction, but dispassionately, with the eye of one who was soon to be a professional; I was less interested in visions of ultimate tomorrows and more in seeing how Messrs Bester, Pohl, Knight, Sheckley, Dick etc, carried off their tricks. One of my stories was published – for a fee of $5, I think – by an amateur magazine called _Different,_ operated by a poetess named Lilith Lorraine. Harry Harrison asked me to do an article about fandom for a science fiction magazine he was editing, and I turned in a competent journalistic job and was paid $30. That was in September 1953. I sent a short story called 'Gorgon Planet' off to a magazine called _Nebula,_ published in Scotland by Peter Hamilton, and in January 1954 he notified me that he would use it, and sent me his check for $12.60.
That same month I sold a novel to a major American publisher. The earlier sales could be brushed aside as inconsequential – two weak short stories accepted by obscure magazines, and one specimen of mere journalism – but the novel was something else. I was not yet nineteen years old, and I was a professional writer. I had crossed the threshold.
That novel! Its genesis went back almost three years. When I was editor of my high-school newspaper in 1951 a book appeared for review, a science-fiction novel for boys, published by the Thomas Y. Crowell Company, an old-line New York firm. Steeped as I was in Wells and Heinlein and Stapledon and such, I reviewed this clumsy, naive book scornfully, demolishing it so effectively that in the summer of 1953 the publishing company invited me to examine and criticize, prior to publication, the latest manuscript by that author. I read it and demolished it too, with such thoroughness that the book was never published. This time the Crowell editor asked me to the office and said, in effect, 'If you know so much about science fiction, why don't you try a novel for us yourself?' I accepted the challenge.
I had attempted a novel once before, at the age of thirteen. It began as two short stories, but I subsequently combined them, elaborated, padded most shamefully, and ended up with an inch-thick manuscript that must have been one of the least coherent hodgepodges ever committed to paper. The outline of the book I suggested to Crowell in September 1953 was better, but not much. It concerned the trip of four young space cadets to Alpha Centauri on a sort of training cruise. No plot, not much action. The cadets are chosen, leave for space, stop at Mars and Pluto, reach Alpha Centauri, become vaguely entangled in a revolution going on there, become disentangled and go home. Some novel.
Every weekend that autumn I wrote two or three chapters, working swiftly despite the pressures of college. When eight chapters were done I submitted them and received an encouraging note urging me to complete the book. It was done by mid-November: nineteen chapters, 145 pages of typescript. I sent it in, heard nothing for two months, and on a Sunday in January 1954, received a stunning telephone call from the Crowell editor: they were sending me a contract for my novel. Of course, some changes would be required before it could be published.
In March I was sent a severe four-page letter of analysis. Anticlimax after anticlimax, they said; first part of book fine, last half terrible. Though immensely discouraged, I set to work rewriting, trying to build complications and a resolution into my rudimentary story. On 5 June this revision came back to me: I had allowed my main protagonist to achieve his goal by default rather than by positive action, and the publishers wouldn't let me get away with that. I promised to spend the summer considering ways to restructure the book; meanwhile Crowell would consult an outside reader for suggestions and evaluations.
The summer passed. I did no writing, though I began vaguely to hatch a completely new plot turning on my hero's climactic conversion to the revolutionary party. At the end of October the long-awaited reader's report on the manuscript landed in the mailbox of my campus apartment. It made the job I had done on that unpublished book the year before look like praise. What was wrong, I learned, was that I really didn't know how to write. I had no idea of characterization or plotting, my technique was faulty, virtually everything except my typing was badly done. If possible, the reader said, I should enroll in a writing course at New York University.
A year earlier, I might have been crushed; but by the autumn of 1954 I had sold a couple of competent if uninspired short stories, I had written five or six more that seemed quite publishable to me (ultimately, I sold them all), and I felt that I had a fairly firm technical grasp on the art of fiction, however faulty the execution of my novel might be at the moment. Instead of abandoning the project, I spent three hours considering what I could do to save it, and in the afternoon I telephoned my editor to tell her that I proposed a total rewrite based on the conversion-to-revolution theme. By this time she must have come to doubt her original faith in my promise and talent, but she told me to go ahead.
I knew this was my last chance. The first step was to throw out the first nine chapters, which had survived intact through all the earlier drafts. They were good, solid chapters –it was the end of the story that was weak, not the beginning – but they had little relevance to my new theme. I compressed them into two pages and got my characters off to the Alpha Centauri system as fast as I could. In six weekends of desperate work the new novel, wholly transformed, was done. And on 2 January 1955 – one year almost to the hour since I had been notified that a contract would be offered me – I received a telegram: CONGRATULATIONS ON A WONDERFUL REVISION JOB ALL SET TO GO.
_Revolt on Alpha C_ was published in August 1955, to generally indifferent reviews. ('inept and unreal . . . a series of old-hat adventures,' said the _New York Times.)_ Perhaps that was too harsh a verdict: the book is short, innocent, a little foolish, but not contemptible. It remained in print, in its Crowell edition, for seventeen years, earning modest but steady royalties until the printing was exhausted. A paperback edition published in 1959 still seems to enjoy a healthy life, having been through seven or eight printings so far, and in 1972 the book was reissued on two microfiche cards as part of the Xerox Micromedia Classroom Libraries series. This strange persistence of a very young author's very unimportant first novel does not delude me into thinking I must have created a classic unrecognized in its own day, nor do I believe it has much to do with my latter-day prominence in science fiction. That _Revolt on Alpha C_ remains in print after nearly twenty years is no more than an odd accident of publishing, but one that I find charming as well as profitable. My father never ceases to ask if the book still brings in royalties, and he is as wonderstruck as I that it does.
I was launched. On the strength of having sold a novel and a few short stories, I was able to get an agent, Scott Meredith, and he has represented me now for two decades. (There are writers and publishers who will tell you that drawing and quartering is too gentle a fate for him, and there are other writers who have been with him longer than I, with every intention of continuing the relationship until time's end. I think every agent evokes a similarly wide spectrum of responses.) I sent my agent all the unsold short stories in my file, and, assuming that manuscripts bearing his sponsorship would sell far more readily than ones coming in unsolicited from an unknown writer, I awaited a flow of publishers' checks. The flow was a bit sluggish, though. Two trifling stories sold to minor magazines in June 1954 and February 1955 for a total of $40.50; in May 1955 came $49.50 for a rather more elaborate piece. But several quite ambitious stories, which I thought worthy of the leading magazines of the time, failed to sell at all, from which I began to draw a sinister conclusion: that if I intended to earn a livelihood writing fiction, it would be wiser to use my rapidly developing technical skills to turn out mass-produced formularized stories at high speed, rather than to lavish passion and energy on more individual works that would be difficult to sell.
In the summer of 1955, just as that sombre insight was crystallizing in me, Randall Garrett appeared in New York and rented a room in the hotel near Columbia University where I was living. Garrett was about eight years older than I, and had had some two dozen stories published, including several in _Astounding,_ the premiere magazine of the era. Alone in a strange city, down on his luck, he struck up a curious friendship with me. We were markedly different in personal habits and rhythms, in philosophy, in background; but somehow these differences were a source of vitality rather than disharmony in the collaborative partnership that swiftly evolved. We complemented one another. Garrett was an established professional writer, but his discipline had collapsed and he was writing very little; I was unknown but ambitious, and could force an entire short story out of myself at a single sitting. Garrett had had a scientific education; mine was literary. Garrett was an efficient storyteller, but his prose was mechanical; I had trouble constructing internally consistent plots, but I wrote smoothly and with some grace. Garrett's stories rarely delved into character; I was already concerned, as much as I could be at the age of twenty, with emotional and psychological depth. We began to work together.
Until then, I had submitted all my stories by mail or else through my agent. Garrett took me to editorial offices. I met John Campbell of _Astounding,_ Bob Lowndes of the esteemed but impoverished _Science Fiction Stories,_ Howard Browne of _Amazing,_ Larry Shaw of the newly founded _Infinity._ Editors, Garrett said, bought more readily from writers they had met than from strangers who had only postal contact with them, and lo! it was so. I sold five stories in August 1955, three in September, three in October, six in November, nine in December. Many of these were collaborations with Garrett, but quite a few were stories I did on my own, capitalizing on contacts I had made with his help. Suddenly I was something more than a beginner, here in my final year of college: I was actually earning a living, and quite a good living, by writing. I think the partnership with Garrett accelerated the progress of my career by several years.
Unfortunately there were negative aspects. Once I had assumed, naively, that if I merely wrote the best stories that were in me, editors would recognize their merits and seek my work. Now I was coming to see that there was a quicker road to success – to live in New York, to visit editors regularly, learn of their issue-by-issue needs and manufacture fiction to fit them. I developed a deadly facility; if an editor needed a 7500-word story of alien conquest in three days to balance an issue about to go to press, he need only phone me and I would produce it. Occasionally I took my time and tried to write the sort of science fiction I respected as a reader, but usually I had trouble selling such stories to the better markets, which reinforced my growing cynicism. By the summer of 1956 – by which time I had graduated from college and had married – I was the complete writing machine, turning out stories in all lengths at whatever quality the editor desired, from slam-bang adventure to cerebral pseudo-philosophy. No longer willing to agonize over the gulf between my literary ambitions and my actual productions, I wrote with astonishing swiftness, selling fifteen stories in June of 1956, twenty the following month, fourteen (including a three-part serial, done with Garrett, for _Astounding)_ the month after that.
This hectic productivity was crowned at the World Science Fiction Convention in September 1956, when I was voted a special Hugo as the most promising new writer of the year. The basis for the award could only have been my ubiquity, since most of what I had published was carefully-carpentered but mediocre, and much was wholly opportunistic trash. It is interesting to note that the writers I defeated for the trophy were Harlan Ellison, who at the time had had only one or two dismal stories published, and Frank Herbert, whose impressive _Under Pressure_ had appeared in _Astounding_ the year before. A week after the convention I went with my bride, Barbara, to the first Milford Science Fiction Writers' Workshop, an awesome assembly of titans – Theodore Sturgeon, Fritz Leiber, Cyril Kornbluth, Lester del Rey, Damon Knight, Frederik Pohl, James Blish, William Tenn, and a dozen more of equal stature. Ellison and I were the only neophytes present. Harlan had not yet begun to show a shadow of his future abilities, and he made an easy whipping-boy for the patriarchs, but I was a different matter: self-contained, confident, quite sure of what I was doing and why.
Del Rey and a few others tried to shake my cynicism and persuade me to aim higher than sure-thing potboilers, but _it_ was clear that potboilers were what I wanted to write, and no one could argue with my success at hammering out penny-a-word dreadfuls. I was only a boy, yet already my annual income was beyond that of anyone in the field except Asimov, Heinlein, Clarke, and Bradbury, those long- enshrined demigods. What I dared not say was that I had opted to write mechanical junk because I had no faith, any longer, in my ability to write anything better. It had been my experience that whenever I assayed the kind of fiction that Sturgeon or Leiber or Kornbluth wrote, I had trouble getting it published. My craftsmanship was improving steadily, in the narrow sense of craft as knowing how to construct a story and make it move; possibly some fatal defect of the soul, some missing quality, marred my serious work, so that it was idle of me, I thought, to try to compete with the Sturgeons and Leibers. I will leave art to the artists, I said quietly, and earn a decent living doing what I do best.
By the end of 1956 I had more than a million published words behind me. I lived in a large, handsome apartment in what was then a desirable neighbourhood on Manhattan's Upper West Side. I was learning about fine wines and exotic foods and planning a trip to Europe. The collaboration with Garrett had long since ended, but the impetus he had given me was sufficient and reliable. (A few, notably Horace Gold of _Galaxy,_ swore at me for ruining a potentially important talent, but Horace bought my artfully aimed _Galaxy-type_ potboilers all the same.) My fellow writers viewed me with alarm, seeing me as some sort of berserk robot that would fill every page of every magazine with its output; they deplored my utter lack of literary ambition, yet accepted me as one of their number, and I formed strong friendships within the close-knit science fiction fraternity. And I wrote, and I sold, and I prospered, and with rare exceptions abandoned any pretence at literary achievement. I wanted to win economic security – to get enough money into the bank so that I would be insulated against the financial storms that had buffeted most of the writers I knew, some of the greatest in the field among them. Lester del Rey pointed out to me that simply on the money-making level I was going about things the wrong way. The stuff I was writing earned me a cent or two a word and then dropped into oblivion, while stories written with more care, with greater intensity of purpose, were reprinted again and again, earning their authors fees far beyond the original sale. I knew that this was so, but I preferred to take the immediate dollar rather than the hypothetical future anthology glory.
So it went through 1957 and 1958. I grew a beard and acquired other, less superficial, stigmata of sophistication. I journeyed to London and Paris, to Arizona and California, treating myself at last to the travels I had not had in boyhood. I learned the lore of the investment world and made some cautious and quite successful forays into the stock market, seeking always the financial independence that I believed would free me from the karmic wheel of high-volume hackmanship.
Not everything I wrote was touched by corruption. I still loved science fiction for its soaring visionary expansiveness, for its mind-liberating power, and however dollar-oriented I became I still yearned to make some valuable contribution to the field, and felt guilty that the stuff I was churning out was the sort of thing I had openly scorned in my fan-magazine critical essays seven or eight years before. I recall in particular a Sunday afternoon party at Harlan Ellison's Manhattan apartment in 1957 where I talked shop with Cyril Kornbluth, Algis Budrys, James Blish, and one or two other sf writers of their level, and went home in an abyss of self-contempt because these men, my friends, were trying always to publish only their best while I was content to do my worst. Whenever I felt the sting, I put aside hackwork and tried to write honest fiction.
Scattered through my vast output of the late 1950's, then, are a good many quite respectable stories, not masterpieces – I was still very young, and much more callow than most people suspected – but decently done jobs. Occasionally even now they find their way into anthologies. They were my comfort in those guilt-ridden days, those stories and the novels. In longer lengths I was not so commercially-minded, and I genuinely hoped to achieve in books what was beyond me in the magazines. There were few publishers of science fiction novels then, however: the market consisted, essentially, of three houses, Doubleday, Ballantine, and Ace. With the leading writers of the day keeping the first two well supplied with books, I found no niche for myself, and turned of necessity to Donald Wollheim's Ace Books. This small company published scores of novels a year in a rather squalid format, and was constantly searching for new writers to meet its hunger for copy. The shrewd and experienced Wollheim worked miracles on a tiny budget and produced an extraordinarily broad list, ranging from juvenile action stories to superb novels by Philip K. Dick, A. E. van Vogt, Clifford D. Simak, Isaac Asimov, and other luminaries. Wollheim saw potential in me, perhaps as a mass-producer of action fiction and perhaps as something more than that, and encouraged me to offer him novels. He purchased the first, _The Thirteenth Immortal,_ late in 1956, and I wrote nine more for him, I think, in the next seven years.
My Ace novels would be fruitful material for somebody's thesis. The first was melodramatic, overblown, a little absurd, yet sincerely conceived; its faults are those of its author's youth, not his cynical approach toward his trade. The second, _Master of Life and Death_ (1957), was something of a _tour de force,_ a maze of plot and sub-plot handled, I think, with some dexterity. _Invaders from Earth_ (1958), the third, attempts a sophisticated depiction of psychological and political realities. I liked those two well enough to allow them to be reprinted a decade later. _Stepsons of Terra_ (1958) was an intricate time-paradox novel with a certain van Vogtian intensity. On the evidence of these four books alone I would seem an earnest and ambitious young writer striving constantly to improve. But the rest of the novels I wrote for Wollheim were slapdash adventure stories, aiming no higher than the least of his line; I had learned there was little money and less prestige in doing books for Ace, and without those rewards I was content to do the minimum acceptable job. (A few of my later Ace books were better than that, but they were aimed at better markets and went to Wollheim only after others had rejected them.) I know that Wollheim was disappointed in the trend my work for him had taken, but I was too far gone in materialism to care.
During the high-volume years I wrote a good deal that was not science fiction – crime stories, a few westerns, profiles of movie stars, and other odds and ends. Some of this work came to me on assignment from my agent, and some I sought because my rate of productivity was now so high that the science fiction field could not absorb all the wordage I was capable of turning out. I had the conviction, though – shared by a surprisingly large number of science fiction writers – that to write sf was the One True Task, and any other kind of writing was mere hack-work done to pay the bills. This was a legitimate enough attitude when held by people like James Blish or William Tenn, who in their early days were forced to write sports fiction and other trivia because the sf market was so tiny; but it was a bit odd for me to feel that way when virtually everything I wrote, sf or not, was pounded out in the same cold-blooded high-velocity manner. Still, I did feel that way, and whatever my private feelings about the quality of most of my science fiction at that time, I still saw it as a higher endeavour than my westerns and crime stories.
Then, late in 1958, the science fiction world collapsed. Most of the magazines for which I was writing regularly went out of business as a result of upheavals in distribution patterns, and those that survived became far more discriminating about what they would publish. My kind of mass production became obsolete. To sustain what had become a comfortable standard of living I found it necessary to leave the cozy, incestuous science fiction family and look for work in the general New York publishing scene.
The transition was quick and relatively painless. I was facile, I was confident, and my friends had friends. I hired out to any editor who would undertake to pay on time; and, though I continued to write some science fiction in 1959 and 1960, my records for those years show all sorts of strange pseudonymous stories and articles: 'Cures for Sleepless Nights', 'Horror Rides the Freeway', 'I Was a Tangier Smuggler', 'Hot Rod Challenge', 'Buried Billions Lie in Wait', and so many others that it strains my own credulity. I recall writing one whole piece before lunch and one after lunch, day in, day out; my annual output climbed well above a million words in 1959 and went even higher in 1960 and 1961.
These were years of wandering in the wilderness. I was earning more money than I had in science fiction, and I had no problems of guilt, for in pouring out this grotesque miscellany I did not need to flagellate myself with the knowledge that I was traducing a literature I loved. On the other hand, I had no particular identity as a writer. In the past, when people asked me what I did, I had answered that I wrote science fiction; now, working anonymously in twenty different sub-literate markets, I had no ready reply, so I went on saying I was a science fiction writer. In truth I did have the occasional story in _Galaxy_ or _Astounding,_ and an Ace book now and then, to make the claim legitimate. I was mainly a manufacturer of utilitarian prose, though, churned out by the yard. It was stupefyingly boring, and, as the money piled up, I invested it shrewdly and talked of retiring by the time I was thirty, living on my dividend income, and spending my days travelling, reading, and studying. Already I was doing a good bit of that. In the winters my wife and I fell into the habit of going to the West Indies, where we became skin-divers and explored coral reefs. In the summers we made other journeys – Canada in 1959, Italy in 1960, the American Northwest in 1961. I was working only four or five hours a day, five days a week, when at home, which left me ample leisure for my private interests – contemporary literature and music, art, ancient history. There was an almost total split between my conscienceless commercialized working-hours self and the civilized and fastidious man who replaced him in early afternoon. I was still only about twenty-five years old.
Unexpectedly the seeds of a new writing career began to sprout. One of my few science fiction pieces of 1959 was a little novel for children, _Lost Race of Mars,_ published by the notable house of Holt, Rinehart and Winston. (My earlier connection with Crowell had fallen apart in 1956, after their rejection of my proposed successor to _Revolt on Alpha_ C, and this was my first contact with a major publishing house since then.) _Lost Race of Mars_ was short and simple, but it was an appealing book; the _New York Times_ chose it as one of the hundred best children's books of the year, and the publisher expressed eagerness to do more of my work. _(Lost Race_ is still in print and selling well, both in hardcover and a paperback edition.) I had visited Pompeii while in Italy in 1960, and now I saw a way of capitalizing on my interest, strong since childhood, in antiquity and its remains: I suggested a book for young readers on the excavation of Pompeii.
The people at Holt, Rinehart and Winston considered the idea for quite a while but ultimately declined it. Henry Morrison, who then was handling my affairs at the Scott Meredith agency and who since has become an important agent in his own right, told me he thought the project would fare better if I wrote not about one ancient site but several – say, Chichén Itzá and Angkor and Babylon as well as Pompeii – and he even offered me a title for the expanded book, _Lost Cities and Vanished Civilizations._ When I agreed he sold the book, on the basis of a brief outline, to a Philadelphia house of which I knew nothing, Chilton Books.
With my agent's help I began to emerge from that wilderness of anonymous potboilerei. I began to work in book-length non-fiction, and displayed gifts for quick, comprehensive research and orderly uncluttered exposition. For a minor paperback company called Monarch, now defunct, I did books on the American space program, the Rockefeller family, and the life of Sir Winston Churchill; and for Chilton, in the summer of 1961, I wrote my lost-cities book.
None of this was art, but it was far from despicable work. I used secondary sources and wrote with journalistic speed, but what I produced was clear, generally accurate, an honest kind of popularized history. Chilton liked _Lost Cities_ and hastened to accept my next proposal, for a book on underwater archaeology. Early in 1962 a suggestion for a young readers' book on great battles found favour at the old-line house of G. P. Putnam's Sons. In April of that year _Lost Cities and Vanished Civilizations_ was published and – to my amazement, for I thought of it as no more than a competent rehash of other writers' books – was chosen as one of the year's five best books for young people by an annual awards committee in the field of juvenile publishing, and was selected by the Junior Literary Guild, an important book club. Once again I found myself launched.
Many of New York's leading hardcover publishing houses were willing, on the strength of the success of _Lost Cities,_ to give me contracts for non-fiction juvenile books on whatever subject happened to interest me. As rapidly as I dared I severed my connections with my sleazy magazine outlets and ascended into this new, astoundingly respectable and rewarding career. Chilton took another general archaeology book, _Empires in the Dust._ Holt, Rinehart and Winston accepted a biography of the great Assyriologist, Austen Henry Layard. The New York Graphic Society commissioned a book on American Indians, and Putnam one on the history of medicine.
The rhythm of my life changed dramatically. I still wrote in the mornings and early afternoons – wrote at almost the same incredible velocity as when I had been doing tales of Tangier smugglers – but now I spent the after-hours time taking notes in libraries and museums, and I began to assemble a vast private reference library at home. Although my early non-fiction books had been hasty compilations out of other popularizations, I swiftly became more conscientious, as though to live up to the high opinion others had formed of those early books; I went to primary sources whenever possible, I visited actual sites, I did intensive research in many ways. The results were visible. Within a year or two I was considered one of the most skilled popularizers of the sciences in the United States, with publishers eagerly standing in line as my changing interests took me from books on Antarctica and ancient Egypt to investigations of scientific hoaxes and living fossils. For the first time since I had become a professional writer, nearly a decade earlier, I won my own respect.
I maintained a tenuous link with science fiction, largely social, since then as now my closest friends were science fiction writers. I attended parties and conventions, and kept up with what was being published. But of actual science fiction writing I was doing very little. There seemed no commercial reason to get back into sf, even though it had recovered considerably from its 1958 swoon; I had more work than I could handle in the lucrative juvenile non-fiction hardcover field. Only the old shame remained to tweak me: I had served science fiction badly in my 1955–8 days, and I wanted to atone. When Frederik Pohl became editor of _Galaxy_ he suggested that I do short stories for him and offered me absolute creative freedom: I could write what I pleased and, within reason, he undertook to buy it. In such an arrangement I could blame neither editorial shortsightedness nor constricting editorial policies for the quality of what I wrote: I was my own master. In the summer of 1962 I offered Pohl a short story, 'To See The Invisible Man', inspired by Borges, which was out of an entirely different artistic universe from anything I had written in my first go-round in science fiction – a mature, complex story. He published it and, over the next couple of years, half a dozen more of similar ambitious nature, and, bit by bit, I found myself drawn back into science fiction, this time not as a producer of commodities but as a serious, dedicated artist who turned away from more profitable work to indulge in sf out of love.
During those years – 1962 to 1965 – when I dabbled in science fiction for sheer diversion only, science fiction was undergoing radical changes. The old pulp-magazine rigidities were dissolving. New writers were everywhere: Brian Aldiss, J. G. Ballard, Roger Zelazny, Samuel R. Delany, R. A. Lafferty, Michael Moorcock, and a dozen more. In the bad old days one had to be an established writer of mighty stature, a Bester or a Blish or a Sturgeon, to 'get away' with any sort of literary adventurousness; most editors rightly thought that their readers were hostile to unusual modes of narrative, and nearly everyone wrote in an interchangeable manner, unquestioningly adopting universal conventions of style and construction. Suddenly the way of telling stories was released from convention. The familiar old robots and starships were being put through strange and fascinating new paces. Pulp-magazine requirements for neat plots and 'upbeat' positive resolutions were abandoned. I had been only too willing, in 1957 and thereabouts, to conform to the prevailing modes, for it seemed quixotic to try to do otherwise. Now an army of younger, or at any rate newer, writers had boldly overthrown the traditional rules, and, a trifle belatedly, I joined the revolution.
Even after I returned to science fiction, the non-fiction books remained my chief preoccupation. For one thing, to go back to the mass production of sf would be to defeat the purpose of returning; for another, I was so overwhelmed with non-fiction contracts stretching two and three years into the future, that there was no question of a full-time resumption of sf. The non-fiction was becoming ever more ambitious and the books took longer; in the summer of 1965 I spent months working on one title alone, which I had never done before. (It was a book on the Great Wall of China – no mere cut-and-paste job, but an elaborate and unique synthesis of all available knowledge about the Wall.) Then, too, science fiction had become more permissive but there was still not much money to be had in writing it, and I was continuing to pursue my goal of economic independence, which mandated my centering my career in other fields.
One gigantic item of overhead had entered my life. Early in 1962 I had purchased an imposing house – a mansion, in fact – in a lovely, almost rural enclave near the northwest corner of New York City. I had always lived in apartments; now I joined the landed classes, and had my own lawn and garden, my own giant oak trees, my own wild raccoons wandering about at night (in New York!). There was room for all my books and all I was likely to acquire for many years to come. The third floor of the house, a separate four-room suite, became my working area, and we filled the rest of the place with books and paintings and _objets d'art._ It was a magnificent house, beautiful and stately, and not at all costly in terms of my income at the time. What was costly was the upkeep, taxes and cleaning and heat and all, running to many thousands of dollars a year; though I still intended to retire from full-time high-volume writing as soon as possible, I recognized that by buying the house I had postponed that retirement by at least five years.
The non-fiction books grew ever more demanding as – driven by vanity, I suppose, or by intellectual pride, or merely by the feeling that it was time for my reach to begin exceeding my grasp – I tackled bigger and bigger projects. Though I still was doing books for readers in their teens, a biography of Kublai Khan and one of Socrates, a book on bridges and one on coral reefs, I was aiming primarily for older readers in much of what I did, and endeavouring now to deal with subjects that had had no serious examinations in recent times. The Great Wall book was the first of these; and early in 1966 I embarked on a far more arduous task, a book called _The Golden Dream,_ a study of the obsessive quest for the mythical land of El Dorado. Working an impossible, brutal schedule, pouring out thousands of words a week, I knew more than a little about the psychology of obsession, and the book, 120,000 words long, was surely the finest thing I had ever done. It was published in an appropriately handsome edition by the Bobbs-Merrill Company, was treated with respect by reviewers, and, I grieve to report, dropped into oblivion as fast as any of my hackwork. The book earned me no income beyond the small initial advance in the United States, was never published at all in Great Britain, and achieved only one translation, in France. I was disappointed but not discouraged; it would have been agreeable to grow rich on the book, but this was secondary to the joy and challenge of having written it. I was learning to love my work for its own sake, regardless of its fate in the marketplace. Growing up, that is.
About the time of _The Golden Dream_ I inaugurated still another aspect of my career by asking the publisher of some of my non-fiction juveniles to let me edit a science fiction anthology. Here at last I could put to some practical use all those years of collecting and reading sf; I had built a superb science fiction library, with literally every magazine ever published and most of the books. The anthology, _Earthmen and Strangers,_ was released in the autumn of 1966. I found editing so much to my taste that I sought other anthology contracts and ultimately was devoting as much time to editing as to my own writing.
In that same period – 1965–6 – I built close associations with the two major science fiction houses of the era, Ballantine and Doubleday. When I first became a professional writer these houses were the exclusive preserves of the Clarkes and Heinleins and Sturgeons and Asimovs and Bradburys, and seemed unattainable to the likes of me; now, still having not much of a reputation in science fiction but solidly established outside the field and confident of my skills, I found no difficulty convincing Betty Ballantine of Ballantine and Larry Ashmead of Doubleday to publish my sf. (Even though I considered myself a very part-time science fiction writer in those days, I was still prolific enough to require two regular publishers.) To Ballantine I gave _To Open the Sky,_ a pseudo-novel constructed from five novelettes I had written for Fred Pohl's _Galaxy._ To Doubleday I offered _The Time Hoppers,_ an expansion of one of those ambitious short stories of my youth that I had had so much trouble placing in 1954. They were both good, middle-of-the-road science fiction, not exactly of Hugo quality but several notches above anything I had published in the field before.
Ballantine also agreed to do a collection of my short stories; and, in January 1966, I proposed a new novel, a book called _Thorns,_ telling Mrs Ballantine, 'Much of the texture of the story will rely on background details that can't be sketched in advance. I hope you can gather enough of my intentions from the outline to go ahead with it. What I have in mind is a psychological sf novel, somewhat adventurous in style and approach and characterization, and I think I can bring it off. It's worth trying, at any rate.' She agreed to the gamble.
I spent the next few months writing the El Dorado book, and in June I fell into a mysterious illness. All energy went from me and I lost close to twenty pounds – though I was slender to begin with – in a few weeks. I had not been ill since finishing with the standard childhood maladies, indeed was not even prone to minor upsets, and this was a startling event to me. The symptoms answered well to leukemia and other dire things, but turned out to be only a metabolic change, a sudden hyperactivity of the thyroid gland. Such thyroid outbreaks, I learned, are often caused by the stress of prolonged overwork, and I think the forced marches of El Dorado had much to do with this one. I took it as a warning: I was past thirty and it was time to think realistically about slowing down. Though I had enough book contracts to keep me busy for two or three years, I resolved to reduce my output and gradually to make drastic reductions in the time I devoted to work.
Though greatly weakened, I wrote steadily – but at a slower pace – through the infernally hot summer of 1966, while at the same time planning _Thorns_ and doing preliminary research for another major non-fiction work, a study of the prehistoric Mound Builder cultures of the central United States. I was still gaunt and haggard when I attended the annual science fiction convention in Cleveland at the beginning of September, but the drug therapy for my thyroid condition was beginning to take hold, and immediately after the convention I felt strong enough to begin _Thorns._ The title describes the book: prickly, rough in texture, a sharp book. I worked quickly, often managing twenty pages or more a day, yet making no concessions to the conventions of standard science fiction. The prose was often oblique and elliptical (and sometimes shamefully opaque in a way I'd love to fix retroactively); the action was fragmented in the telling; the characters were angular, troubled souls. Midway in the job I journeyed out to Pennsylvania to attend a party at Damon Knight's Milford Workshop. I knew nearly all the writers there, and they knew me. They all knew how prosperous I was, and some were aware that I had achieved worthwhile things with my non-fiction, but they couldn't have had much respect for me as a writer of science fiction. They might admire my professionalism, my productivity, my craftsmanship – but to them I was still that fellow who had written all that zap-zap space-opera in the 1950's. Their gentle and not-so-gentle comments hardly troubled me, though, for I knew I was no longer that mass-producer of garbage, and sooner or later they would all know it too. While at Milford I glanced at an Italian science fiction magazine and found a harsh review of one of my early Ace novels, recently published in Italy. Badly done and wordy, the critic said – _malcondotto e prolisse._ Perhaps it was. The next day, when I went home to finish _Thorns,_ Malcondotto and Prolisse joined the cast of characters.
I regained my health by the end of the year and eventually made a full and permanent recovery. I withdrew, bit by bit, from my lunatic work schedule: having written better than a million and a half words for publication in 1965, I barely exceeded a million in 1966, and have never been anywhere near that insane level of productivity since. Though I still wrote daily except when travelling, I worked less feverishly, content to quit early if I had had a good morning at the typewriter, and I began alternating science fiction and nonfiction books to provide myself with periodic changes of rhythm. I looked forward to 1967 with some eagerness – and with much curiosity, too, for that was the year in which my first really major science fiction, _Thorns_ and _The Time Hoppers_ and a novella called 'Hawksbill Station', would finally be published. Would they be taken as signs of reform and atonement for past literary sins, or would they be ignored as the work of a writer who by his own admission had never been much worth reading?
I began the year by writing a short story, 'Passengers', for Damon Knight's new _Orbit_ anthology series. He asked for revisions, minor but crucial, five times, and though I grumbled I saw the wisdom of his complaints and did the rewriting. I wrote a novel for Doubleday, _To Live Again,_ which surpassed anything I had done in complexity of plot and development of social situation. I expanded 'Hawksbill Station' into a novel. I did my vast Mound Builder book, bigger even than El Dorado, a book that was as much a study of the myth-making process as it was an exploration of American Indian culture. (When it appeared in 1968, as _Mound Builders of Ancient America: The Archaeology of a Myth,_ many reviewers, even those in the archaeological journals, assumed I was myself an archaeologist, and I received flattering if embarrassing invitations to lecture, to teach, and to write reviews. The book was greeted enthusiastically by professional archaeologists and has become a standard reference item, to be found in most libraries. Having said so many uncomplimentary things about my own writing in these pages, I think I've earned the right to be a bit boastful about this one.) There were three other big projects in this year of supposedly reduced output: the novels _The Masks of Time_ and _The Man in the Maze_ and another Goliath of a non-fiction work, _The Longest Voyage,_ an account of the first six circumnavigations of the world.
I was, in truth, riding an incredible wave of creative energy. Perhaps it was an overcompensation for my period of fatigue and illness in 1966, perhaps just the sense of liberation and excitement that came from knowing I was at last writing only what I wanted to write, as well as I could do it. In any event I look back in wonder and awe at a year that produced _To Live Again, Masks of Time, Man in the Maze,_ two 150,000-word works of history, several short stories, and – I have as much trouble believing this as you – no less than seven non-fiction books for young readers, each in the 60,000-word range. No wonder my peers regarded me as some sort of robot: I have no idea myself how I managed it all, working five hours a day five days a week, with time off for holidays in Israel and the West Indies and a week at Montreal's Expo 67.
_Thorns_ was published in August of 1967. All of Ballantine's science fiction titles were then automatically being distributed free to the members of the two-year-old Science Fiction Writers of America, and so all my colleagues had copies in hand at the time of that year's sf convention. Many of them had read it, and – as I hoped – it shook their image of my work. At least a dozen of my friends told me, with the frankness of true friendship, that the book had amazed them: not that they thought me incapable of writing it, but rather that I would be willing to take the trouble. It seemed such a radical break from my formularized science fiction of the 1950's that they thought of it as the work of some entirely new Robert Silverberg. I was pleased, of course, but also a little pained at these open admissions that I had been judged all these years by the basest of what I had written between 1955 and 1958. _Thorns_ was not all that much of a breakthrough for me; it represented only a plausible outgrowth of what I had begun to attempt in 1962's short story, 'To See the Invisible Man', and in the work that followed it over a period of four years.
Even before the publication of _Thorns_ I found my position in the American science fiction world undergoing transformations. In the summer of 1967 I had become President of the Science Fiction Writers of America, succeeding Damon Knight, founder of the organization. The job was not an award for literary merit but rather a tribute to the experience I had had in building a career and dealing with publishers. Certainly I was well qualified for the job, and I felt no hesitation about accepting it, especially since the organization would have collapsed if I had declined – no one else was willing to take it on. Doubtless if I had run against some writer whose work was more highly regarded than mine, James Blish or Poul Anderson or Philip José Farmer, I would have been defeated; but willy-nilly I ran unopposed, gladly letting myself in for a year of drudgery on behalf of my fellow writers. At least _Thorns_ soon showed the rank-and-file of the membership that their new President would not disgrace the organization.
_Thorns_ did not universally give delight. Those who found pleasure in my old straightforward action stories were appalled by this dark, disturbing book. One of my dearest friends, an old-line writer conservative in his tastes, explicitly accused me of a calculated sellout to the 'new wave' of science fiction – of writing a deliberately harsh and freaky book to curry favour with the influential leaders of the revolution within science fiction. That charge was particularly painful to me. Having blithely sold out so many times as a young man to any editor with the right price in his hand, I was hurt to find myself blamed for selling out again, this time to the opposite camp, when I finally wrote something that grew from my own creative needs instead of the market's demands. Such criticisms were rare, though. _Thorns_ was nominated both for the Hugo and for the Science Fiction Writers' Nebula trophy – the first time anything of mine reached the final ballot in either contest.
They won no awards, nor did 'Hawksbill Station', which was also up for a Nebula; but the critics were re-evaluating my place in science fiction, invariably invoking my seamy early work before getting around to saying how much better a writer I was nowadays. 1968 promised to be a rewarding year. It was less than six weeks old, though, when I awakened at half past three one frigid winter morning to the glare of an unaccustomed light in the house. Burglars have broken in, I thought, groping toward wakefulness – but no, there were no burglars. The glare I saw was fire.
So out into the miserable night we went and watched the house burn. Papers stored in the attic, I think, had ignited. My wife and I carried our four cats and a flock of kittens to the dubious safety of the basement, and I seized the manuscript of my current book and a few ancient artifacts and cached them in the garage; then the firemen refused to let us return to the building, and we took refuge in the house across the way. By dawn it was over. The roof was gone; the attic had been gutted; my third-floor office was a wreck; and the lower floors of the house, though unburned, were awash in water rapidly turning to ice. A priest from a nearby Catholic college appeared and, unbidden, took several Volkswagen-loads of our houseplants to safety in his cabin, lest they freeze in the unprotected house. Then he returned and offered consolation, for I was in a bad way. No Catholic I, but I had felt the hand of some supernatural being pressing against me that night, punishing me for real and imagined sins, levelling me for overweening pride as though I had tried to be Agamemnon.
Friends rallied round. Barbara performed prodigies, arranging to have our belongings taken to storage (surprisingly, most of our books and virtually all the works of art had survived, though the structure itself was a ruin) and negotiating with contractors. I was not much good for anything for days – stupefied, God-haunted, broken. We moved to a small, inadequate rented house about a mile away as the immense job of reconstruction began. I bought a new typewriter, reassembled some reference books, and, after a few dreadful weeks, began once more to work in strange surroundings.
In nine months the house was ready to be occupied again, and by the spring of 1969 the last of the rebuilding was done and the place was more beautiful than ever – an exact replica of its former self, except where we had decided on improvements. But I was never the same again. Until the night of the fire I had never, except perhaps at the onset of my illness in 1966, been touched by the real anguish of life. I had not known divorce or the death of loved ones or poverty or unemployment, I had never experienced the challenges and terrors of parenthood, had never been mugged or assaulted or molested, had not been in military service (let alone actual warfare), had never been seriously ill. The only emotional scars I bore were those of a moderately unhappy childhood, hardly an unusual experience. But now I had literally passed through the flames. The fire and certain more personal upheavals some months earlier had marked an end to my apparent immunity to life's pain, and drained from me, evidently forever, much of the bizarre energy that had allowed me to write a dozen or more books of high quality in a single year. Until 1967, I had cockily written everything in one draft, rolling white paper into the machine and typing merrily away, turning out twenty or thirty pages of final copy every day and making only minor corrections by hand afterwards. When I resumed work after the fire I tried to go on that way, but I found the going slow, found myself fumbling for words and losing the thread of narrative, found it necessary in mid-page to halt and start over, pausing often to regain my strength. It has been slower and slower ever since, and I have only rarely, and not for a long time now, felt that dynamic sense of clear vision that enabled me to write even the most taxing of my books in wild joyous spurts. I wasted thousands of sheets of paper over the next three years before I came to see, at last, that I had become as other mortals and would have to do two or three or even ten drafts of every page before I could hope to type final copy.
I hated the place where we settled after the fire – it was cramped, dirty, confused, ugly – but the rebuilding job called for thousands of dollars beyond the insurance settlement, and I had to go on writing regardless of externals. With most of my reference library intact but in storage for the duration, I was forced back into virtual fulltime science fiction, the nonfiction temporarily impossible for me. One of the first things I wrote, in the early days of the aftermath, was a curiously lyrical novella, 'Nightwings', to which I added a pair of sequels some months later to constitute a novel. Later in the year came a novel for young readers, _Across a Billion Years,_ almost unknown among my recent works – a rich, unusual book that never found an audience. There was a short story, 'Sundance', a display of technical virtuosity, my favourite among all my myriad shorter pieces. And, in my despair and fatigue, I managed somehow to write a bawdy comic novel of time travel, _Up the Line._ The fire had shattered me emotionally and for a time physically, but it had pushed me, I realized, into a deeper, more profound expression of feelings. It had been a monstrous tempering of my artistic skills.
In September of 1968 I went to California for the science fiction convention – my third visit to that state, and I was struck once again by its beauty and strangeness. I was toastmaster at the convention's awards banquet, a last-minute replacement for the late Anthony Boucher, and for five hours toiled to keep a vast and restless audience amused – a fascinating, almost psychedelic experience. November saw me back in my restored house, working on the biggest of all my non-fiction books, an immense exploration of the Zionist movement in the United States. The publishers invested a huge sum of money in it, and planned to promote it to best-seller status, but, as usual, nothing came of it but good reviews: I was destined never to win wide attention for my long non-fiction works.
My science fiction, though, was gathering acclaim. _Masks of Time_ failed by only a few votes to win a Nebula, as did the novella 'Nightwings'. But 'Nightwings' did take a Hugo at the St Louis convention in 1969. In the spring of that year I wrote a novel, _Downward to the Earth,_ which was in part inspired by a journey to Africa (and in which were embedded certain homages to Joseph Conrad) and in part by my own growing sense of cosmic consciousness: I had never been a religious man, had never belonged to any organized church, but something had been set ticking in me by the fire, a sense of connections and compensating forces, and _Downward to the Earth_ reflected it. _Galaxy_ purchased it for serialization and New American Library for book publication. In the autumn – slowly, with much difficulty – I wrote _Tower of Glass,_ for Charles Scribner's Sons, the publishers of Hemingway and Wolfe and Fitzgerald, now experimenting with science fiction. _Galaxy_ bought that one too. And at the end of the year I wrote my strangest, most individual book, _Son of Man,_ a dream-fantasy of the far future, with overtones of Stapledon and Lindsay's _Voyage of Arcturus_ and a dollop of psychedelia that was altogether my own contribution. It was becoming extremely hard for me to get words on paper, despite this long list of 1969's accomplishments, and, with the expenses of the fire behind me, I was again talking of retirement. Not total retirement – writing was a struggle, but _having written_ was a delight – but at least a sabbatical of some months, once I had dealt with the contractual obligations I had taken on for the sake of rebuilding my home.
The paradox of this stage of my career manifested itself ever more forcefully in 1970: I felt continual growth of my art, my power, my vision, and simultaneously it became constantly more difficult to work. I tired more easily, I let myself be distracted by trifles, and when I did write I was over-finicky, polishing and polishing so that on a good day I was lucky to get nine or ten pages written. Still an immense output, but not what I had grown accustomed to pulling from myself in the vanished days of indefatigable productivity. Nevertheless it was an active year. I did _The WorldInside,_ a novel composed of loosely related short stories set within a single great residential tower; I think it and _Tower of Glass_ (another story of a giant erection!) are closer to pure science fiction, the exhaustive investigation of an extrapolative idea, than anything else I have written. I did A _Time of Changes,_ more emotional than most of my work and heavily pro-psychedelic. I did _The Second Trip,_ a rough and brutal novel of double identity, and I wrote the last of my major non-fiction books, _The Realm of Prester John,_ which I regard as a genuine contribution to scholarship. (Doubleday published it and no one bought it.)
By now it was clear that the science fiction world had forgiven me for the literary sins of my youth. My short story 'Passengers' won a Nebula early in 1970. _Up the Line_ and one of the 'Nightwings' series were on the ballot also, though they failed to win. In the summer I was American Guest of Honour at the World Science Fiction Convention in Heidelberg, a little to my surprise, for though I was beginning to think I would someday be chosen for this greatest of honours in science fiction, I had assumed it was at least ten years in the future. I was a triple Hugo nominee that year too, but came away, alas, with a bunch of second and third-place finishes. Another quite improbable boyhood fantasy was eerily fulfilled for me in 1970. When I was about sixteen and _Galaxy_ was the newest and most controversial of science-fiction magazines, I diverted myself one day with an amiable daydream in which I was the author of three consecutive serials in that magazine – an awesome trick, since the authors of _Galaxy's_ first five novels were Simak, Asimov, Kornbluth and Merril, Heinlein, and Bester. But there I was in 1970 with _Downward to the Earth, Tower of Glass,_ and most of _The World Inside_ running back-to-back, and _Time of Changes_ following them in 1971. I remembered my old daydream and felt a little disbelieving shiver.
My new working habits were entrenching themselves: revise, revise, revise. Projects that might have taken me two weeks in 1965 took three months in 1970. I refused to sign new contracts, knowing that I no longer had much control over the length of time it took me to finish anything, and I could not therefore guarantee to meet delivery dates. Nonfiction in particular I was phasing out; I had had a good run in that career for a decade, but the burden of research now was more than I cared to carry, and the failure of my big books to have much commercial success had eventually had a depressing effect. Now that I was in my full stride in science fiction, working at the top of my form and enjoying public favour, I wanted to devote as much of my dwindling literary energies to that field as I could.
Strangely, it was becoming impossible for me to take the stuff of science fiction seriously any more – all those starships and androids and galactic empires. I had come to believe that the chances that mankind would reach and colonize the planets of other stars were very slight indeed, and the stories set on such worlds now seemed idle fantasy to me, not serious projection. So too with many of the other great themes of science fiction: one by one they became unreal, though they continued to have powerful metaphorical and symbolic value for me. I discovered that much of what I was writing in 1971 was either barely sf at all _(The Book of Skulls)_ or was a kind of parody of science fiction ('Good News from the Vatican', 'Caliban', and other short stories) or borrowed a genuine science fiction theme for use in an otherwise 'straight' mainstream novel _(Dying Inside)._ This realization inspired flickers of new guilt in me. I no longer had to apologize, certainly not, for shortcomings of literary quality; but was this new Silverberg really serving the needs of the hard-core science fiction audience? Was he providing the kind of sincerely felt fiction about the future that the readers still seemed to prefer, or was he doing fancy dancing for his own amusement and that of a jaded elite?
The pattern of awards in the field reinforced these doubts. I was getting nominated by twos and threes every year now for the Hugo and the Nebula; indeed, I have by now amassed more final-ballot nominations than any other writer. In 1972 the Science Fiction Writers of America favoured me with two Nebulas, an unusual event, for my novel _A Time of Changes_ and my short story 'Good News from the Vatican' – but the writers have relatively sophisticated tastes, and I have fared far less well with the Hugos, awarded by a broader cross-section of the sf readership. Though nominated every year, my books and stories have finished well behind more conservative, 'safer' works. This causes me no serious anguish or resentment, for I have hardly been neglected in the passing around of honours in the sf world, but it does lead me to brood a bit in idle hours. Not that it affects what I write: I am bound on my own course and will stay to it. I wish only that I could be my own man and still give pleasure to the mass of science fiction readers.
In 1971 I at last achieved the partial retirement of which I had been dreaming for so many years. The press of contracts abated, and in late spring I simply stopped writing, not to resume until autumn. I had never, not since early college days, gone more than four weeks away from my typewriter; now I was away from it five whole months, and felt no withdrawal symptoms at all. I read, swam, loafed; now and then I would work on anthology editing for an hour or so in the morning, for such editing was becoming increasingly important to me, but essentially I was idle all summer. A more complete break with the old Silverberg could not have been imagined. To underscore the transformation I had spent some weeks just before the holiday revising an early novel of mine, _Recalled to Life,_ for a new edition. When I wrote it, in 1957, I had exaggeratedly high regard for it, seeing it as a possible Hugo nominee and hoping _it_ would gain me a place with Ballantine or Doubleday or some other major publishing house. Looking at this masterpiece of my youth fourteen years later, I was appalled at its crudity, and repaired it as best I could before letting it be reissued. That experience gave me a good yardstick to measure my own growth.
Further transformations of my life, unexpected ones, lay in wait for me. My wife and I were native New Yorkers, and, however extensively we travelled, we always returned to New York, the home base, after a few weeks. We loved the city's vitality, its complexity, the variety of experience it offered, and we had money enough to insulate ourselves from its inconveniences and perils. Our rebuilt house was more than a dwelling to us, it was a system of life, an exoskeleton, and we assumed we would live in it the rest of our lives. But New York's deterioration and decline was driving away our friends. Two by two they trooped away, some to distant suburbs, many to California; and by the autumn of 1971 we found ourselves isolated and lonely in a city of eight million. New York now was dangerous, dirty, ever more expensive; taxes were rising alarmingly and the amenities we prized, the restaurants and galleries and theatres, were beginning to go out of business. We were held fast by pride and pleasure in our house – but did we want to find ourselves marooned in our magnificent fortress while everything dissolved about us? Timidly we began talking about joining the exodus. It still seemed unthinkable; we toyed with the notion of moving to California the way loyal Catholics might toy with the idea of conversion to Buddhism, enjoying the novelty and daring of such an outlandish idea, but never taking it seriously. In October, 1971 we flew to San Francisco for a reunion with many of our transplanted Eastern friends; we said we were considering moving, and they urged us to come. It was impossible to give up our house, we said. We went back to California in November, though, still hesitating but now willing to look, however tentatively, at areas where we might find a comparable place to live. And just after the turn of the year we discovered ourselves, to our amazement, boarding a plane for a sudden weekend trip west to see a house that a friend had located for us.
That house turned out not to work – it was too big even for us, and too decayed – but before the weekend was over we had found another, strange and beautiful, an architectural landmark in a park-like setting, and we placed a bid on it and after some haggling the bid was accepted, and, as if in a dream, we put our cherished New York place up for sale and made arrangements to move West. It all happened so swiftly, in retrospect – less than six months from the moment the temptation first struck to the day we arrived, with tons of books and furniture, in golden California, in the new El Dorado.
California, then. A new life at the midpoint. For reasons of climate, my 1971 scheme of working autumn and winter and taking a holiday in spring and summer did not seem desirable, though I still wanted to work only half the time. I hit on a plan of working mornings, normally a cloudy time of day here, and giving myself the afternoons free, with frequent total interruptions of work for short holidays away from home. This has worked well for me. My output continues to decline: 1971 saw me write about a quarter of a million words, 1972 only some 115,000, or about what I would have done in an average month a decade earlier. Since _Dying Inside_ in 1971 I have written no novels, though doubtless that datum will be obsolete before this essay is published: my major work in California has been a novella, 'Born With the Dead', but a novel soon will be upon me, I think. Mainly I have written short stories, ostensibly science fiction, though the definition has required some stretching; they are strange and playful pieces, qualities evident in the titles of the two story collections I have made of them: _Unfamiliar Territory_ and _Capricorn Games._
Though one good quiver of the San Andreas Fault could destroy all I have built in a moment, I am at present in a comfortable situation, invulnerable to the demands of the marketplace, able to write what I choose and have it published by people I respect. The work comes slowly, partly because I revise so much, partly because the temptations of lovely California are forever calling me from my desk, partly because the old pressures–to prove myself artistically, to make myself secure financially – no longer operate on me. I keep close to nature, regularly visiting the mountains and deserts nearby and, when at home, labouring in my well-stocked and ever-expanding garden; I read a good deal, I edit anthologies of original material that bring me into contact with younger writers, I maintain many friendships both within and outside the science fiction cosmos, and, as the mood takes me, I pursue such old interests – music, archaeology, the cinema, whatever – as still attract me. Though I may eventually write more non-fiction, if only for the sake of learning more about the natural environment here by studying it systematically in preparation for a book, I expect that such writing as I do henceforth will be almost exclusively science fiction, or what passes for science fiction in my consciousness these days. I still respond to it as I did as a child for its capacity to open the gates of the universe, to show me the roots of time. I have little admiration for most of the science fiction I read today, and even less for the bulk of what I wrote myself before 1965, but I do go on reading it however short it falls of my ideal vision of it, and I do go on writing it in my fashion, pursuing an ideal vision there too and always falling short, but coming closer, coming closer now and then, close enough to lead me to continue.
**Alfred Bester:**
**my affair with science fiction**
I'm told that some science fiction readers complain that nothing is known about my private life. It's not that I have anything to conceal; it's simply the result of the fact that I'm reluctant to talk about myself because I prefer to listen to others talk about themselves. I'm genuinely interested, and also there's always the chance of picking up something useful. The professional writer is a professional magpie.
Very briefly: I was born on Manhattan Island 18 December 1913, of a middle-class, hard-working family. I was born a Jew but the family had a _laisser faire_ attitude toward religion and let me pick my own faith for myself. I picked Natural Law. My father was raised in Chicago, always a raunchy town with no time for the God bit. Neither had he. My mother is a quiet Christian Scientist. When I do something that pleases her she nods and says, 'Yes, of course. You were born in Science'. I used to make fun of her belief as a kid and we had some delightful arguments. We still do, while my father sits and smiles benignly. So my homelife was completely liberal and iconoclastic.
I went to the last Little Red Schoolhouse in Manhattan (now preserved as a landmark) and to a beautiful new high school on the very peak of Washington Heights (now the scene of cruel racial conflicts). I went to the University of Pennsylvania in Philadelphia where I made a fool of myself trying to become a Renaissance man. I refused to specialize and knocked myself out studying the humanities and the scientific disciplines. I was a maladroit on the crew and football squads, but I was the most successful member of the fencing team.
I'd been fascinated by science fiction ever since Hugo Gernsback's magazines first appeared on the stands. I suffered through the dismal years of space opera when science fiction was written by the hacks of pulp Westerns who merely translated the Lazy X ranch into the Planet X and then wrote the same formula stories, using space pirates instead of cattle rustlers. I welcomed the glorious epiphany of John Campbell whose _Astounding_ brought about the Golden Age of science fiction.
Ah! Science fiction, science fiction! I've loved it since its birth. I've read it all my life, off and on, with excitement, with joy, sometimes with sorrow. Here's a twelve-year-old kid, hungry for ideas and imagination, borrowing fairy tale collections from the library – _The Blue Fairy Book, The Red Fairy Book, The Paisley Fairy Book –_ and smuggling them home under his jacket because he was ahamed to be reading fairy tales at his age. And then came Hugo Gernsback.
I read science fiction piecemeal in those days. I didn't have much allowance so I couldn't afford to buy the magazines. I would loaf at the newsstand outside the stationery store as though contemplating which magazine to buy. I would leaf through a science fiction magazine, reading rapidly, until the proprietor came out and chased me. A few hours later I'd return and continue where I'd been forced to leave off. There was one hateful kid in summer camp who used to receive the _Amazing Quarterly_ in July. I was next in line and he was hateful because he was a slow reader.
It's curious that I remember very few of the stories. The H. G. Wells reprints, to be sure, and the very first book I ever bought was the collection of Wells' science fiction short stories. I remember 'The Captured Cross Section' which flabbergasted me with its arresting concept. I think I first read 'Flatland by A. Square' as an _Amazing_ reprint. I remember a cover for a novel titled, I think, _The Second Deluge._ It showed the survivors of the deluge in a sort of second ark gazing in awe at the peak of Mt Everest now bared naked by the rains. The peak was a glitter of precious gems. I interviewed Sir Edmund Hillary in New Zealand a few years ago and he never said anything about diamonds and emeralds. That gives one furiously to think.
Through high school and college I continued to read science fiction but, as I said, with increasing frustration. The pulp era had set in and most of the stories were about heroes with names like 'Brick Malloy' who were inspired to combat space pirates, invaders from other worlds, giant insects and all the rest of the trash still being produced by Hollywood today. I remember a perfectly appalling novel about a Negro conspiracy to take over the world. These niggers, you see, had invented a serum which turned them white, so they could pass, and they were boring from within. Brick Malloy took care of those black bastards. We've come a long way, haven't we?
There were a few bright moments. Who can forget the impact of Weinbaum's 'A Martian Odyssey'? That unique story inspired an entire vogue for quaint alien creatures in science fiction. 'A Martian Odyssey' was one reason why I submitted my first story to Standard Magazines; they had published Weinbaum's classic. Alas, Weinbaum fell apart and degenerated into a second-rate fantasy writer, and died too young to fulfill his original promise.
And then came Campbell who rescued, elevated, gave meaning and importance to science fiction. It became a vehicle for ideas, daring, audacity. Why, in God's name, didn't he come first? Even today science fiction is still struggling to shake off its pulp reputation, deserved in the past but certainly not now. It reminds me of the exploded telegony theory; that once a thoroughbred mare has borne a colt by a non-thoroughbred sire she can never bear another thoroughbred again. Science fiction is still suffering from telegony.
Those happy golden days! I used to go to secondhand magazine stores and buy back copies of _Astounding._ I remember a hot July weekend when my wife was away working in a summer stock company and I spent two days thrilling to 'Slan' And Heinlein's 'Universe'! What a concept, and so splendidly worked out with imagination and remorseless logic! Do you remember 'The Destroyer'? Do you remember Lewis Padgett's 'Mimsy Were The Borogroves'? That was originality carried to the fifth power. Do you remember . . . But it's no use. I could go on and on. The _Blue,_ the _Red_ and the _Paisley Fairy Books_ were gone forever. _,_
After I graduated from the university I really didn't know what I wanted to do with myself. In retrospect I realize that what I needed was a _Wanderjahr,_ but such a thing was unheard of in the States at that time. I went to law school for a couple of years, just stalling, and to my surprise received a concentrated education which far surpassed that of my undergraduate years. After thrashing and loafing, to the intense pain of my parents who would have liked to see me settled in a career, I finally took a crack at writing a science fiction story which I submitted to Standard Magazines. The story had the ridiculous title of 'Diaz-X'.
Two editors on the staff, Mort Weisinger and Jack Schiff, took an interest in me, I suspect mostly because I'd just finished reading and annotating Joyce's _Ulysses_ and would preach it enthusiastically without provocation, to their great amusement. They told me what they had in mind. _Thrilling Wonder_ was conducting a prize contest for the best story written by an amateur, and so far none of the submissions was worth considering. They thought 'Diaz-X' might fill the bill if it was whipped into shape. They taught me how to revise the story into acceptable form and gave it the prize, $50. It was printed with the title, 'The Broken Axiom'. They continued their professional guidance and I've never stopped being grateful to them.
I think I wrote perhaps a dozen acceptable science fiction stories in the next two years, all of them rotten, but I was without craft and experience and had to learn by trial and error. I've never been one to save things, I don't even save my mss, but I did hold on to the first four magazine covers on which my name appeared. _Thrilling Wonder Stories_ (15¢). On the lower left hand corner is printed 'Slaves of the Life Ray, a startling novelet by Alfred Bester'. The feature story was 'Trouble on Titan, A Gerry Carlyle Novel by Arthur K. Barnes'. Another issue had me down in the same bullpen, 'The Voyage to Nowhere by Alfred Bester'. The most delightful item is my first cover story in _Astonishing Stories_ (10¢). 'The Pet Nebula by Alfred Bester'. The cover shows an astonished young scientist in his laboratory being confronted by a sort of gigantic, radioactive seahorse. Damned if I can remember what the story was about.
Some other authors on the covers were: Neil R. Jones, J. Harvey Haggard, Ray Cummings (I remember that name), Harry Bates (his too) Kelvin Kent (sounds like a house name to me), E. E. Smith, Ph.D. (but of course), and Henry Kuttner with better billing than mine. He was in the lefthand _upper_ corner.
Mort Weisinger introduced me to the informal luncheon gatherings of the working science fiction authors of the late thirties. I met Henry Kuttner (who later became Lewis Padgett), Ed Hamilton, and Otto Binder, the writing half of Eando Binder. Eando was a sort of acronym of the brothers Earl and Otto Binder. E. and O. Earl died but Otto continued to use the well-known _nom de plume._ Malcolm Jameson, author of navy-orientated space stories, was there, tall, gaunt, prematurely grey, speaking in slow, heavy tones. Now and then he brought along his pretty daughter who turned everybody's head.
The vivacious compère of those luncheons was Manley Wade Wellman, a professional Southerner full of regional anecdotes. It's my recollection that one of his hands was slightly shrivelled, which may have been why he came on so strong for the Confederate cause. We were all very patient with that; after all, our side won the war. Wellman was quite the man-of-the-world for the innocent thirties; he always ordered wine with his lunch.
Henry Kuttner and Otto Binder were medium sized young men, very quiet and courteous, and entirely without outstanding features. Once I broke Kuttner up quite unintentionally. I said to Weisinger, 'I've just finished a wild story that takes place in a spaceless, timeless locale where there's no objective reality. It's awfully long, 20,000 words, but I can cut the first 5,000.' Kuttner burst out laughing. I do too when I think of the dumb kid I was. Once I said most earnestly to Jameson, 'I've discovered a remarkable thing. If you combine two story-lines into one the result can be tremendously exciting.' He stared at me with incredulity. 'Haven't you ever heard of plot and counterplot?' he growled. I hadn't. I discovered it all by myself.
Being brash and the worst kind of intellectual snob, I said privately to Weisinger that I wasn't much impressed by these writers who were supplying most of the science fiction for the magazines, and asked him why they received so many assignments. He explained, 'They may never write a great story but they never write a bad one. We know we can depend on them.' Having recently served my time as a magazine editor I now understand exactly what he meant.
When the comic book explosion burst, my two magi were lured away from Standard Magazines by the Superman Group. There was a desperate need for writers to provide scenarios (Wellman nicknamed them 'squinkas') for the artists, so Weisinger and Schiff drafted me as one of their writers. I hadn't the faintest idea of how to write a comic book script, but one rainy Saturday afternoon Bill Finger, the star comics writer of the time, took me in hand and gave me, a potential rival, an incisive, illuminating lecture on the craft. I still regard that as a high point in the generosity of one colleague to another.
I wrote comics for three or four years with increasing expertise and success. Those were wonderful days for a novice. Squinkas were expanding, there was a constant demand for stories, you could write three and four a week and experiment while learning your craft. The scripts were usually an odd combination of science fiction and 'Gangbusters'. To give you some idea of what they were like, here's a typical script conference with an editor I'll call Chuck Migg, dealing with a feature I'll call 'Captain Hero'. Naturally, both are fictitious. The dialogue isn't.
'Now listen,' Migg says, 'I called you down because we got to do something about Captain Hero.'
'What's your problem?'
'The book is closing next week and we're thirteen pages short. That's a whole lead story. We got to work one out now.'
'Any particular slant?'
'Nothing special, except maybe two things. We got to be original and we got to be realistic. No more fantasy.'
'Right.'
'So give.'
'Wait a minute, for Christ's sake. Who d'you think I am, Saroyan?'
Two minutes of intense concentration, then Migg says, 'How about this? A mad scientist invents a machine for making people go fast. So crooks steal it and hop themselves up. Get it? They move so fast they can rob a bank in a split second.'
'No.'
'We open a splash panel showing money and jewellery disappearing with wiggly lines and – Why no?'
'It's a steal from H. G. Wells.'
'But it's still original.'
'Anyway, it's too fantastic. I thought you said we were going to be realistic.'
'Sure I said realistic but that don't mean we can't be imaginative. What we have to – '
'Wait a minute. Hold the phone.'
'Got a flash?'
'Maybe. Suppose we begin with a guy making some kind of experiment. He's a scientist but not mad. This is a straight, sincere guy.'
'Gotcha. He's making an experiment for the good of humanity. Different narrative hook.'
'We'll have to use some kind of rare earth metal; cerium, maybe, or – '
'No, let's go back to radium. We ain't used it in the last three issues.'
'All right, radium. The experiment is a success. He brings a dead dog back to life with his radium serum.'
'I'm waiting for the twist.'
'The serum gets into his blood. From a lovable scientist he turns into a fiend.'
At this point Migg takes fire. 'I got it! I got it! We'll make like King Midas. This doc is a sweet guy. He's just finished an experiment that's gonna bring eternal life to mankind. So he takes a walk in his garden and smells a rose. Blooie! The rose dies. He feeds the birds. Wham! The birds plotz. So how does Captain Hero come in?'
'Well, maybe we can make it Jekyll and Hyde here. The doctor doesn't want to be a walking killer. He knows there's a rare medicine that'll neutralize the radium in him. He has to steal it from hospitals and that brings Captain Hero around to investigate.'
'Nice human interest.'
'But here's the next twist. The doctor takes a shot of the medicine and thinks he's safe. Then his daughter walks into the lab and when he kisses her she dies. The medicine won't cure him any more.'
By now Migg is in orbit. 'I got it! I got it! First we run a caption : IN THE LONELY LABORATORY A DREADFUL CHANGE TORTURES DR – whatever his name is – HE IS NOW DR RADIUM ! ! ! Nice name, huh?'
'Okay.'
'Then we run a few panels showing him turning green and smashing stuff and he screams : THE MEDICINE CAN NO LONGER SAVE ME **!** THE RADIUM IS EATING INTO MY BRAIN ! ! I'M GOING MAD, HA-HA-HA ! ! ! How's that for real drama?'
'Great.'
'Okay. That takes care of the first three pages. What happens with Dr Radium in the next ten?'
'Straight action finish. Captain Hero tracks him down. He traps Captain Hero in something lethal. Captain Hero escapes and traps Dr Radium and knocks him off a cliff or something.'
'No. Knock him into a volcano.'
'Why?'
'So we can bring Dr Radium back for a sequel. He really packs a wallop. We could have him walking through walls and stuff on account of the radium in him.'
'Sure.'
'This is gonna be a great character, so don't rush the writing. Can you start today? Good. I'll send a messenger up for it tomorrow.'
The great George Burns, bemoaning the death of vaudeville, once said 'there just ain't no place for kids to be lousy any more'. The comics gave me an ample opportunity to get a lot of lousy writing out of my system.
The line '... knocks him off a cliff or something' has particular significance. We had very strict self-imposed rules about death and violence. The Good Guys never deliberately killed. They fought, but only with their fists. Only villains used deadly weapons. We could show death coming – a character falling off the top of a high building _Aiggghhh!_ – and we could show the result of death – a body, but always face down. We could never show the moment of death; never a wound, never a rictus, no blood, at the most a knife protruding from the back. I remember the shock that ran through the _Superman_ office when Chet Gould drew a bullet piercing the forehead of a villain in a _Dick Tracy._
We had other strict rules. No cop could be crooked. They could be dumb but they had to be honest. We disapproved of Raymond Chandler's corrupt police. No mechanical or scientific device could be used unless it had a firm foundation in fact. We used to laugh at the outlandish gadgets Bob Kane invented (he wrote his own squinkas as a rule) for _Batman and Robin_ which, among ourselves, we called Batman and Rabinowitz. Sadism was absolutely taboo; no torture scenes, no pain scenes. And, of course, sex was completely out.
_Holiday_ tells a great story about George Horace Lorrimer, the awesome editor-in-chief of the _Saturday Evening Post,_ our sister magazine. He did a very daring thing for his time. He ran a novel in two parts and the first installment ended with the girl bringing the boy back to her apartment at midnight for coffee and eggs. The second installment opened with them having breakfast together in her apartment the following morning. Thousands of indignant letters came in and Lorrimer had a form reply printed 'the _Saturday Evening Post_ is not responsible for the behaviour of its characters between installments.' Presumably our comic book heroes lived normal lives between issues; Batman getting bombed and chasing ladies into bed, Rabinowitz burning down his school library in protest against something.
I was married by then and my wife was an actress. One day she told me that the radio show, _Nick Carter,_ was looking for scripts. I took one of my best comic book stories, translated it into a radio script and it was accepted. Then my wife told me that a new show, _Charlie Chan,_ was having script problems. I did the same thing with the same result. By the end of the year I was the regular writer on those two shows and branching out to _The Shadow_ and others. The comic book days were over, but the splendid training I received in visualization, attack, dialogue and economy stayed with me forever. The imagination must come from within; no one can teach you that. The ideas must come from without, and I'd better explain that.
Usually, ideas don't just come to you out of nowhere; they require a compost heap for germination, and the compost is diligent preparation. I spent many hours a week in the reading rooms of the New York public library at 42nd Street and Fifth Avenue. I read everything and anything with magpie attention for a possible story idea; art frauds, police methods, smuggling, psychiatry, scientific research, colour dictionaries, music, demography, biography, plays . . . the list is endless. I'd been forced to develop a speed-reading technique in law school and averaged a dozen books per session. I thought that one potential idea per book was a reasonable return. All that material went into my Commonplace Book for future use. I'm still using it and still adding to. it.
And so for the next five or six years I forgot comics, forgot science fiction and immersed myself in the entertainment business. It was new, colourful, challenging and – I must be honest – far more profitable. I wrote mystery, adventure, fantasy, variety, anything that was a challenge, a new experience, something I'd never done before. I even became the director on one of the shows, and that was another fascinating challenge.
I did write one straight science fiction show. It was called, I think, _Tom Corbett, Space Cadet._ It was a very low budget show and most of the action was played in bookfold sets. Even the doors weren't practical, they were painted and you had to dissolve before a character made his exit. I quit for an amusing reason. In those early days of tv there wasn't any standard form for typing scripts; each show had its own particular requirements. _Corbett,_ for reasons which I never could understand, insisted that all stage directions be typed in lower case and all dialogue in caps :
(Corbett enters office)
CORBETT: | YOU WANTED TO SEE ME, SIR?
---|---
CAPTAIN: | YES. AT EASE, CORBETT. SIT DOWN.
| (Corbett sits)
CAPTAIN: | WHAT I'M GOING TO TELL YOU IS TOP SECRET.
CORBETT: | SIR?
CAPTAIN: | IT's MUTINY.
CORBETT: | (Leaping up in consternation) NO !
CAPTAIN: | (Quietly) KEEP YOUR VOICE DOWN.
I just couldn't stand the characters continually shouting at each other.
Eventually a very slow and insidious poison began to diminish my pleasure; it was the constraints of network censorship and client control. There were too many ideas which I was not permitted to explore. Management said they were too different, the public would never understand them. Accounting said they were too expensive to do, the budget couldn't stand it. One Chicago client wrote an angry letter to the producer of one of my shows – 'Tell Bester to stop trying to be original. All I want is ordinary scripts'. That really hurt. Originality is the essence of what the artist has to offer. One way or another we must produce a new sound.
But I must admit that the originality compulsion can often be a nuisance to myself as well as others. When a concept for a story develops, a half-dozen ideas for the working-out come to mind. These are explored and dismissed. If they came that easily they can't be worthwhile. 'Do it the hard way,' I say to myself, and so I search for the hard way, driving myself and everybody around me quite mad in the process. I pace interminably, mumbling to myself. I go for long walks. I sit in bars and drink, hoping that an overheard fragment of conversation may give me a clue. It never happens but all the same, for reasons which I don't understand, I do get ideas in saloons.
Here's an example. Recently I was struggling with the pheromone phenomenon. A pheromone is an external hormone secreted by an insect, an ant, say, when it finds a food source. The other members of the colony are impelled to follow the pheromone trail, and they find the food, too. I wanted to extrapolate that to a man and I had to do it the hard way. So I paced and I walked and at last I went to a bar where I was nailed by a dumb announcer I knew who drilled my ear with his boring monologue. As I was gazing moodily into my drink and wondering how to escape, the hard way came to me. 'He doesn't _leave_ a trail,' I burst out. 'He's impelled to _follow_ a trail.' While the announcer looked at me in astonishment I whipped out my notebook and wrote 'Death left a pheromone trail for him; death in fact, death in the making, death in the planning.'
So, out of frustration, I went back to science fiction in order to keep my cool. It was a safety valve, an escape hatch, therapy for me. The ideas which no show would touch could be written as science fiction stories and I could have the satisfaction of seeing them come to life. (You must have an audience for that.) I wrote perhaps a dozen and a half stories, most of them for _Fantasy & Science Fiction_ whose editors, Tony Boucher and Mick McComas, were unfailingly kind and appreciative.
I wrote a few stories for _Astounding,_ and out of that came my one demented meeting with the great John W. Campbell, Jr. I needn't preface this account with the reminder that I worshipped Campbell from afar. I had never met him; all my stories had been submitted by mail. I hadn't the faintest idea of what he was like, but I imagined that he was a combination of Bertrand Russell and Ernest Rutherford. So I sent off another story to Campbell, one which no show would let me tackle. The title was 'Oddy and Id' and the concept was Freudian, that a man is not governed by his conscious mind but rather by his unconscious compulsions. Campbell telephoned me a week later to say that he liked the story but wanted to discuss a few changes with me. Would I come to his office? I was delighted to accept the invitation despite the fact that the editorial offices of _Astounding_ were then the hell and gone out in the boondocks of New Jersey.
The editorial offices were in a grim factory that looked like and probably was a printing plant. The 'offices' turned out to be one small office, cramped, dingy, occupied not only by Campbell but by his assistant, Miss Tarrant. My only yardstick for comparison was the glamourous network and advertising agency offices. I was dismayed.
Campbell arose from his desk and shook hands. I'm a fairly big guy but he looked enormous to me, about the size of a defensive tackle. He was dour and seemed preoccupied by matters of great moment. He sat down behind his desk. I sat down on the visitor's chair.
'You don't know it,' Campbell said, 'you can't have any way of knowing it, but Freud is finished.'
I stared. 'If you mean the rival schools of psychiatry, Mr Campbell, I think – '
'No I don't. Psychiatry, as we know it, is dead.'
'Oh come now, Mr Campbell. Surely you're joking.'
'I have never been more serious in my life. Freud has been destroyed by one of the greatest discoveries of our time.'
'What's that?'
'Dianetics.'
'I never heard of it.'
'It was discovered by L. Ron Hubbard, and he will win the Nobel peace prize for it,' Campbell said solemnly.
'The peace prize? What for?'
'Wouldn't the man who wiped out war win the Nobel peace prize?'
'I suppose so, but how?'
'Through dianetics.'
**'** I honestly don't know what you're talking about, Mr Campbell.'
'Read this,' he said, and handed me a sheaf of long galley proofs. They were, I discovered later, the galleys of the very first dianetics piece to appear in _Astounding._
'Read them here and now? This is an awful lot of copy.'
He nodded, shuffled some papers, spoke to Miss Tarrant and went about his business, ignoring me. I read the first galley carefully, the second not so carefully as I became bored by the dianetics mishmash. Finally I was just letting my eyes wander along, but was very careful to allow enough time for each galley so Campbell wouldn't know I was faking. He looked very shrewd and observant to me. After a sufficient time I stacked the galleys neatly and returned them to Campbell's desk.
'Well?' he demanded. 'Will Hubbard win the peace prize?'
'It's difficult to say. Dianetics is a most original and imaginative idea, but I've only been able to read through the piece once. If I could take a set of galleys home and – '
'No,' Campbell said. 'There's only this one set. I'm rescheduling and pushing the article into the very next issue. It's that important.' He handed the galleys to Miss Tarrant. 'You're blocking it,' he told me. 'That's all right. Most people do that when a new idea threatens to overturn their thinking.'
'That may well be,' I said, 'but I don't think it's true of myself. I'm a hyperthyroid, an intellectual monkey, curious about everything.'
'No,' Campbell said, with the assurance of a diagnostician, 'You're a hyp-O-thyroid. But it's not a question of intellect, it's one of emotion. We conceal our emotional history from ourselves although dianetics can trace our history all the way back to the womb.'
'To the womb!'
'Yes. The foetus remembers. Come and have lunch.'
Remember, I was fresh from Madison Avenue and expense-account luncheons. We didn't go to the Jersey equivalent of Sardi's, '21', or even P. J. Clark's. He led me downstairs and we entered a tacky little lunchroom crowded with printers and file clerks; an interior room with blank walls that made every sound reverberate. I got myself a liverwurst on white, no mustard, and a coke. I can't remember what Campbell ate.
We sat down at a small table while he continued to discourse on dianetics, the great salvation of the future when the world would at last be cleared of its emotional wounds. Suddenly he stood up and towered over me. 'You can drive your memory back to the womb,' he said. 'You can do it if you release every block, clear yourself and remember. Try it.'
'Now?'
'Now. Think. Think back. Clear yourself. Remember! You can remember when your mother tried to abort you with a button hook. You've never stopped hating her for it.'
Around me there were cries of 'BLT down, hold the mayo. Eighty-six on the English. Combo rye, relish. Coffee shake, pick up.' And here was this grim tackle standing over me, practising dianetics without a licence. The scene was so lunatic that I began to tremble with suppressed laughter. I prayed. 'Help me out of this, please. Don't let me laugh in his face. Show me a way out.' God showed me. I looked up at Campbell and said, 'You're absolutely right, Mr Campbell, but the emotional wounds are too much to bear. I can't go on with this.'
He was completely satisfied. 'Yes, I could see you were shaking.' He sat down again and we finished our lunch and returned to his office. It developed that the only changes he wanted in my story was the removal of all Freudian terms which dianetics had now made obsolete. I agreed, of course; they were minor and it was a great honour to appear in _Astounding_ no matter what the price. I escaped at last and returned to civilization where I had three double gibsons and don't be stingy with the onions.
That was my one and only meeting with John Campbell and certainly my only story conference with him. I've had some wild ones in the entertainment business but nothing to equal that. It reinforced my private opinion that a majority of the science fiction crowd, despite their brilliance, were missing their marbles. Perhaps that's the price that must be paid for brilliance.
One day, out of the clear sky, Horace Gold telephoned to ask me to write for _Galaxy_ which he had launched with tremendous success. It filled an open space in the field; _Astounding_ was hard science, _Fantasy & Science Fiction_ was wit and sophistication, _Galaxy_ was psychiatry-orientated. I was flattered but begged off, explaining that I didn't think I was much of a science fiction author compared to the genuine greats.
'Why me?' I asked, 'you can have Sturgeon, Kornbluth, Asimov, Heinlein.'
'I've got them,' he said, 'and I want you.'
'Horace, you're an old script writer so you'll understand. I'm tied up with a bitch of a tv show starring a no-talent. I've got to write continuity for him, quiz sections for him to emcee and dramatic sketches for him to mutilate. He's driving me up the wall. His agent is driving me up the wall. I really haven't got the time.'
Horace didn't give up. He would call every so often to chat about the latest science fiction, new concepts, what authors had failed and how they'd failed. In the course of these gossips he contrived to argue that I was a better writer than I thought and to ask if I didn't have any ideas that I might be interested in working out.
All this was on the phone because Horace was trapped in his apartment. He'd had shattering experiences in both the European and Pacific Theatres during World War II and had been released from the service with complete agoraphobia. Everybody had to come to his apartment to see him, including his psychiatrist. Horace was most entertaining on the phone; witty, ironic, perceptive, making shrewd criticisms of science fiction.
I enjoyed these professional gossips with Horace so much that I began to feel beholden to him; after all, I was more or less trapped in my workshop, too. At last I submitted perhaps a dozen ideas for his judgment. Horace discussed them all, very sensibly and realistically, and suggested combining two different ideas into what ultimately became _The Demolished Man._ I remember _one_ of the ideas only vaguely; it had something to do with extra sensory perception but I've forgotten the gimmick. The other I remember quite well. I wanted to write a mystery about a future in which the police are armed with time machines so that if a crime is committed they could trace it back to its origin. This would make crime impossible. How then, in an open story, could a clever criminal outwit the police?
I'd better explain 'open story'. The classic mystery is the closed story or whodunnit. It's a puzzle in which everything is concealed except the clues carefully scattered through the story. It's up to the audience to piece them together and solve the puzzle. I had become quite expert at that. However, I was carrying too many mystery shows and often fell behind in my deadlines, a heinous crime, so occasionally I would commit the lesser crime of stealing one of my scripts from Show A and adapting it for Show B.
I was reading a three-year-old Show A script for possible theft when it dawned on me that I had written all the wrong scenes. It was a solid story but in the attempt to keep it a closed puzzle I had been forced to omit the real drama in order to present the perplexing results of the behind-the-scenes action. So I developed for myself a style of action-mystery writing in which everything is open and known to the audience, every move and counter-move, with only the final resolution coming as a surprise. The technique is a commonplace today. This is an extremely difficult form of writing; it requires you to make your antagonists outwit each other continually with ingenuity and resourcefulness.
Horace suggested that instead of using time machines as the obstacle for the criminal I use ESP. Time travel, he said, was a pretty worn out theme, and I had to agree. ESP, Horace said, would be an even tougher obstacle to cope with, and I had to agree.
'But I don't like the idea of a mind-reading detective,' I said, 'it makes him too special.'
'No, no,' Horace said. 'You've got to create an entire Esper society.'
And so the creation began. We discussed it on the phone almost daily, each making suggestions, dismissing suggestions, adapting and revising suggestions. Horace was, at least for me, the ideal editor, always helpful, always encouraging, never losing his enthusiasm. He was opinionated, God knows, but so was I, perhaps even more than he. What saved the relationship was the fact that we both knew we respected each other; and our professional concentration on the job. For professionals the job is the boss.
The writing began in New York. When my show went off for the summer, I took the ms out to our summer cottage on Fire Island and continued there. I remember a few amusing incidents. For a while I typed on the front porch. Wolcott Gibbs, the _New Yorker_ drama critic, lived up the street and every time he passed our cottage and saw me working he would denounce me. Wolcott had promised to write a biography of Harold Ross that summer and hadn't done a lick of work yet. I. F. (Izzy) Stone dropped in once and found himself in the midst of an animated discussion of political thought as reflected by science fiction. Izzy became so fascinated that he asked us to take five while he ran home to put a fresh battery in his hearing-aid.
I used to go surf-fishing every dawn and dusk. One evening I was minding my own business, casting and thinking of nothing in particular when the idea of using typeface symbols in names dropped into my mind. I reeled in so quickly that I fouled my line, rushed to the cottage and experimented on the typewriter. Then I went back through the ms and changed all the names. I remember quitting work one morning to watch an eclipse and it turned cloudy. Obviously somebody up there didn't approve of eclipse-breaks. And so, by the end of the summer, the novel was finished. My working title had been 'Demolition'. Horace changed it to _The Demolished Man._ Much better, I think.
The book was received with considerable enthusiasm by the _Galaxy_ readers, which was gratifying but surprising. I hadn't had any conscious intention of breaking new trails, I was just trying to do a craftsmanlike job. Some of the fans' remarks bemused me. 'Oh, Mr Bester! How well you understand women.' I never thought I understood women. 'Who were the models for your characters?' They're surprised when I tell them that the model for one of the protagonists was a bronze statue of a Roman emperor in the Metropolitan museum. It's haunted me ever since I was a child. I read the emperor's character into the face and when it came time to write this particular fictional character I used my emperor for the mould.
The _reclame_ of the novel turned me into a science fiction somebody and people were curious about me. I was invited to gatherings of the science fiction Hydra Club where I met the people I was curious about; Ted Sturgeon, Jim Blish, Tony Boucher, Ike Asimov, Avram Davidson, then a professional Jew wearing a yarmulka, and many others. They were all lunatic (So am I. It takes one to spot one.) and convinced me again that most science fiction authors have marbles missing. I can remember listening to an argument about the correct design for a robot which became so heated that for a moment I thought Judy Merril was going to punch Lester del Rey in the nose. Or maybe it was _vice versa._
I was particularly attracted to Blish and Sturgeon. Both were soft-spoken and charming conversationalists. Jim and I would take walks in Central Park during his lunch hour (he was then working as a public relations officer for a pharmaceutical house) and we would talk shop. He was very serious. Although I was an admirer of his work I felt that it lacked the hard drive to which I'd been trained, and I constantly urged him to attack his stories with more vigour. He never seemed to resent it, or at least was too courteous to show it. His basic problem was how to hold down a PR writing job and yet write creatively on the side. I had no advice for that. It's a problem which very few people have solved.
Sturgeon and I used to meet occasionally in bars for drinks and talk. Ted's writing exactly suited my taste which is why I thought he was the finest of us all. But he had a quality which amused and exasperated me. Like Mort Sahl and a few other celebrities I've interviewed – Tony Quinn is another – Ted lived on crisis and if he wasn't in a crisis he'd create one for himself. His life was completely disorganized, so it was impossible for him to do his best work consistently. What a waste !
I'd written a contemporary novel based on my tv experiences and it had a fairly decent reprint sale and at last sold to the movies. My wife and I decided to blow the loot on a few years abroad. We put everything into storage, contracted for a little English car, stripped our luggage down to the bare minimum and took off. The only writing materials I took with me were a portable, my Commonplace Book, a thesaurus and an idea for another science fiction novel.
For some time I'd been toying with the notion of using the _Count of Monte Cristo_ pattern for a story. The reason is simple; I'd always preferred the anti-hero and I'd always found high drama in compulsive types. It remained a notion until we bought our cottage on Fire Island and I found a pile of old _National Geographics._ Naturally I read them and came across a most interesting piece on the survival of torpedoed sailors at sea. The record was held by a Philippine cook's helper who lasted for something like four months on an open raft. Then came the detail that racked me up. He'd been sighted several times by passing ships which refused to change course to rescue him because it was a Nazi submarine trick to put out decoys like this. The magpie mind darted down, picked it up, and the notion was transformed into a developing story with a strong attack.
_The Stars My Destination_ (I've forgotten what my working title was) began in a romantic white cottage down in Surrey. This accounts for the fact that so many of the names are English. When I start a story I spend days reading maps and telephone directories for help in putting together character names – I'm very fussy about names – and in this case I used English maps and directories; I'm compelled to find or invent names with varying syllables; one, two, three, and four. I'm extremely sensitive to tempo. I'm also extremely sensitive to word colour and context. For me there is no such thing as a synonym.
The book got under way very slowly and by the time we left Surrey for a flat in London I had lost momentum. I went back, took it from the top and started all over again, hoping to generate steam pressure. I write out of hysteria. I bogged down again and I didn't know why. Everything seemed to go wrong. I couldn't use a portable but the only standard machines I could rent had English keyboards. That threw me off. English ms paper was smaller than the American and that threw me off. And I was cold, cold, cold. So in November we packed and drove to the car ferry at Dover, with the fog snapping at our ass all the way, crossed the Channel and drove south to Rome.
After many adventures we finally settled into a penthouse apartment on the Piazza della Muse. My wife went to work in Italian films. I located the one (1) standard typewriter in all Rome with an American keyboard and started in again, once more taking it from the top. This time I began to build up momentum, very slowly, and was waiting for the hysteria to set in. I remember the day that it came vividly.
I was talking shop with a young Italian film director for whom my wife was working, both of us beefing about the experimental things we'd never been permitted to do. I told him about a note on synesthesia which I'd been dying to write as a tv script for years. I had to explain synesthesia – this was years before the exploration of psychedelic drugs – and while I was describing the phenomenon I suddenly thought, 'Jesus Christ! This is for the novel. It leads me into the climax'. And I realized that what had been holding me up for so many months was the fact that I didn't have a fiery finish in mind. I must have an attack and a finale. I'm like the old Hollywood gag – 'start with an earthquake and build to a climax'.
The work went well despite many agonies. Rome is no place for a writer who needs quiet. The Italians _fa rumore_ (make noise) passionately. The pilot of a Piper Cub was enchanted by a girl who sunbathed on the roof of a mansion across the road and buzzed her, and me, every morning from seven to nine. There were frequent informal motorcycle rallies in our piazza and the Italians always remove the mufflers from their vehicles; it makes them feel like Tazio Nuvolare. On the other side of our penthouse a building was in construction and you haven't heard _rumore_ until you've heard stonemasons talking politics.
I also had research problems. The official US library was woefully inadequate. The British Consul library was a love and we used it regularly, but none of their books was dated later than 1930, no help for a science fiction writer needing data about radiation belts. In desperation I plagued Tony Boucher and Willy Ley with letters asking for information. They always came through, bless them, Tony on the humanities – 'Dear Tony, what the hell is the name of that Russian sect that practised self-castration? Slotsky? Something like that.' – Willy on the disciplines – 'Dear Willy, how long could an unprotected man last in naked space? Ten minutes? Five minutes? How would he die?'
The book was completed about three months after the third start in Rome; the first draft of a novel usually takes me about three months. Then there's the pleasant period of revision and rewriting; I always enjoy polishing. What can I say about the material? I've told you about the attack and the climax. I've told you about the years of preparation stored in my mind and my Commonplace Book. If you want the empiric equation for my science fiction writing, for all my writing, in fact, it's :
I must enlarge on this just a little. The mature science fiction author doesn't merely tell a story about Brick Malloy vs The Giant Yeastmen from Gethsemane. He makes a statement through his story. What is the statement? Himself, his own dimension and depth. His statement is seeing what everybody else sees but thinking what no one else has thought, and having the courage to say it. The hell of it is that only time will tell whether it was worth saying.
Back in London the next year I was able to meet the young English science fiction authors through Ted Carnell and my London publisher. They gathered in a pub somewhere off the Strand. They were an entertaining crowd, speaking with a rapidity and intensity that reminded me of a debating team from the Oxford Union. And they raised a question which I've never been able to answer : Why is it that the English science fiction writers, so brilliant socially, too often turn out rather dull and predictable stories? There are notable exceptions, of course, but I have the sneaky suspicion that they had American mothers.
John Wyndham and Arthur Clarke came to those gatherings. I thought Arthur rather strange, very much like John Campbell, utterly devoid of a sense of humour and I'm always ill-at-ease with humourless people. Once he pledged us all to come to the meeting the following week; he would show slides of some amazing underwater photographs he had taken. He did indeed bring a projector and slides and show them. After looking at a few I called, 'Damn it, Arthur, these aren't underwater shots. You took them in an aquarium. I can see the reflections in the plate glass'. And it degenerated into an argument about whether the photographer and his camera had to be underwater too.
It was around this time that an event took place which will answer a question often asked me : Why did I drop science fiction after my first two novels? I'll have to use a flashback, a device I despise, but I can't see any other way out. A month before I left the States my agent called me in to meet a distinguished gentleman, senior editor of _Holiday_ magazine, who was in search of a feature on television. He told me that he'd tried two professional magazine writers without success, and as a last resort wanted to try me on the basis of the novel I'd written about the business.
It was an intriguing challenge. I knew television but I knew absolutely nothing about magazine piece-writing. So once again I explored, experimented and taught myself. _Holiday_ liked the piece so much that they asked me to do pieces on Italian, French and English tv while I was abroad, which I did. Just when my wife and I had decided to settle in London permanently, word came from _Holiday_ that they wanted me to come back to the States. They were starting a new feature called 'The Antic Arts' and wanted me to become a regular monthly contributor. Another challenge. I returned to New York.
An exciting new writing life began for me. I was no longer isolated in my workshop; I was getting out and interviewing stimulating people in interesting professions. Reality had become so colourful for me that I no longer needed the therapy of science fiction. And since the magazine imposed no constraints on me, outside of the practical requirements of professional magazine technique, I no longer needed a safety valve.
I wrote scores of pieces, and I confess that they were much easier than fiction, so perhaps I was lazy. But try to visualize the joy of being sent back to your old university to do a feature on it, going to Detroit to test-drive their new cars, covering the NASA centers, taking the very first flight of the Boeing 747, interviewing Sophia Loren in Pisa, De Sica in Rome, Peter Ustinov, Sir Laurence Olivier (they called him Sir Larry in Hollywood), Mike Todd and Elizabeth Taylor, George Balanchine. I interviewed and wrote, and wrote, and wrote, until it became cheaper for _Holiday_ to hire me as Senior Editor, and here was a brand new challenge.
I didn't altogether lose touch with science fiction; I did book reviews for _Fantasy & Science Fiction_ under Bob Mills' editorship and later Avram Davidson's. Unfortunately, my standards had become so high that I seemed to infuriate the fans who wanted special treatment for science fiction. My attitude was that science fiction was merely one of many forms of fiction and should be judged by the standards which apply to all. A silly story is a silly story whether written by Robert Heinlein or Norman Mailer. One enraged fan wrote in to say that I was obviously going through change of life.
Alas, all things must come to an end. _Holiday_ failed after a robust twenty-five years; my eyes failed, like poor Congreve's; and here I am, here I am, back in my workshop again, immured and alone, and so turning to my first love, my original love, science fiction. I hope it's not too late to rekindle the affair. Ike Asimov once said to me 'Alfie, we broke new trails in our time but we have to face the fact that we're over the hill now.' I hope not, but if it's true I'll go down fighting for a fresh challenge.
What am I like? Here's as honest a description of myself as possible. You come to my workshop, a three-room apartment, which is a mess, filled with books, mss, typewriters, telescopes, microscopes, reams of typing paper, chemical glassware. We live in the apartment upstairs and my wife uses my downstairs kitchen for a storeroom. This annoys me; I used to use it as a laboratory. Here's an interesting sidelight. Although I'm a powerful drinker I won't permit liquor to be stored there; I won't have booze in my workshop.
You find me on a high stool at a large drafting table editing some of my pages. I'm probably wearing flimsy pyjama bottoms, an old shirt and am barefoot; my customary at-home clothes. You see a biggish guy with dark brown hair going grey, a tight beard nearly all white and the dark brown eyes of a sad spaniel. I shake hands, seat you, hoist myself on the stool again and light a cigarette, always chatting cordially about anything and everything to put you at your ease. However, it's possible that I like to sit higher than you because it gives me a psychological edge. I don't think so, but I've been accused of it.
My voice is a light tenor (except when I'm angry; then it turns harsh and strident) and is curiously inflected. In one sentence I can run up and down an octave. I have a tendency to drawl my vowels. I've spent so much time abroad that my speech pattern may seem affected, for certain European pronunciations cling to me. I don't know why. GA-rahj for garage, the French _r_ in the back of the throat, and if there's a knock on the door I automatically holler, _'_ _Avanti!'_ a habit I picked up in Italy.
On the other hand my speech is larded with the customary profanity of the entertainment business, as well as Yiddish words and professional phrases. I corrupted the WASP _Holiday_ office. It was a camp to have a blond junior editor from Yale come into my office and say, 'Alfie, we're having a _tsimmis_ with the theater piece. That _goniff_ won't rewrite'. What you don't know is that I always adapt my speech pattern to that of my vis-à-vis in an attempt to put him at his ease. It can vary anywhere from burley (burlesque) to Phi Beta Kappa.
I try to warm you by relating to you, showing interest in you, listening to you. Once I sense that you're at your ease I shut up and listen. Occasionally I'll break in to put a question, argue a point, or ask you to enlarge on one of your ideas. Now and then I'll say 'wait a minute, you're going too fast. I have to think about that.' Then I stare into nowhere and think hard. Frankly, I'm not lightning, but a novel idea can always launch me into outer space. Then I pace excitedly, exploring it out loud.
What I don't reveal is the emotional storms that rage within me. I have my fair share of frustrations and despairs, but I was raised to show a cheerful countenance to the world and suffer in private. Most people are too preoccupied with their own troubles to be much interested in yours. Do you remember Viola's lovely line in _Twelfth Night?_ 'And with a green and yellow melancholy, she sat like Patience on a monument, smiling at grief.'
I have some odd mannerisms. I use the accusing finger of a prosecuting attorney as an exclamation point to express appreciation for an idea or a witticism. I'm a 'toucher', hugging and kissing men and women alike, and giving them a hard pat on the behind to show approval. Once I embarrassed my boss, the _Holiday_ editor-in-chief, terribly. He'd just returned from a junket to India and, as usual, I breezed into his office and gave him a huge welcoming hug and kiss. Then I noticed he had visitors there. My boss turned red and told them 'Alfie Bester is the most affectionate straight in the world.'
I'm a faker, often forced to play the scene. In my time I've been mistaken for a fag, a hardhat, a psychiatrist, an artist, a dirty old man, a dirty young man, and I always respond in character and play the scene. Sometimes I'm compelled to play opposites – my fast to your slow, my slow to your fast – all this to the amusement and annoyance of my wife. When we get home she berates me for being a liar and all I can do is laugh helplessly while she swears she'll never trust me again.
I do laugh a lot, with you and at myself, and my laughter is loud and uninhibited. I'm a kind of noisy guy. But don't ever be fooled by me even when I'm clowning. That magpie mind is always looking to pick up something.
**A BRIEF BIOGRAPHY**
Born: | 18 Dec. 1913
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Height : | 6 ft. 1 in.
Weight: | 180 lbs.
Hair: | Brown-grey
Eyes: | Brown
Beard: | Grey-brown
Scars: | None
Tattoos : | None
I.Q. (1928): | 119
Born 18 December 1913 in New York City and raised there. New York public schools and the University of Pennsylvania, class of 1935. Suffered from what later came to be known as TMA – too many aptitudes. By a fluke sold a story and dropped all the other A's. Professional writer ever since; fiction, comics, radio, tv, interviewer, editor. Not very good to start with but hopefully have improved. Never learned how to spell.
Married in 1936 and still married to the same lady, which is some sort of record these days. She is an actress who is now a swingin' vice-president of an advertising agency. No children (outside of myself). We decided never to give hostages to fortune. Also we were way ahead of Women's Lib. Little remains to be told. I have my ups and downs; big money, no money, but by some strange freak of luck my wife is earning most when I'm earning least. Perhaps we should have become professional gamblers; there's a gambler's streak in all artists. We stake our lives on everything we produce.
Ever since I became aware of writers and writing, I've been irritated by writers who claim to have been cooks, lumberjacks, and sandhogs; and whose photographs show them as unshaven persons in hairy sweaters at the tiller of a sloop. It's almost as though writers in America suffer from a terror of being thought effete . . . at least that class of writers who are of the masculine persuasion.
I have never been a cook, a lumberjack, a sandhog, or even a soda-jerk. I've been a writer all my life, and I don't give a damn who knows it. If required, I can produce my favourite photograph of myself, an epicene portrait of a burly gent in Edwardian waistcoat standing on the fire escape of a tenement, brandishing a beautifully furled umbrella. This is symbolic of my background and my taste.
I come from a middle class family, was born on 'The Rock', as genuine New York locals call Manhattan Island, and was raised on The Rock where I was the worst stick-ball player on Post Avenue between Dyckman Street and 204th Street. I went to George Washington High School and, much to my regret, did not play a part in any of the sex scandals that were the gossip of the cafeteria.
I went to the University of Pennsylvania where I was the worst center in the history of Penn football squads, and where I did not discover sulfanilamide. I was given this newfangled drug for an experiment in vital staining in physiology research, and reported that sulfa was useless. I also studied music composition and orchestration, and offended my classmates by arriving stinking to high heaven from dissection work in the comparative anatomy lab. It was a triumph and vindication for me when we first visited Leopold Stokowski's studio. It stank to high heaven from cooked cauliflower.
I began writing when I graduated from college in 1935, only because I'd tried law and medicine, given them up, and was floundering around, wondering what to do with myself. I sold a few stories of the old pulp science fiction sort to _Thrilling Wonder_ magazine (Ugh!), and then came the advent of comic books. This fantastic phenomenon exploded into a million dollar industry overnight, and there was a desperate search for writers who could be trained to turn out scenarios for the artists. I wrote stories for comic book heroes with unlikely names like 'the Green Lantern', 'the Star Spangled Kid', and 'Captain Marvel'. These were the days before sex and sadism polluted the comics, and we had wonderful training in visualization, and tight, crisp, action writing.
With this preparation it was only natural for me to shift to radio script writing. For years I wrote _Charlie Chan, Nick Carter, The Shadow_ and other shows. When the shift to television came, I went along with it, but rather reluctantly. Radio had been a tough, demanding craft, without room for fakers. Television was quite the opposite. It was around this time that I began writing science fiction again, solely for release from an entertainment medium which I disliked. After a few stories for _Astounding_ and _Fantasy & Science Fiction_ I was persuaded by Horace Gold of _Galaxy_ to write _The Demolished Man_ which had and still has a renown that amazes me.
After three years of tv writing I became so disgusted that I wrote a corrosive novel about the business called _Who He?_ (reprinted as _The Rat Race),_ took the money from the movie sale, got the hell out of the country and lived abroad for a couple of years. Some people claim I had to get out, but, alas, the book wasn't that corrosive. I only wish it had been. It must always be the mission of the writer to excite and astonish and, if possible, to infuriate.
While I was abroad I wrote another science fiction novel, _The Stars My Destination,_ and did some magazine pieces for _Holiday Magazine_ who insisted on bringing me back, kicking and screaming, to write their entertainment column. Since then I've been writing magazine features, occasional tv hour specials, and am in hock to my publisher for a couple of books.
I collect XIXth century scientific apparatus, am the world's worst amateur astronomer, am a trustee of my town on Fire Island where I'm also the world's worst surf-fisherman. Last year I took a refresher course in physiology at the graduate school at Washington Square and discovered that science has passed me by, so this year I'm studying bookbinding at the YWCA.
Almost everybody I know has a secret ambition to be a writer. I am a writer, and, typically, I have a secret ambition to be a scientist. I want to win the Nobel Prize for discovering something like 'Bester's Binomial' or 'Bester's Syndrome' or 'The Fissure of Bester'. I don't think I ever got over that sulfa goof.
**Harry Harrison:**
**the beginning of the affair**
What I find interesting, something I had never considered before I began to write this particular piece of prose, is that for many years I never had any sort of strong drive to become a writer. As a child I never had much of a strong drive to 'become' any particular sort of adult, which gave me many moments of depression and guilt. My interests were many and I began far more projects than I ever completed. At some time during high school I seem to have reached the decision that my fate was in the arts, but whether in writing or painting I could not be sure. I remember a period of mental coin-flipping where art won and I slanted my plans in that direction. Not that plans could be slanted very much, other than choosing which service to enter. Mine was the draftee generation; halfway through high school when the war began – with war not college waiting upon graduation.
The war did many good things for me, though I certainly did not appreciate them at the time. First, and most important, it kicked into existence a strong sense of survival that has been of great service since. It also terminated my childhood, a fact that I was certainly not grateful for at the time since growing up can be a painful process. I also learned to drink and curse, the universal coin of military life, but, more important, I was robbed of three years of my life without satisfactory return. At least I believed so for a long time, which is the same thing, and this gave me that singular capacity for solitary work, the drive to get it done, without which the freelance cannot succeed. That I am a writer now I can blame almost completely on science fiction.
I was a single child, always solitary and bookish, reading constantly from some tender age. _Loneliness_ is a word that has deepfelt meaning to me. Until the age of twelve I did not have a single friend among my classmates, nor did I belong to any gang or pack. Totally without companions I was absorbed in reading. Through the telescope of time I cannot quite make out if bookishness prevented personal relationships, or if rejection by my peer group drove me to books. I do know that I cannot remember a time when I could not read. The Queens Borough Public Library was quite a long walk from one of the apartments where we lived (these were depression days and we moved very often, dodging creditors and greedy landlords) so, in order not to waste valuable reading time, I cultivated the ability to walk and read at the same time, glancing up only when I came to a curb. But these library books, ten or twenty a week, were there to back up the gaps in my reading time when I could not read the pulps. While I would root through the shelves in the library with catholic taste, both fiction and non-fiction, my taste in the pulps was very exacting. No straight detective fiction, no westerns (and certainly no love-westerns, that awful mismatch that was poor Eisenhower's favourite reading) and no general fiction pulps. War, air war, railroad and science fiction, with science fiction heading the list as best of all. And of course the hero-centred pulps, but only when the hero was of a science fictional nature. Doc Savage, Operator 5, The Spider; wonderful stuff. And science fiction, always science fiction. As my tastes and enthusiasms changed and modified during the years sf was the single thing I was true to. John Buchan had a very long run for his money, while the works of C. S. Forester are the only ones to match the sf interest for longevity. Through everything the single unchanging pivot of my life was always science fiction. This is obvious only by hindsight; at the time I had other things to think about.
Coming out of the army was a traumatic experience and years passed before I could understand why. It seems very obvious now. I was a sergeant, I had been a gunnery instructor, a truck driver, an armourer, a power-operated turret and computing gunsight specialist, a prison guard with a loaded repeating shotgun to guard my charges when we went out in the garbage trucks, and a number of other interesting things. Though I loathed the army I was completely adjusted to it. I could not return to the only role I knew in civilian life, that of being a child.
Some months passed, lubricated alcoholically by what we called the 52–20 Club. Among the benefits received by those who survived the war was a mini-pension of $20 a week for 52 weeks while the veteran theoretically sought employment Employment was easy to dodge and the twenty bucks was enough for beer money. Days passed easily. But in a few months the fall term was due to begin and I resolved to end the lotus eating and go to college. Despite the feeling of anger that the girls and draft dodgers I had graduated high school with would be getting their degrees soon after I entered. (The term 'draft dodger' is used here in the military sense, not the civilian one. We members of the civilian army looked with envy and applause upon anyone smart enough or sick enough to miss the draft. When on furlough we helped friends still undrafted to avoid the fate we suffered. We jeered openly at anyone sucker enough to volunteer. It might have been different in the navy or the marines – but let us not forget they were drafting marines too in those days.)
Fate, and an overburdened educational system, saw to it that I enrolled at Hunter College which, up until this point, had been for girls only. In the mythology of New York City the Hunter girls had always been known for (a) their brains and (b) their ugliness. For once rumour proved to be fact. It was a sort of horrifying and warped mirror image of the army where everyone had been male and stupid. I did not finish out the term.
Hunter did one good thing for me, for it was there I met John Blomshield. I was in John's watercolour class and he spotted some infinitesimal drop of talent in my work. When I left Hunter I continued as a private pupil of his. I must say either too much or too little about this admirable man so, sadly, it will have to be too little. He was a master painter, a mannerist five centuries late for his school, an incomparable draftsman, a great portraitist, a professional artist and a civilized man of the world. The first one I had ever met. He had studied in Paris after the First World War and knew all of the people who to me were just names on canvases or book spines. He was a man of culture and gave me a glimpse of a world I had never known. The results of this exposure were not immediate – but the seed was planted. I continued to study with him privately, went to a number of art schools, and within the year I was a hardworking comic book hack who still spent three afternoons a week drawing the antique at the Metropolitan Museum of Art.
Within a few years I became a practising and moderately successful commercial artist, eventually to have a studio of my own with three other artists who depended upon me to draw in the accounts. Illustrations, lettering, book jackets, anything, but mostly those hack comic books. This sort of operation was called a 'factory' for obvious reasons. I draw the curtain gratefully over those years. I know that all those hours at the drawing board have helped my visual sense in my writing; I just wish there had not been so many of them. In the end I had stopped most of the drawing and began packaging comics instead. For a fee I would assemble a complete comic book ready for the engraver. The fee was so small I had to write most of the book myself, and even ink some of the stories. Which was fine since I was building a new career. When the comic business folded I used my editorial experience to move into editing pulps, which were gasping their last at the time. Then into writing. And it was the science fiction that got me into writing as I have said before.
I had been an sf reader since the age of seven, an active fan since thirteen when I wrote my first letter to an sf magazine, and an enthusiast all of the time. We were now into the fifties, that false spring of science fiction success. New magazines were being started every day and New York was the centre of the science fiction world. Everyone was there. If they didn't live there then by God they came through town to see the editors. There was an organization called the Hydra Club, which was the focal point of professional sf activity, and I had the pleasure of being a member – eventually even attaining the heights of the presidency. (Before being hurled down and out of the club completely in a power struggle. Many pros were ex-fans and fannish ways die hard.) I was Harry the artist because I did sf book jackets and magazine illustrations. But at heart I was Harry the fan and wallowing in a fannish dream of glory. Just look at who was there.
Many of the meetings took place in Fletcher Pratt's apartment, a wonderful great place on 58th Street. He was collaborating with L. Sprague de Camp at the time. Fletcher was chess champion of the club until Fred Brown took the crown away from him. I drew a game with Fred once; he always beat me after that. It was at Fletcher's place that we all met Olaf Stapledon when he came to the city. Fred Pohl was one of the pillars of the club, as was Judy Merril. So Cyril Kornbluth came too. The list of the others reads like a history of modern science fiction. Damon Knight, Frank Belknap Long, Lester del Rey, Phil (William Tenn) Klass, H. Beam Piper, Richard Wilson, Isaac Asimov, Sam Merwin, Bruce Elliot, Jerry Bixby, Doc Lowndes, Groff Conklin, Ted Sturgeon, George O. Smith, Hans Santesson, Willy Ley, Katherine McLean, Danny Keyes. Editors too, Marty Greenberg and Dave Kyle who founded Gnome Press, Larry Shaw, Horace Gold, Sam Merwin; Tony Boucher would stop by when in town. I'm sure I have forgotten some here, I apologize in advance, but what an overabundance of riches !
While science fiction was having a fine time and going along at a great clip my personal world was getting a bit unstable. An unfortunate first marriage was breaking up and the comic book business was going on the rocks, I lost fifty pounds during a massive throat infection and Damon Knight was editing _Worlds Beyond. Yes,_ they all do fit together. I was doing illustrations for Damon's magazine, some of the best I have ever done because Damon worked me very hard, and I would have illustrated almost all of the third issue if I hadn't been laid low by the infection. I was too sick to draw, but was able to write since a trembling finger on the typewriter key does not show.
Until this time I had done some comic script writing, some fillers for those same magazines, and a page or two of experimental work. I almost had to, being surrounded by writers and hearing writing talk so much. But my interest was still with the graphic arts. Bedridden, I wrote a story, and when I eventually staggered out I went to see Damon to pick up my assignments and to ask him what I should do with the story. He read it, quickly as I remember, and said that he would give me a hundred dollars for it. I was getting fifteen bucks a shot for the illos (or was it five?) so the price was certainly right. My story appeared in issue three of Damon's magazine, instead of the missing illos, and as soon as it was published the magazine promptly folded. At this time Fred Pohl had his own literary agency and he took me on as a client and my next sale was to him. He anthologized the story I had sold Damon. This sort of success was not repeated for a long time.
This first story had a terrible title ('I Walk Through Rocks', which Damon promptly changed to 'Rock Diver', a distinct improvement) but was well slanted commercially. I was a commercial artist, wasn't I? If I was going to write, I was going to write in the same way. I used a tried and true sf device, the matter penetrator, and expanded upon its possible uses. I borrowed a classic western plot, the claim jumpers, as a vehicle. I had many years to go before I attempted to practise the art of writing in addition to the craft.
Not that I intend to knock the craft. John Blomshield always said that painters should be like masons; learn to lay bricks before you build a house. I believe the same is true of writing. Since all my drawings were done on assignments by editors, ('Harrison, I want a three by four of an eight tentacled monster squashing a girl with big tits in a transparent space suit, line and none of your zip-a-tones or damn Benday, twelve bucks by tomorrow afternoon,') I went to editors to find out what kind of writing they wanted. What they wanted I wrote.
Men's adventures.
I WENT DOWN WITH MY SHIP,
I CUT OFF MY OWN ARM,
MAGRUDER – THE WOODEN CANNON GENERAL
Confessions.
HE THREW ACID IN MY FACE,
MY IRON LUNG BABY,
MY HUSBAND GAMBLED MY BODY AWAY
And more. I don't regret a one of them. I learned to write clearly, I learned to communicate with the reader, I learned to write to deadline, I learned a lot of things. And I stopped writing this sort of repetitive, unrewarding hack just as soon as I could.
Because I really wanted to write fiction, and particularly science fiction. I could not write it in New York because I did not have the sitting and thinking time, nor the correct atmosphere, nor the money to afford the sitting and thinking. I needed out. I also had the wife, Joan, who made everything possible, and a newborn son. She will read these words first, as she first reads everything I write. The baby, Todd, is now in college and the movie projectionist in the local fleapit and he will read them next. Moira, a very grown up fifteen, will read them about the same time. All of their lives and all of my work was made possible by fleeing New York.
Bruce Elliott, who had written commissioned stories for me when he was down and I was editing, was at this time managing editor of _Pic_ and _Picture Week_ and other dime pocket magazines, and said sure when I went to him for a job. I had my choice of art director or copy editor. I took the art job since I could do it in my sleep, and often did, and I wanted to save my writing energies for articles in the evenings. I worked until I had saved a bit of money; then I quit and we went to Mexico.
In those days everyone came to New York and no one left it, so everyone said I was psycho. Joan was about the only exception – and she was the only one who would suffer if I were cracked. Todd just smiled; a very happy baby. We sold our air conditioner (a New Yorker's most prized possession) and everything else we could not take with us, loaded up our Ford Anglia with cot and mattress and bags and left. We never came back. After a year in Mexico we moved to England, then to Italy. We returned for a few months to stay in Long Beach, New York, when Moira was born (I have very little good to say about the doctors in the Naples area), but left as soon as we could for Denmark. The children grew up there, a country made for children, then after a year in England we came to California where this is being written. If things go as planned we will be living in England again by the time you read this. Charles Monteith, who is, I am happy to say, my friend and editor as well as Brian's, once told me 'Harry, you are the most peripatetic fellow I know'. Perhaps he was right.
And what has this got to do with science fiction? Everything. With a few exceptions everything I wrote after going to Mexico was science fiction. Everything I sold was science fiction and every penny I earned was from science fiction. This has both a good and bad side. Good; I could write the thing I enjoyed most. Bad; until I learned my art I was a commercial writer and wrote with my market in mind at all times. This is a fine way to begin but no way to end. I wrote and sold the same novel four times before I had the nerve, pushed on reluctantly by Joan's encouragement, to write a book just for myself. But this was to come much later. At the time I was the most unliterary of writers, which also has both a good and bad side to it. The good is that I had no pretensions of art, no unattainable goals, no ambitions to make any kind of particular mark in writing. I was literarily very naive; in 1950 I took A _Passage_ to _India_ out of the library by mistake, thinking by a quick glimpse at the spine that it was by C. S. Forester. This 'mistake' was more than compensated for by the pleasure of discovering Forster's novel for myself; I still reread it at least once a year. My background of literary appreciation was only that of New York City elementary and high school, a singularly weak reed in the thirties and early forties. For me reading has always been primarily a pleasure, the reading of good fiction the greatest pleasure of all.
The bad side of this coin is that thinking myself an 'unliterary' person I took a very long time to write anything other than beginning, middle, end, action-moved, plot-supported, sexless, hardcore science fiction. But agonizing decisions about my work were still years in the future. I had yet to write my first novel. In fact I did not really think of it as a novel but rather a serial for _Astounding,_ the most important magazine in the history of science fiction.
Cuautla is a market town one hundred kilometres south of Mexico City. In 1956 there were no American tourists there and only a handful of gringo residents. Our furnished house was rented for thirty dollars a month, the full time maid was just five dollars for the same month, with food and drink in the equivalent price range. Despite this the money finally vanished and for a moment there we thought that everyone in New York had been right. Patience, worrying, writing and tequila at 75¢ a litre bridged this period and some money began to come in. Soon there was enough so that I could consider writing on speculation, that is doing fiction in the hopes it would eventually sell, rather than hacking out confessions and men's adventures to order. A few years earlier I had practised writing narrative hooks, just the hooks without stories to follow. (This is a pulp writer's term– which may very well have its roots in Grub Street – for the copy on the first page of manuscripts. Something lively, fascinating or intriguing must be on this half a page to 'hook' the editor, and eventually the reader, into turning to the second page of manuscript.) I had written one that intrigued me so much that I had to write the story to find out what the hook meant. This hook forms the first four paragraphs of _The Stainless Steel Rat,_ originally written as a novelette, later expanded into a novel that was eventually to have two sequels. From small acorns . . .
So I wrote the story and mailed it off to John Campbell who bought it for _Astounding._ Emboldened by this sale, while eating regularly with a little money in the bank, I outlined an idea for a novel and sent it to John. His response was warm and immediate and generous, the response he gave to all writers. He liked the idea and suggested ways of developing it that I had not considered. Buoyed up by his enthusiasm I began writing _Deathworld._
My ambition might be said to have outrun my talent because I wanted to do more than simply sell a story for money. I felt that there was too much empty writing in _Astounding,_ stories that just talked and described and never moved on any level. I wanted to bring the 'action' back into science fiction. (How times change. Today there is far too much pulp action and I lean in the opposite direction.) But I didn't want forced action, so I needed a plot where the movement was an integral part. I started the book slowly. I worked on it during the year we lived in Mexico, continued it in England and Italy, then brought it back to the United States. Still unfinished. Completely unsure of myself I sent John 30,000 words, about half the final length, and he responded with a deep editorial grumble that he thought he was reading the entire thing and looked forward to finishing it. I finished it. At that time I did not know John's habit of not writing acceptance notes since my previous sale to him had been done through an agent. (A terrible agent, given the bounce when we returned from Italy.) Off the book went to him and some days passed and a thin envelope arrived in the mail. There was no letter in it. Just a cheque for $2,100.
This was a very clear message. I could write and sell short stories and now I had sold the serialized version of my novel to the best paying and most prestigious magazine in the field. One half of the money would be a bit of financial security, the other half was spent on one way tickets for us all to Denmark.
Written so long after the fact all of this seems to have a sort of destiny to it, as though there were no other course, that this was the wisest possible decision, that money flowed in steadily to support my advancing career. Nothing could be further from the truth, everything happened by happinstance. We settled in Cuautla, Mexico because that is the town where the paved road ends. We went to England because a fan flight had been organized for the first world sf convention outside of North America. We went to Bromley because we met an English fan at the convention who lived there. We stayed there some months because money ran out and we couldn't pay our bill to leave the loathsome residential hotel where we were staying. With the bill paid we moved to a Pakistani rooming house in London because we had met Pakistani friends of Hans Santesson's. It was a very cold winter and when an old friend wrote us he would be going to Italy it did not take much more temptation to join Gary. (This was Gary Davis, World Citizen Number One.) I wrote a last true confession story to buy our way out of England and we went to join him on Capri, the island made famous by the song – which was about all we knew about it – where he had friends. It was some time before Gary showed up, followed closely by the police since he had entered the country illegally and without a passport. But we were well settled in by that time and could only wish him luck when he was arrested and sent off to the concentration camp at Frascati. But Gary had been staying in France with Dan Barry, another American expatriate, who came to join us in Capri a few months later. Dan is the well known artist who had just started doing the comic strip _Flash Gordon_ for King Features. He needed a writer and since there were very few ex-comic artist American science fiction script writers living in Europe I got the job. Then we went back to the United States to find a decent doctor and to bounce my agent, next to Denmark, because we had a friend there named Preben Zahle whom we had met in Mexico when I heard him trying to explain his automobile problems in French to a mechanic and I aided him with some translation. Preben was a very fine painter who also acted as consulting art director on _Tidens Kvinder,_ the leading Danish woman's magazine. Through his good offices I wrote some travel articles for the magazine and even collaborated with Joan on an article about travelling with children. About which we had amassed a good bit of empirical information. We had planned only to visit Denmark, but we liked it so much we stayed six years.
There is no way that this series of events could have been predicted. Certainly most things were not planned in advance. I have always felt that they just happened and, after happening, became part of the record. But perhaps there is more to the sequence than the blind workings of chance. When Brian read the first draft of this memoir he questioned my attitude. He felt that there was more than a touch of destiny in this. He has caused me to pause and think a bit.
I think he is right – if he will agree that we hold destiny in our hands and work to shape it. There are more ways _not_ to become a freelance, self-employed author than there are ways to attain this goal. The writers who do get through the obstacle course must climb over and under some strange things and put up with a good deal of assorted miseries along the way. If they are blocked they find a new path through the maze. The path can be very long and very difficult and the reading public should be aware that when they read about authors, as they are reading about them in this volume, they are reading about the victors. Among the fallen are writers with just as much talent – occasionally more – who are victims of what Cyril Connolly called 'the enemies of promise'. The best way to irritate a freelance is to tell him how _lucky_ he is to be able to lead the kind of life he does.
In this series, meeting Dan Barry was, in a way, the most fortunate event, because writing the daily and Sunday _Flash Gordon_ scripts gave me a solid bit of income that almost covered basic living expenses. Up until this point we had been broke too often; my camera and Joan's gold bracelet made many trips into and out of hock shops. (I learned early never to hock something at anything like its true value, not that uncle will allow this to happen in any case. The camera was worth $150 so I would hock it for $15. This made it worth getting out of pawn. To be held in reserve not only for taking the occasional picture but for another rainy day of hocking.) It is hard to write when hungry, cold and broke. Even harder to write when (as happened in Italy) the only money to hand was 100 lire (7 pence) which made the decision difficult whether to buy one more air mail stamp with it to write once more to the moron agent (the one I later dumped) for some money, or to buy milk for the baby with it.
Salvation came from two directions. Joan got credit from the local grocer for food and Hans Santesson sent some money in advance against a story yet to be written. A friend in need, he bought a number of robot stories from me for _Fantastic Universe_ which he edited. A good number of them were written after receipt of the money. They were later gathered into a collection titled _War With the Robots_ which was published in America, England, Germany, Italy and Spain.
Too much boom and bust is hard on the writer, not to mention his wife and children. For ten years I wrote the scripts for _Flash_ until I was so choked up with loathing for comics I could not type a word more. This is my hangup and not the fault of either Dan or Flash. They underwrote the slow years of getting established, until I could actually live on the income from my books and stories.
I was writing more, becoming more critical at the same time, more aware of what I was doing, aware that I was working to a pulp formula in my novels but too fearful to change a winning technique. _Deathworld_ has pulp motion and plot because that was all I knew. _The Stainless Steel Rat_ was clobbered together from two _Astounding_ novelettes and seemed a good idea at the time to make money. _Sense of Obligation,_ my next _ASF_ serial, was a slightly disguised _Deathworld._ The next was _The Ethical Engineer_ which appeared in paperback as _Deathworld 2_ – which is certainly clear enough description. The serials were popular with _ASF_ readers who always voted me a bonus, and the same for my short stories as well. The fans liked them too : _Deathworld_ lost out to a Heinlein novel for a Hugo. Bantam was selling an awful lot of copies of the paperback versions. The world was quite happy with my work; I wasn't. It was a strange time to get a critical conscience. I wanted to write better and I wanted to use different material.
Salvation came through the good offices of Joseph Heller and Brian Aldiss. I read _Catch-22_ which crystallized my thinking, and had met Brian a few years earlier. In addition to his friendship, which I value above all others, I appreciate his literary and critical skills. Brian is a prose stylist, one of the best writing in the English language and certainly the best in science fiction. As different as we are as writers and in background, we are in agreement on so many other things that friendship and collaboration come naturally. (It was after reading his sf literary criticism that I realized I wanted to read more like it. From this came the idea for _SF Horizons,_ the first critical sf magazine, which we published and edited.) We met often, at conventions and even stranger places, and we talked a good deal – oh yes how we talked and still do !
It was a little late, but my literary education had begun. Proximity to England helped because there is not as much snobbism there about sf being some kind of inferior form of fiction. For the first time in my life I met writers and critics who were not sf writers – yet who respected the sf field of literary endeavour. Brian himself, of course, was for endless years a full time literary editor who would later publish best-selling mainstream novels. Kingsley Amis, whose critical look at sf gets more than a nod in the title of this book, Bruce Montgomery, Robert Conquest and Geoff Doherty are the names that spring instantly to mind. If they were going to respect sf as part of literature I would have to look at it that way myself.
Heller and Voltaire demonstrated to me that some things are so awful that they can only be approached through the medium of humour. I had already written at least one humorous short story that was well received and anthologized, 'Captain Honario Harpplayer, RN', and I felt I could do more. All of my experimentation so far had been in the short story, since the time investment there is obviously much less than the novel. This was both good and bad because the 'experimental' did not do very well, not in these dark days of the early sixties. The quotes are around experimental there because my stories were nothing of the kind. They just fell outside the classic pulp taboos that still dominated the field.
The story of one of them 'The Streets of Ashkelon', is typical. I wrote this for a Judy Merril anthology of 'dangerous' ideas that was never published due to the publisher going broke. The story came back and went out, and returned rather quickly from all the American markets. It was too hot to handle since it had an atheist in it. This is the truth. Even my good friend, Ted Carnell, would not take it for the more liberal British _New Worlds._ I asked Brian if he had any idea what could be done with it. He had some critical remarks about the priest's characterization, which I agreed with, but said as well that he would like to use it in an anthology he was doing for Penguin. (And I must add that Carnell, once he knew the story would be anthologized, felt it would be all right to use in his magazine. This might indicate that his spine needed stiffening, but if so it indicates as well that the American editors had no spines at all.) The story has a happy ending in that it was eventually anthologized three times in the United States and translated into Swedish, Italian, Russian, Hungarian – and twice into French.
It was this sort of experience that made me hesitant to put the time into an entire novel that might not sell. At that period a novel a year was the most I could do and the thought of losing a year's book income was not to be considered. Until then all of my novels had been serialized in _ASF,_ bringing in nearly three thousand dollars with the bonus, and the novel I had in mind was certainly not for Campbell.
Eventually the artist triumphed over the businessman, ears became numb to the sound of hungry children crying in the background, and I contacted Damon Knight. Damon was acting as an sf literary scout for Berkley Books and I was sure he would be _simpatico_ to my needs. I sent him the first (and only) chapter I had written of an experimental novel titled _If You Can Read This You Are Too Damn Close._ With it were some one page character sketches and a few words about the kind of novel I wanted to attempt. Damon liked it and went to bat for me and extracted a $1,500 advance from Berkley. Taking a deep breath I climbed to my office in the attic, looked out across the frozen Øresund to the snowy shore of Sweden, and began writing _Bill, the Galactic Hero._
It was a shaking experience. I was doing less than half my normal wordage every day and greatly enjoying myself – at the time. Laughter all day at the typewriter – how I _do_ enjoy my own jokes – instant depression when I came down for dinner. Upon rereading, the stuff seemed awful. Or awfully way out; there had never been anything like it in sf before. Then back the next day for some more chuckling and suffering. Joan was a pillar of strength at this period, reading the copy and laughing out loud and saying it was great and get on with the job and stop muttering to yourself. I got on with it, finished it, had it typed and mailed off to Damon.
Who rejected it saying what I had here was an adventure story loused up with bad jokes. Take the jokes out and it would be OK.
Although everything eventually ended happily this was one or life's low moments. Tom Dardis, the editor-in-chief at Berkley, seemed to like the book, but he did not want to go over the head of his paid adviser. It was Tim Seldes, the Doubleday sf editor, who broke the ice. He greatly enjoyed the book and said he would buy it for that firm. Cheered on by this assurance Berkley agreed to publish it as well. In England, Hilary Rubinstein was then editor for Gollancz and he read, enjoyed and bought it as well. (Thus beginning a long and enduring relationship for he is now a literary agent, the best in Britain, and mine of course.) Fred Pohl bought the serial rights for _Galaxy_ and Mike Moorcock did the same in Britain for _New Worlds,_ then in the first flower of its new personality after Mike had replaced Carnell as editor.
Here was a message of some kind. Sf was growing and contained within its once pulp boundaries new and different markets. _Bill_ was positively not an _ASF_ serial and had not even been submitted there. (In later years I discovered that my judgment had been correct in this at least. One day John Campbell asked me why I had written this book. I said I would tell him if he told me why he had asked. His answer was that he had seen my name on the paperback and bought it – as if he did not have enough sf to read ! – and had hated it. I made some sort of waffling answer and worked hard to change the subject.) I felt that there must be a bigger market out there than I had imagined and perhaps I could now write for myself and please readers at the same time. This was a momentous discovery and marked a new period in my writing. Not that I didn't do the familiar to stay alive. _Deathworld 3_ and a number of Stainless Steel Rat books were still in the future, but I found I could experiment with new ideas and forms and still hope to sell them as well. This has a happy ending in that I now usually write only the kind of novels I want to write and enjoy good serial and book sales.
There was no sudden change in my life. The work crawled out and the checks crept in. I was still earning about as much as a non-union elevator operator in New York. Except I was not living in New York. At that time the dollar went much further in Europe than it does now. We lived well and enjoyed life. Each winter we would go skiing in Norway, every summer camping across the continent to Italy. We could only do this because, just ten years ago, the prices were unbelievable by present inflated standards. Norway? One person, round trip by overnight ferry with sleeping cabin from Copenhagen to Oslo, then train to the ski resort plus seven days room and full board there cost £20 for an adult. Half price for children. Camping for the four of us, in our rebuilt VW bus, formerly a Copenhagen taxi, averaged £5 a day. Including all gas, camping fees, booze, film, food – with dinner in a restaurant every night for four. Life was as pleasant as it ever can be, considering the number of dark things that are sweeping down the river towards us at all times, and I had a major book I wanted to do.
I had been working on it for some time, a total of five years of preparation in fact, just digging out the material to make an intelligent estimate of what life would be like in the year 2000 AD. At this time there were no popular nonfiction books on the dangers of overpopulation, overconsumption, pollution and allied problems. But there was a great deal of talk and speculation in the scientific journals that interested me greatly. Overpopulation had been a recurrent theme in sf for years, but the overpopulated future had always been the far future and was about as relevant to life today as E. E. Smith's Lensman.
My basic idea was a simple one : set a novel in the year 2000 just a few decades away, when the reader, and certainly the reader's children, would be around to see what the world would be like. But in setting a novel so close in time I had to extrapolate every detail of our lives and see that I got it as right as possible. I also wanted to write a more realistic novel than I had ever attempted before. I went to the specialists, the demographers and the petrologists and agronomists, and read a great number of very thick books. It took a great deal of time to write the novel, which was the longest I had ever done, since, in addition to getting my facts right, I had to write the realistic story of life in that world. I smile as I remember that I wanted to rush the book into print before the growing interest in these problems faded. As it was, _Make Room! Make Room!_ came out too early and vanished with a dull whiffling sound. But it was the only novel on these topics and when the world at large became aware of these problems it was bought for movie adaptation and turned into the film _Soylent Green_ – which, at times, bears a slight resemblance to the book.
If you want a neat commentary on contemporary values consider this; I discovered that MGM had been looking at this book for five years as a possible film. But they did not think the theme of overpopulation was an important enough one to shoot. However, when a cannibalism twist was added to the script they saw real possibilities and went ahead with contracts.
All of this was still in the future. What was coming next was a physical change in my life which, for better or worse, would affect my work. We left Denmark – for a number of reasons. We were tired of renting and wanted to buy – but could not in Denmark. Nor were we sure we wanted a permanent home there. There were more things like this; the children spoke more Danish than English, I had trouble talking English myself at times, a problem for a writer, there were family crises in the States. No one thing, but they all added up to a move. We weren't quite ready for the States yet and considered buying a house in England. We went there and rented for a year, but it came to nothing. It was a rather stern furnished house in a grimly middle class neighbourhood. This did not bother us, but horrified many English writer friends who had spent their life substances fleeing such a deadend place. But I shall always be fond of Banstead Road South in Sutton, Surrey. The year was fun and I wrote _The Technicolor Time Machine_ in the dining room which I took over for a study.
We have a Greek friend who owns some ships and he gave us a most reasonable passage on one of his vessels. We brought the decaying VW bus along as deck cargo, or ballast, since we had been offered only $100 as a trade-in on a new car. After a stop in New York we left, pursued by a blizzard, and crossed the continent looking for a warm spot to settle. (Reverse gear fell out of the bus which was fine since we planned only a one-way trip.) California seemed the best bet, for climate and work, and we reached San Diego in a rainstorm. Next day the sun shone and we came down a hill in the boondocks near the Mexican border and saw the house for us. We bought it a few weeks later.
This memoir is being written in that same house, a number of books, stories and seven years later. But the house is on the market and within a few months we hope to be out and moved back to England. Though this piece is about writing, and the writing of science fiction at that, I feel that some explanation of a move of this sort is needed. It is not a simple thing because too many factors are involved. Part of it, surely, is dissatisfaction with life in a country that could commit the crimes of Vietnam and not be ashamed. Or living under a government headed by the man whom Harry Truman called 'a shifty-eyed goddamn liar', a man who appears to have done his best to destroy the democratic form of government. I will try not to complain and say not why I am leaving, but what I am going to. No paradise ! I doubt if you will find a single Briton who claims that. It is a country we know well and respect, where life has a different pace that helps both my work and my existence, where there are many friends, where there is a fullness of things to do and see and enjoy and an entire continent of more of the same just a few miles away. A writer lives by ingesting from life and from books. I can read the same books anywhere. But when I walk out of my front door into the arid, sidewalkless streets of Southern California – or into the streets of Oxford or London, I am entering totally different worlds. Nor is the choice mine alone. Joan and the children are eager for the move.
I have never had great expectations for my work so I can truthfully say I am pleased with it and the way it is going. I keep saying that I don't enjoy editing, but I must or I would not still be doing it. Happily I do a lot of it with Brian, such as the annual best sf series, _The Astounding-Analog Reader_ and others. But writing is where the real action is and I know I have good books lurking in my future. (I can't see the form or idea for any of them now, which is perhaps another reason I am going to England.) I have learned to write and am still learning to write all of the time. My books have been well received and translated into a number of languages – eighteen at the last count – and I sell more and more of them every year. I have learned that when I put time, effort and love into a book or a story the final product is worth the effort. I hope to have more of all of these elements in England, but you will have to judge the resulting work on its own merits. I know where I have been, I know where I am, I know where I am going. I could not always say this; that I can now is a victory of sorts.
**Damon Knight:**
**knight piece**
I was born in Baker, Oregon, at midnight on 19 September 1922, the only child of Frederick Stuart Knight and Leola Damon Knight. They seem to have decided that I would be a writer even before I was born – at any rate, my father once told me he had chosen my name, Damon Francis Knight, to be a euphonious by-line on the model of 'Stuart Edward White'.
On both sides, my ancestors were middle-west Protestants. My father was taught that drinking, smoking, dancing and card-playing were all sinful; he relaxed his views on all but the first when he was grown. On my mother's side most of the men were ministers; I have a crayon portrait of her grandfather, a stern-featured man with a full beard and shoulder-length hair; and one of his wife, an even sterner crone who looks thirty years older, even allowing for the evident fact that she has lost all her teeth.
My father ran away from the South Dakota farm where he grew up when he was sixteen, went to the West Coast and put himself through college by washing dishes. He met my mother when she was an elementary schoolteacher in Bingen, Washington, and they were engaged; then he went off to teach for four years in a rural school in the Philippines. She broke off the engagement, but when he came back they were married anyhow.
In photographs he brought back from the Philippines my father is slender, but when I knew him his paunch and deep chest made him look stocky. He was forty when I was born, and my mother was thirty-five. It was her third pregnancy; the first two children were stillborn girls.
My father was a frustrated newspaperman, and taught a journalism class in the high school whose principal he became in 1928; his heroes were Irvin S. Cobb and Will Rogers. He was a shy man who could not express his emotions. Although he had run away from the farm, he always loved farming and believed in hard physical labour; he was concerned about me because I didn't sweat enough. He owned a farm left him by his father and kept tenants on it, hoping that when I grew up I might want to go and live there. He described it as good farmland but when I went there with him as a child and tramped over it with the tenant, it was nothing to my eye but a sea of dried mud.
When I was five or six my mother had what was called a nervous breakdown, as a consequence of which she remained nervous and jumpy and one eye bulged a little. I associate this with a recollection of driving in the country with my mother and passing a field below the road where there was a wrecked car and some people were groaning. I remember imitating their noises, thinking it was funny. After that she never drove, and my parents never went out together to visit or had company in our house. I accepted this without question then, but it is a mystery to me now. Although she got a little queer when she was older, and thought the family doctor, a much younger man, was secretly in love with her, there was nothing whatever wrong with her mind when I was a child. My mother was an affectionate and demonstrative woman who laughed easily. She spent hours reading to me from the Thornton W. Burgess books, and we both laughed until we cried over the adventures of Peter Rabbit.
It may be that neither of my parents was very sociable to begin with. I never saw any sign that either of them felt the lack of company. My father had his lodge meetings, about which he was forbidden to tell us anything, my mother pretended to believe that the Masons took off all their clothes and ran around in their little aprons. Both my parents went to the Riverside Church (non-denominational) every Sunday; they took me, too, until I said I didn't want to go any more. They never reproached me for this.
In the summers, while my father took more education courses in summer schools, my mother and I went to the seaside resort town of Newport, where her stepmother kept a boarding house called the Damon House. The stepmother was a wrinkled, shrewd old woman, famous for her table. (It was wasted on me – I wouldn't eat seafood.)
Neither my mother nor I knew how to swim. I had lessons at the Natatorium, but couldn't get over my paralyzing fear of water, and although the teacher told my mother I had swum a few strokes, in order to earn me a promised reward, it was a lie.
We spent the long afternoons on the beach. There were dunes of golden sand that broke away in chunks, nascent sandstone, and I pretended that it was gold and that I was rich. At low tide there were cratered rocks covered with barnacles that closed and squirted if you touched them. There were miles of flat sand on which to run trailing a stick or a long strand of kelp. A little farther up the coast there was another beach, accessible only at low tide, where the sand was covered with the polished shells of periwinkles. And somewhere there must have been crabs, for I remember bringing home a bucket of them, and waking up later to find them all over the walls.
I loved this place, and looked forward to it all year with longing and disbelief. There was a little library shaped like a lighthouse; on the boardwalk I remember on one side a jewellery shop whose windows were filled with polished pieces of moss agate and jasper; on the other, a candy shop in whose window the taffy machine endlessly revolved its shining arms. The taffy was hard and brittle; you bought it in porous chunks broken off with a hammer, and it crunched and melted sublimely in the mouth.
Although both my parents were teachers, neither one had any great habit of reading, and there were few books in the house. There was a copy of _Anthony Adverse,_ which my father had abstracted from the school library as too racy (but if so, I could never find the good parts), and a volume of Philippine fairy tales which I still own, and that was nearly all. We had a little illustrated dictionary, not Webster's, with fascinating colour plates of fruits and national flags. I remember my father reading a historical novel which had something in it about Greek fire, and my mother reading a modern novel called _if i have four apples._ In each case the event was memorable because it was unparalleled. Both of them had great respect for writing, however, or for any creative work, and often said that I was going to be an artist. I drew pictures from the time I could hold a pencil, and in my teens made some clumsy attempts at painting without any instruction.
Hood River, Oregon where my father was principal of the high school for twelve years, is a little town on the confluence of the Hood River and the Columbia. The climate is mild and wet. Two snow-capped mountains are visible from Hood River – Mt Hood and Mt Adams. The town is built on a hillside as steep as San Francisco's, and from walking up and down it to and from school, I could go as fast uphill as I could on level ground.
Although toward the end I longed to get away, looking back I can see that Hood River was not a bad place for children to grow up. The streets were ours for bicycling and skating; we even skated down Creamery Hill, reaching a velocity that would have crushed us to bits if we had run into anything, but we never did. In the summer evenings we gathered in a group of ten or twenty to play hide-and-seek or king of the hill or red-light. I remember the purple twilights and the scent of lilacs and the lonesome sound of 'Allee-allee-all's-in-free'. We played until it was too dark to see, and after; we hated to go in to bed.
Because of my slow physical development I began losing touch with my contemporaries when I was about eight, and got most of my ideas about life out of books. When I tried to apply these to the world around me, I was usually disappointed. _Boy's Life,_ for example, published a series of stories about a bunch of boys who had a secret club, with mysterious hailing signs and so on. I organized one like it on my block, but when I chalked the assembly symbol on the sidewalk, the other members went on riding their tricycles. Later I tried to organize another club which would assemble model airplanes and sell them for profit, but when I tried my first kit, I put it together wrong. In shop, I could not plane a board smooth or wash the paint out of a brush. I continued to read novels, especially English novels, because England was a long way away and I could believe that life was different there.
We had our local halfwit, a man named Warren Chaffee who could not talk without slobbering but was good at mechanical things and could fix toys for children; he had a hauling service and once turned in a bill to my parents that read: '2 kums 2 goes @ 50 _¢_ a went'. There was a retarded child across the street, a boy named Petie, on whom cruel jokes were played and who was brutal to others. Next door to him lived a girl named Zella Hendricks and her little brother, with both of whom I practised what we called 'doing naughty things' – feeling each other's bodies inexpertly under our clothes – until their mother caught me with my hand down the brother's shirt, looking for God knows what. When she came over to talk to my mother about it I tried to hold the door shut.
Our house in Hood River was one of two identical white frame cottages on adjoining lots in a neighbourhood of much older houses. It had a living room and dining room symbolically separated by a false beam; two bedrooms, bath and kitchen. When I grew too old to sleep in my mother's room my father hired a carpenter to help him and built on another room beside the back porch. The walls were made of Fir-Tex, a thick felt-like substance made of compacted wood fibres, and the floor and woodwork, at my request, were painted black. The living room and bedrooms had kalsomined walls which my father renewed every year or two, using an enormous brush. There was no central heating; the kitchen was kept warm by the old wood-burning range, and the living room first by a potbellied stove and later by an oil-burning space heater that was not much bigger. On cold nights we took hot-water bottles to bed with us.
Our street was just outside a respectable residential area and was not far from being a slum, although that never occurred to me then. Up the hill, separated from our house by their back garden and ours, was the elegant home of Mr Breckinridge, the school superintendent, whose daughter Ada May was my playmate up until the time she started wearing high heels and lipstick. Around us were old two-storey houses in various stages of decay; the one next door even had a barn, weathered grey like the house. The children who lived there were barefoot and ragged and had the grey-brown faces of the poor, but they were active, alert and good humoured. The oldest, a boy of sixteen or so, drew cartoons that were better than mine – and I was not modest about my drawing – then dropped them on the ground; I never could understand why he valued them so little. I kept everything, and counted my possessions like a miser. I knew and cherished every marble; when a boy cozened me into playing 'for keeps' and took four of mine, I wept. I was inconsolable when a tree surgeon cut off the low horizontal limb of the cherry tree in the front yard, the one I had used to climb on.
As I grew older I played with younger children, sometimes joined by another outcast, a boy even older than myself. As time advanced I lost even these playmates and fell back on solitary pleasures entirely. I attacked the Hood River library in various ways, by authors – all of Dickens, all of Dumas – then by subject – all the pirate books – and finally, at random. One of my pleasant memories is of some illness when the librarian sent me out a pile of books, all by authors new to me. I read children's books and fairy tales, but I also read romantic novels and novels of manners that I only half understood. I read a novel called _V.V.'s Eyes_ which had belonged to an uncle of mine, and found that he had written in the margin encouraging comments such as 'Go it, V.V!' This was my first experience with the defacers of books. For years I could not bring myself to make any mark in a book, even when I had it for review; now I do it, but always with a feeling of guilt, and I use a soft pencil in case anyone should want to erase what I write.
In the thirties I became aware that there were such things as pulp magazines. There were _Spicy Adventure_ and _Spicy Mystery,_ which I did not dare buy, even in the dingy little second-hand store at the bottom of a side street in town. There were air-war magazines which I did buy and devoured. One story concerned a squadron leader who was having headaches and whose hair was falling out; it turned out that a German agent had been concealing a capsule of radium under his pillow.
Then I saw and bought an issue of something called _Aamazing Stories._ It was bigger than other pulps, about 8½ x 11, and the cover, in sick pastels, showed two helmetted and white-suited men aiming rifles at a bunch of golliwogs. This was the August-September 1933 issue, and the cover story was 'Meteor-Men of Plaa' by Henry J. Kostkos. That was the beginning.
The illustrations in _Amazing,_ the work of a man named Leo Morey, were sketchy, grey, and ill-defined, but that somehow increased the mysterious and alien feeling. In Gernsback's magazines, especially the back issues, I admired the work of Frank R. Paul in other ways, but I got from it much the same satisfaction. Paul's drawings look a little quaint now, because of the knee-breeches and the statuesque poses, but he was endlessly fertile at inventing strange landscapes and filling them with the flora and fauna of strange worlds. These cover paintings and illustrations served as focal points for daydreaming. They supplied the visual information which the stories, as a rule, did not give, and they helped an adolescent reader dream himself into the world of the story.
Not all the science fiction magazines were available in Hood River, and I could not always afford to buy them, but when we made our annual family trip to Portland it was not only Jantzen's Beach (the amusement park) that drew me, it was the second-hand stores with their stacks of _Science Wonders_ and _Amazings._
On one visit I found a new magazine on the news-stands, one I had never even heard of before – _Astounding Stories._ Back in our hotel room I developed a fever; it turned out that I had the measles and we were all quarantined. My parents must have been chagrined, but I was blissful, lying there reading 'The Son of Old Faithful' by Raymond Z. Gallun.
At home, along one wall over my bed, I had shelves put up, and presently these shelves were filled with science fiction magazines. I read and reread every story, including those I didn't understand. I read the editorials and the readers' letters; I read the ads. I read the stories believing that something like them must be true. I yearned to go to Barsoom, and I spread my arms to the red planet, but nothing happened. I tried to calculate if I was likely to live to see the year 2000. I haunted libraries and bookshops, and seized on any book whose title made it sound as if it might be science fiction. See me as a desperate limpet, sucking my nourishment out of books.
In the mid-thirties _Wonder Stories_ was being edited by Charles Hornig, under whom the magazine displayed a marked interest in sadism. I did not know the word, but could not help noticing that the stories emphasized torture.
Sex and sadism were the formula of a string of pulp magazines published at that time, with titles such as _Terror Tales_ and _Horror Stories._ The _Spicy_ magazines _(Spicy Mystery, Spicy Adventure,_ etc.) used this formula in a much milder form, along with conventional titillation, and the editor of two sf magazines, _Dynamic_ and _Marvel,_ tried it briefly.
I also sampled several of the series-hero pulps, such as _The Spider, Doc Savage_ and _The Shadow._ Images from these stories have stayed with me all my life. In an issue of _Operator_ #5, the villain used a sinister drug, graphically described and depicted on the cover, to destroy his victims' will. The stuff looked like viscous green ink; I can still see and taste it. These pulps did not satisfy my thirst for fantastic adventures; the stories were too muddled to let me identify with their heroes, and I always dropped them after one or two tries. But I read with fascination all the _Saint_ novels of Leslie Charteris. The Saint was precisely everything I was not and longed to be – grown up, strong, handsome, fearless, cool in the presence of women. I yearned after Leslie Howard in _The Scarlet Pimpernel,_ too, and I read my way through eight or ten of Raphael Sabatini's novels. These were particularly satisfying, because the heroine always misunderstood the hero and had to apologize later.
Presently I invented a scheme whereby my father would subscribe to the science fiction magazines in my name, and I would pay him back monthly out of my allowance. _Wonder Stories_ promptly dwindled and expired, and _Astounding_ began to trim its edges, changed illustrators, and became inferior to its old self in every way. I read it faithfully, all the same.
In high school I became the cartoonist for the school paper, the _Guide._ It was a mimeographed paper, well produced under my father's direction, and it won some state prizes. My cartoons appeared weekly for nearly three years, and by the time I graduated I was an expert in an art that turned out to be dying.
In the late thirties science fiction magazines began to pick up again after a few years in the doldrums. _Astounding_ became livelier under a new editor, John W. Campbell Jr. _Wonder_ had been converted into _Thrilling Wonder_ and was bad but interesting because novel. There was a rash of new magazines. Campbell brought out _Unknown,_ which instantly enthralled me. Among the other new magazines were two called _Super Science_ and _Astonishing,_ both edited by Frederik Pohl, and in one or the other was a regular listing of magazines published by fans. I sent for some, and got into correspondence with Bob Tucker, the editor of _Le Zombie._ I did some cartoons for him. I published my own fanzine, _Snide._ From this other correspondence followed, including some with Richard Wilson, Donald A. Wollheim and Robert W. Lowndes, all New York fans, members of a group that called itself the Futurian Society.
I wrote and illustrated _Snide_ myself, and manufactured a hundred copies or so on a tray hektograph I had been given for Christmas. The cover of the first issue showed a man with a briefcase running after a rocket ship which had just taken off; he was shouting, 'Hey, wait ! '
When _Astounding_ came into full flower in the late thirties, with stories in every issue by Robert A. Heinlein and L. Sprague de Camp, and beautifully realistic brush-drawn illustrations by Hubert Rogers, I would have given anything to be Campbell, or Heinlein, or Rogers. I sent Campbell stories, and he sent them back with letters of rejection on grey stationery, signed with his looping scrawl. I now know how much more this was than I had any right to expect, but I was frustrated because I couldn't sell the stories and had no idea how to make them better. I drew cartoons and inked them in, and _Amazing_ bought one for three dollars. (A spacesuited man has found a robot in a cavern, and is about to push one of the buttons on its chest; the robot is waving a huge mallet behind its back. Caption : _Wonder what this one does.)_ This success elated me, and I sent _Amazing_ more cartoons and proposals, but they never bought another. Years later, walking down a street in Queens, I saw a copy of _Amazing_ in the gutter, open to my cartoon.
During this time I wrote several short pieces that were published in fanzines, including an article called 'Unite or Fie!' which urged the formation of a national fan organization. That was the extent of my contribution; a fan named Art Widner took up the idea, published correspondence about it, drew up a set of by-laws and saw it into being. This was the National Fantasy Fan Foundation, which later became notorious for its lack of accomplishment.
I kept trying to write science fiction stories, spurred on by one of John W. Campbell's periodic announcements that he would pay $60 for a short story (inconceivable wealth). I could start stories but could not finish them; baffled, I gave the manuscripts to my father with a covering letter to myself and asked him to put them in his safe deposit box. Later I did succeed in finishing two or three new stories and sent them to Robert A. ('Doc') Lowndes, who was then trying to set himself up as an agent. Lowndes sent most of them back with patronizing letters about plot and characterization; then he wrote me that Donald A. Wollheim was putting together the first issue of a new magazine and would print my story 'Resilience' if I would donate it. (Wollheim had no editorial budget for the magazine, and had to fill the whole issue this way.) I of course agreed.
One of my unfinished stories was about a young man who had duplicated himself seven or eight times by means of a matter-duplicator; I was going to have him/them set out in a spaceship of his design to explore the universe, but after the first few pages I didn't know what to write. The story had narcissistic overtones, like David Gerrold's recent novel _The Man Who Folded Himself._
I kept running into incomprehensible responses in other people around me, as when I criticized the new comic strip _Flash Gordon_ because the natives of Mongo spoke English, and a friend of mine said. 'What else would they talk?' I came to believe that somewhere in the outside world, probably in New York, things were altogether different, and Hood River became hateful to me because I couldn't get out of it.
My last year in high school was a nightmare of boredom. When it was over, my father offered to send me to college, but that was the last thing I wanted. We agreed that I would go to Salem for a year and attend the WPA Art Center there. I lived in a boarding house at first, run by an insurance man and his fat, comely, cheerful wife. At her table I ate my first steak and found it unchewable; it was not until years later that I discovered steak did not have to be tough.
While I was in Salem Don Wollheim's first issue of _Stirring Science Stories_ appeared, with my story in it. The printers had changed 'Brittle People' to 'Little People' in the first sentence, rendering the story unintelligible, but I was proud of it anyway.
In Salem I met another science fiction reader – found him working in a second-hand bookstore. He was a blond, moon-faced, blue-spectacled young man named Bill Evans, and we agreed to publish the next issue of _Snide_ together, since he had access to a Ditto machine at school. We did so, and announced payment for stories (½¢ a word) beginning with the next issue, but it never appeared. Bill finished school and went into the Bureau of Standards because that was where Richard Seaton, the hero of E. E. Smith's _Skylark_ series, worked. The last I heard, he was still there.
I began to feel that I had no vocation as an artist, or any desire to go on to school, and when the Futurians invited me to come to New York and live with them, my parents agreed to let me go. The World Science Fiction Convention that year was in Denver, and they drove me there over precipitous mountain roads. It was late at night when they dropped me in front of the hotel, but I found a few fans standing around in the convention room. Sick with embarrassment, I goosestepped toward them and raised my hand in a Nazi salute. They asked me who I was and I told them. 'Ah, Damon Knight,' said Forry Ackerman kindly.
The Futurians, when I met them later, were an odd-looking group. Wollheim was the oldest and least beautiful (Kornbluth once introduced him as 'this gargoyle on my right'). He was, I learned later, almost pathologically shy, but he was the unquestioned leader of the group, and John Michel, who worshipped him, later informed me that Donald's personality was such that he could have any woman he wanted. Lowndes was ungainly and flatfooted; he had buck teeth which made him lisp and sputter, and a hectic glare like a cockatoo's. Michel was slender and looked so much more normal than the rest that he seemed handsome by contrast, although he was pockmarked and balding. He had a high voice and stammered painfully. Cyril Kornbluth, the youngest (a few months younger than I) was plump, pale and sullen. He had narrow Tartar eyes and spoke in a rumbling monotone; he looked ten years older than he was. He liked to play the ogre; at the art auction that weekend he bid fifty cents for a Cartier illustration, got it, and tore it in half. Chester Cohen was about my age, and although he was neurotic and jumpy, a nail-picker (not enough left to bite), he was able to freeze on command and hold a pose indefinitely; once Michel pretended to hypnotize him in the elevator and left him there, to the consternation of the hotel employees. They had to find out who he was and carry him up to his room, where he lay like a corpse until Michel arrived and snapped his fingers.
Heinlein, a handsome man in his thirties, was the guest of honour at the convention, and we glimpsed him and his slender brown wife Leslyn occasionally.
After the convention we divided into two groups; Kornbluth, who had been on a trip to Los Angeles with Cohen, got into one car with Wollheim, Michel and me, leaving Chet to go home with Lowndes. 'I've seen a lot of Chester Cohen', Cyril said. We were travelling by 'wildcat bus' – sharing expenses with a goodnatured man named Jack Inskeep who was driving to Cleveland. On the way, Wollheim expanded on an idea of his that the surface of the earth was composed of strips of solid material about two miles across, with roads running down the middle, the rest being hollow. Kornbluth played up to this, thinking of feeble objections which Wollheim demolished one by one.
In Hill City, Kansas, the car broke down. Hill City was a slight rise in the road, not more than a foot and a half in elevation. The whole town could have been covered by a single good-sized aircraft hangar. The garage where the car was worked on had a calendar on the wall depicting a bosomy young woman who was not Rita Hayworth, although that was the name printed under the picture. The one movie theatre was upstairs in a ramshackle building, reached by an outside stairway; locusts leapt in the tall weeds nearby. Down a side street, we came upon a house behind a white picket fence; in the lawn was a neat sign that read : 'Dr ______ , Physian and Surgon'.
Near Columbus, our driver obligingly stopped so that Cyril could meet his girl, Mary Byers, who lived on a farm with several fierce uncles. We went to a bar, and Inskeep played the pinball machines while Cyril and Mary stared into each other's eyes. In Cleveland he left us and Wollheim took a train, while the rest of us went on by bus.
The Futurians at that time lived in a railroad apartment on 103rd Street. It had four rooms in a row : first the kitchen/bathroom (the tub was under the drainboard), then two small bedrooms for Michel and me, then the living room which was also Lowndes's bedroom. It was bare but sunny and clean. I paid my share of the rent (I don't remember how much, but probably about $7), and was expected to keep my room clean and wash the dishes. Lowndes did the cooking; his speciality was Futurian Chop Suey – noodles, hamburger, and a can of cream of mushroom soup; it was better when it had rotted a day in the refrigerator. I don't now remember what Michel's contribution was.
We had wall newspapers, in which Lowndes published communiques about our campaign against the Enemy (bedbugs). We squirted the mattresses with kerosene, and eventually vanquished them.
All the Futurian apartments, then and later, had names; this one was the Futurian Embassy. Kornbluth stayed over on weekends; he lived with his parents, and so did Wollheim. None of us had any money; for amusement in the evenings, we played poker for stakes of 15¢ each, and drank California wine at 50¢ a gallon. Once or twice when Chet and I were sent out for wine, we bought the cheaper stuff at 35¢ and pocketed the difference. When the game broke up at midnight, we would walk down to Times Square to look at the advertising signs, have a cup of coffee in the Times Square Cafeteria, and walk back.
I adopted all the Futurians' attitudes. They looked down on fannish activity, and so did I. They said they were communists; I said I was a communist. They expressed contempt for Campbell and his stable of writers; I lost interest in _Astounding_ and stopped reading it. They were nearly all native New Yorkers who would have died rather than get on a sightseeing bus; I lived in Manhattan for ten years, and never saw the sights.
My ambition now was to be published among the Futurian writers in their magazines, but except for two sales to Lowndes, I couldn't even do that.
Wollheim's _Stirring Science_ and _Cosmic_ had folded shortly after I got to New York, but Lowndes was editing _Future Fiction_ and _Science Fiction_ (later _The Original Science Fiction,_ as if it were a tavern) while Fred Pohl, still technically a Futurian although he did not have much to do with us, was the editor of _Super Science_ and _Astonishing._
Kornbluth organized something called the Inwood Hills Literary Society, which met once a week either at his house or at ours. It was a forerunner of the Milford Conference; each writer was expected to produce a story every week for criticism. When the group met in the Embassy, everybody there but me was a member, and I had to leave the room. I thought this was a bit thick, since I lived there. When the group met at Cyril's, however, I used the time to write, and gradually my work got a little better.
Kornbluth was writing stories under various pseudonyms for all the Futurian magazines. He was nineteen. One of his unfinished stories, which I found lying around at the Embassy, began with a flashback in the stream of consciousness of an intelligent mouse during intercourse. Another, called 'The Ten-G Pussies' (about cats, raised under ten gravities in a centrifuge, that became so muscular that if they pounced at you they would go right through you), began with a philosophical dialogue about the nature of cuteness.
The Futurians had a written set of by-laws which declared that the club was in session whenever two or more members were present. The Futurians seldom troubled with elections but on one occasion when there was an election,Fred Pohl running against Wollheim for president, we prepared for it the night before by making posters setting forth various drawbacks in Fred's character. I drew a skull-face and a pointing finger, with the legend, 'Uncle Freddie Wants _YOU!'_ I also made a linoleum block print and with it printed dark-blue skull faces on several yards of the roll of toilet paper in the bathroom. Fred came over for the balloting, was magnificently cool, and lost the election.
Shortly after this I painted a pentacle on the floor of one of the rooms, with Greek characters around the rim and in the centre (Kornbluth's idea) the Hebrew characters Resh Sin Vau Pe (RSVP). I also painted a mural depicting three sinister supernatural characters, the central one with his hand thrust suggestively under his robe; we called them Stinky, Shorty, and the Holy Ghost.
Kornbluth played the ogre seldom; his humour was sardonic and sometimes cruel, but he was the least malicious of the Futurians. He told us stories about his relatives. Once, a female cousin stepped into the bathroom after him, locked the door, and said, 'Well?' Cyril replied, 'I'll be through in a moment,' finished washing his hands, and left. He played at being grown-up. One fall day he came in wearing a hat, solemnly explaining that in cold weather a man needed headwear in order to balance the bulkier silhouette of his overcoat. When drunk, he was playful.
Michel was a posturer and poseur; he affected corduroy jackets and trousers, smoked a pipe, talked about his dates. He had had several operations for bone tuberculosis, and had ugly craters in his legs to show for it. He took me on an Elevated tour of New York, borrowed a dollar, and said, 'Don't tell Donald'. He had had three or four stories published, and managed to give the impression that he was the most professional writer of us all.
Lowndes was the one we always found ourselves talking about when he was not there. Often when we were going somewhere together, for no evident reason he would cross the street and walk by himself. Except for me, he was the only gentile in the group. His parents had been fundamentalists who thought even the Sunday comics were sinful, and Lowndes as a small boy had had to crawl under the porch to read them. In his youth he had been in the Civilian Conservation Corps, and his arms and legs remained muscular although the rest of him was flabby. When drunk he lurched hideously, and sometimes passed out with his eyes open.
Wollheim did not drink at all, and his remote brown eyes were always watchful.
I myself looked like the ghost of a blond Charlie Chase. We were a gallery of grotesques, but we were all talented to one degree or another, and we counted on that to save us.
We were anything but a close-knit group, and yet we stood together against the outside world. A Futurian crest, designed by I forget who, had a large flat-headed screw with the legend, _Omnes qui non Futurianes sunt._
I first saw Dick Wilson on the beach at Far Rockaway; he had just been in the water and was red, white and blue. He was a gentle, spade-jawed man with a high and very quiet voice, something like Liberace's. Also at the beach that day were Jessica Gould, Dick's plump, pretty and flirtatious girl friend, and Hannes Bok, who was leaping athletically about.
There were two groups of Futurians, the ones I was living with, and the others whom we called the Compatible People (this referred to a party to which our group had not been invited).
The CP were Frederik Pohl, Richard Wilson and Harry Dockweiler and their wives. They differed from us basically in having money, and jobs and being married.
The Futurians had their own official religion, invented by Wollheim; it was called GhuGhuism, and began with the cracking of the Cosmic Egg. It had Vestal Virgins, whose virginity was perpetually renewed, and other features I have forgotten. Wollheim also invented a private language to write the Gholy Ghible in, but he was the only one who could read it.
None of us knew any girls, or had any way of meeting them, except Wollheim, whose girl-friend, Elsie Balter, was part of our circle. Wollheim's courtship was slow. Elsie, who was older than Donald, was a decidedly plain but beautifully good-natured and kind woman. Wollheim gave Elsie a friendship ring after about five years, and after another three or four they were married. (Telling me about the friendship ring, Elsie said, 'And then, do you know what Donald did? He _kissed me.')_
I see now that if any of the rest of us had gone to work, or to school, we would have met girls in any desired numbers, but this did not occur to us. We once got dressed up and went to a Trotskyist meeting because we had heard the Trotskyists had a lot of horny girls. There were a couple of girls, but they wanted no part of us. Another time we went down to Greenwich Village to Anton Romatka's poetry circle, because Donald said we were the real writers and would command instant respect, but it did not work out that way. I put an orange scarf around my neck and read a sonnet which was received in absolute silence.
The Trotskyists called themselves Trotskyists but we called them Trotskyites, because we were Reds. Actually, the Futurians were very mild parlour radicals who had never even joined the YPCL. (The Young People's Communist League, or 'Yipsl'.) In those days, nearly every educated young person in New York was a red-hot radical, in words if not in deeds. With the Futurians, this took the form of occasional doctrinaire articles in fanzines, and that was all. Our bunch knew very well that if they joined any communist organization they would be put to work, and work was what they were trying to avoid. They showed their solidarity, however, by going to Russian movies occasionally and by listening intently to Shostakovich.
We were too poor to go to movies often, or buy books, or travel, or eat at restaurants, but we were used to that and did not mind it. Our recreation was talking. We played endless word-games – _People_ (a form of. Twenty Questions) and _Tsohg_ (Ghost backwards).
When on rare occasions we did have money enough to go out, it was usually to the Dragon Inn in Greenwich Village, where I ate fried rice because it was the only Chinese food I could stomach. Years later, in the throes of an unhappy love affair, I went to a Chinese restaurant and ordered shrimp chow mein in order to distract my mind.
Wollheim had two parlour tricks. One was to put one arm behind his back, bring the hand up all the way around his face and lay it on his opposite cheek. The other was to put a small flashlight up his nose and turn it on; his whole nose would then light up like a pink cucumber.
Once when we walked him to the subway late at night he gestured me to follow him through the turnstile and then onto the train; we rode in silence to his stop. There I got up to follow him off, but he gestured me, with a grin, to remain. The door closed between us.
We moved so many times that I can't remember the sequence. It was a renter's market then; if we wanted to move we just hired a truck and went, usually owing the last month's rent. On one occasion we had to pay up, though, because Lowndes wrote two letters, one to the landlord wishing him bad luck in the hereafter, and one to Elsie giving our new address, and put them in the wrong envelopes.
Lowndes and Michel and I shared another apartment after the Embassy; it was in Chelsea and was called the Futurian Fortress. At various times Lowndes and Michel, Lowndes and Jim Blish, Michel and Larry Shaw briefly shared apartments.
The Blish/Lowndes place was called 'Blowndsh'. While he lived there, Lowndes had a cat named Charles that hid all his pencils under the bedclothes, and another named Blackout that thought Lowndes was God: whenever it rained and he couldn't go out on the fire escape, he walked over and bit Lowndes.
New York excited me, and I wrote a long free-verse poem which included the line, 'I have known hunger and loneness' (for the metre) and sent it to my mother, and she wrote back in some anxiety that she did not want me to go hungry, and wasn't the money they were sending enough? In fact, I was stretching my monthly allowance to pay for Chet Cohen's existence as well as mine (we were sharing an apartment), and some days what we had for dinner was a can of Campbell's pork and beans; but we never felt poor. When we had money we spent it, and when we were broke we waited till we had money. If we couldn't afford cigarettes, we rolled new ones out of butts.
Lowndes got tired of his unsuccessful agency and turned it over to me. I dutifully trudged around to various editorial offices with my unsaleable manuscripts. In the anteroom of Campbell's office one day I met Hannes Bok, who showed me a cheque for a thousand dollars, then a huge sum : he had just sold Campbell a novel for _Unknown._ Campbell was a portly, bristled-haired blond man with a challenging stare, who told me that he wasn't sure how much longer he would edit _Astounding._ He might quit and go into science. 'I'm a nuclear physicist, you know,' he said, looking me right in the eye.
Fred Pohl had persuaded Popular Publications to publish _Super Science_ and _Astonishing_ in 1940 and had edited both magazines for a year or two; then he had been asked to step down, but had remained as an assistant editor to Alden H. Norton, to whose group at Popular the magazines were added. In 1943 there was a vacancy under Norton, and Fred recommended me to fill it; he also lent me a white shirt to appear in when I applied for the job. I was hired at $25 a week.
Norton was a large, bald, amiable man in his forties, who was responsible for half a dozen pulp magazines. He had two sports magazines, the two science fiction pulps, a detective magazine, and _G-8 and His Battle Aces._ As was customary at Popular, he read all the manuscripts, bought stories and scheduled them; the rest of the work – copy-editing, proofreading, and so on – was done by his assistants: Fred, a young woman named Olga Quadland and me. Each of us had two or three magazines for which he was responsible every month, but _G-8,_ because it was so awful, was rotated among us.
_G-8 and His Battle Aces_ was written entirely by one man, Robert J. Hogan. He wrote the lead 'novel', the short stories, and the departments, and brought in every other month a huge stack of manuscripts which then had to be gone over line by line. A _G-8_ manuscript edited by Fred, which they showed me, had no word of the original text unchanged. The one I did concerned a plot by the Germans in World War I to make their soldiers incredibly fierce by injecting them with rhinoceros juice.
After I had been at Popular a month or so I was transferred to Mike Tilden's department and felt at home there immediately. He was a sloppy, beer-bellied man with a quiet, rumbling voice; he was one of the kindest people I ever knew. He always looked unlaundered. He had troubles at home, financial and otherwise, and was always borrowing small sums from other editors, but never from people who worked for him. Once I passed his door and looked in, to find him sitting with his feet up and his hands in his pockets. 'I'm just sitting here saying shit,' he said.
My number came up, and I went down to the Induction Center in Grand Central Station. Lines of men dressed only in undershorts, socks and shoes moped back and forth across a huge hall. Every expression given by the Creator to the idea 'Man' was there. The whole tour took hours, and by the time I got toward the end of it I was numbed and apathetic. Three psychiatrists interviewed me; the first was intelligent and evidently trained, and wrote on my papers, 'Schizoid. Does not think he would do well in the army, and I am inclined to think he is right'. The second man wrote down 'Split Personality', and the third followed his lead. When I handed my papers to the colonel in charge, he read them and said the magic words, 'Oh, well, he's underweight anyway. Four-F'.
Popular Publications at that time had forty titles and was the largest pulp publisher, followed by Better Publications under various corporate names, then Street and Smith, then a straggle of little companies with eight or ten magazines apiece. A year or so before I started work there, Popular had bought up the assets of the Frank A. Munsey company, including a number of pulp titles. The pulps were still the principal enterprise of the company, and there was no hint that they were coming to the end of their time.
Our offices were roomy and airy, on the next-to-the-top floor of a large office building on East 42nd Street. Each department head ran his own magazines with very little interference, and our work relationships were relaxed and easy.
There were three large editorial departments, run by Norton, Tilden, and Harry Widmer, each employing a secretary and one or two assistant editors, plus two editors who ran a couple of magazines each with a secretary – these were love magazines in both cases, for some reason.
Harry Widmer was a pear-shaped little man with a rum- blossom and a dainty way of moving and speaking. He had a young and pretty wife. There was a story told about him, that he had taken the entire contents of a magazine home to work on over the weekend, as we often did, and had stopped for a few drinks before and after dinner with a friend. When he got home with the friend, barely able to stand, he decided to put the manuscript envelope in the safest place he could think of, which at that moment was the refrigerator. When he woke up the next day, he went to the refrigerator; the envelope was not there. He could not account for this until he realized that next to the refrigerator door there was another, very similar in appearance – the door of the incinerator.
I met Harry Harrison and his wife Evelyn in their vast, dim uptown apartment. Harry was short and at that time slender, a voluble, sputtering little chipmunk of a man whom I liked at once; his wife was taller, intense, rodent-toothed and intelligent. Harry was a commercial artist doing comic-book work at that time and I was told that Evelyn was writing the continuities. Later, Harry surprised me by becoming a writer, and I had the pleasure of buying his first story, which I called 'Rock Diver', for _Worlds Beyond._ Still later he became the editor of _Space_ and _Science Fiction Adventures,_ succeeding Lester del Rey, and he bought stories from me. We have been on the same merry-go-round ever since.
When Fred Pohl went into the army, his place was taken by Ejler Jakobsson, a Finn who had come to this country as a boy and who had been on the Columbia track team. He gave me advice about my love life. I told him about a girl named Sally Green who came to see me occasionally and always borrowed a book when she left. (She later told me she had given these books to the Armed Services.) We necked a lot, but I couldn't get any farther. 'Tell her you love her,' said Jake. I tried this, but she didn't believe me.
New people began coming into our circle. Virginia Kidd was from Baltimore; she was fat but shapely (had an hourglass figure, like a John Held drawing). Her face was soft and pretty. She had had polio as a child and had spent years in bed, having her bad leg rubbed by her parents with cocoa butter. She had been a bar girl in Baltimore, and was a science fiction fan; _Wonder stories_ had printed some of her letters. James Blish had been in the army and was still in uniform when I met him in a bar; he spent the whole time talking about James Joyce. He was dark-haired and thin, and had a disability pension. Larry Shaw was from a Catholic family in Rochester, which he hated. He was a funny-looking little man with upstanding hair and bottle-thick glasses; he spoke with difficulty, his face writhing.
At a party one afternoon I was introduced to Judith Zissman, a quiet, intense young woman who had just moved to New York from Philadelphia. She was eager to know science fiction people, and carried me off to dinner in her cluttered Greenwich Village apartment. Here I met a stocky blonde girl named Edith Liebert, who set out to seduce me. (She told me later that she thought it would be nice to have an affair with someone innocent.) She made overtures which would have been enough for anyone else, but not for me, and it was weeks before we finally got to bed in my apartment. I was so inexpert that I left her unsatisfied, and refused her invitations the next morning when we woke up because I had to go to work.
In the spring of that year I had grown increasingly restless at Popular. I sat with a thick manuscript on my lapboard, a Western novelette by Harry Olmsted, and found that I absolutely could not penetrate it. Olmsted always needed heavy editing, but in order to edit him you first had to find out what he meant, and I couldn't. When this had gone on for some weeks, I gave notice and quit.
I went job-hunting. I tried all the conventional things, read the ads in the _New York Times_ on Sunday, typed up resumés, went to agencies, was sent on interviews. One of the jobs I applied for was on the _Police Gazette,_ where I was interviewed in a crowded room near a table spread with glossy 8½ × 11 photographs of ladies in various costumes. I was asked if I knew anything about the _Gazette,_ and replied that I believed it was the sort of thing that was read in barber shops. I didn't get the job. (I wrote about this in 'On the Wheel'.) I applied for a job as a mimeographer, but was turned down because I was overtrained. I left some of these unsuccessful interviews with a feeling of relief; I wanted and yet didn't want the jobs.
At one point Chester and I were reduced to typing envelopes at a penny each for an addressing service. We quit after two back-breaking hours.
We went down to the Merchant Marine office to apply as yeomen and took the typing test. The standard was forty words a minute; I barely made it. Chester and Larry Shaw actually shipped out later, but I never did; the Merchant Marine ID card came in handy, though, and got me into the Museum of Modern Art at half price. Larry made one trip as a steward's mate, or whatever they call waiters on ships; on the way back he broke his glasses and was relieved of duty.
I met Phil Klass, who was nonviolent but excitable; his voice would begin rather softly and then at a certain point, as if he had shifted gears, would begin to blare as he warmed to his subject. He had a set of comic Jewish gestures and grimaces which through habituation had become almost second nature. When I first knew him he had fallen under the spell of Scott Meredith and was writing a series of commercial sf stories which he published under the name of William Tenn. He was saving his own name for the _New Yorker_ pieces he meant to write later. His brother Mort told me it was hard to get him up in the morning because he could carry on a perfectly rational conversation while sound asleep. Mathematics was the one thing he could not handle in this way, Mort said : if you asked him, 'How much is two and two, Phil?' he would reply, 'Well, now, that's a very interesting question. The Babylonians – '
Still at loose ends, I had signed up for a free class in radio writing and had attended the first session, at which the instructor had told us how he felt about the expression, 'But first – ' when I was notified that my father had had a heart attack. My mother wired money and I flew home.
I found my father convalescing, and stayed a week in the familiar house now grown intolerably small. To stave off boredom, I wrote part of a story called 'The Third Little Green Man', which Ree Dragonette later admired for its action scenes. When the time came to go, my father broke down. My mother nodded me out, and I went.
I sold 'The Third Little Green Man' to Malcolm Reiss of _Planet Stories,_ an editor who is remembered with affection. I sold one or two other stories to the same magazine, but by then Wilbur S. Peacock was the editor. I got into the habit of buying him lunch whenever he bought a story of mine, but I don't know why; I didn't like him much.
I also met Ray Cummings, a really frightful-looking man, cadaverous, grey-faced, dressed all in black with a turned-around collar. He was a survivor from the Gernsback days; he had been a secretary to Thomas Edison, and had filled the early _Wonder Stories_ and _Astoundings_ with long stories such as 'Wandl, the Invader' and 'Brigands of the Moon'. Lowndes had been reprinting these, and I was given the task of illustrating a couple. I also illustrated a long novelette by F. Orlin Tremaine, in which a young man blundered into a lost civilization and became its dictator. I was so indignant over this that I drew the hero in a black leather uniform with jackboots, wearing insignia that I made as close to swastikas as I dared, against a background in which little people were dying in the stench of factories and under the whips of overseers. Nobody noticed.
Theodore Sturgeon came back from the Virgin Islands and took up residence in the Village with L. Jerome Stanton and Rita Dragonette. Jay was a popeyed, dark-haired man with a quiet, slow voice that never stopped; Rita, called Ree, was a tiny brown woman, attractive in spite of some missing molars, who later turned out to have several personality quirks. Sturgeon was my agent for a while; he expressed the belief that since Jay was working for Campbell, manuscripts submitted by him would have the inside track, but it did not work out that way.
Lowndes had remained at Columbia Publications, where he edited all the magazies (including one ingenuously called _Complete Cowboy)_ with the exception of the two love pulps, which were edited by a large woman named Marie Park who later appeared in reducing-salon advertisements headlined, 'I looked like a water buffalo.' She was a southern lady, and went into hysterics one day when she discovered that a Negro illustrator had sat in her chair.
Judy Zissman (born Juliet Grossman) was then in her twenties, a strong, rather shapely and good-looking woman with dark skin and hair. Her teeth were bad; later she had them replaced. She was so full of energy that she could not abide sloth and indifference around her, and she soon stirred us up. She and her husband Danny were Trotskyists from Philadelphia, and Judy in a political argument was a juggernaut. Danny was in the navy, serving aboard a submarine, and Judy struck up a friendship with Johnny Michel. This displeased Wollheim, and presently Judy came to tell us that Wollheim had forbidden Johnny to have anything more to do with her (because she was a Trotskyist) or Jim Blish (because he was thought to be a fascist). Our indignation was acute, and we sat up half the night composing a document in which we read Wollheim, Elsie and Michel out of the Futurian Society. We mimeographed and mailed this out to a fanzine mailing list. Wollheim then filed suit for libel in the state supreme court, naming the seven of us who had signed the document : Judy, Blish, Lowndes, Virginia, Chet, Larry and me. The suit was thrown out of court, with costs charged to Wollheim, but it cost us $100 apiece in legal fees.
Blish and I were rivals at first, and I sniped at him in a mimeographed magazine called ' ', whose missing title was supposed to satirize the meaninglessness of all titles; but his ability to absorb criticism without anger disarmed me and we became friends.
In these magazines Blish and Judy Zissman had a rivalry which was much more bitter and long-lasting. Blish and Virginia Kidd were married in the late forties. Jim, who had been trying to make a living as a freelance writer, went to work as a reader for the Scott Meredith Literary Agency. Presently he got me a job there too.
Scott Meredith, born Feldman, was a small, slight man who as a young writer in Brooklyn had been so poor that he had walked across the bridge to hand-deliver his manuscripts. He and Kornbluth had lived on the same block as children. He had saved his money in the air force, and after the war, in partnership with his brother Sid, had opened the agency, which at first did such a feeble business that the partners had to sweep the place out themselves. This stage did not last long.
Sid's role in the agency was not clear. He had an office of his own and stayed in it most of the time, emerging only to distribute manuscripts and collect finished work, and to deliver an occasional homily about the resemblance of the agency to a shoe factory : 'They have the raw materials, the _leather,_ you know, and they take that and put it through the machines just like we do here, and make shoes.'
Meredith also had a list of professional clients, including P. G. Wodehouse, whom he had acquired by writing him a fan letter, but this end of the business was kept separate from the reading-fee operation, and Scott managed it himself. Later, as the agency grew, he handled only the most important clients personally and the rest were turned over to an employee at what was called the 'pro desk'.
Meredith took full page, back-cover ads each month in _Writers' Digest;_ these ads, which were lively and ingenious, encouraged amateur writers to send us their mss for evaluation at $5 for a short story and $25 for a novel. When the manuscripts came in the morning mail, they were distributed to us, and it was our job to read them and write letters of comment, for which we got $1 out of the $5, and $5 out of the $25. The first letter to a new client always began by explaining that his story was unsaleable because it did not follow the Plot Skeleton. The letter went on to enumerate the parts of the Plot Skeleton, viz: 1 A sympathetic and believable _lead character;_ 2 an urgent and vital _problem;_ 3 _complications_ caused by the lead character's unsuccessful attempts to solve the problem; 4 the _crisis_ (this element was added by Blish); 5 the _resolution,_ in which the lead character solves the problem by means of his own courage and resourcefulness.
In a concluding paragraph the letter pointed out which of the elements were missing (ordinarily all of them were) and invited the client to try again. Subsequent letters grew more detailed. We really tried to help the clients, and in one or two cases I think we succeeded.
We had considerable latitude. The introductory letter always used the formula, 'I'm sorry I can't give you a better report, but–' and then the news about the plot skeleton. Blish once got a manuscript that was so awful that he ended the sentence 'it stinks', and then wrote 'Sincerely yours'. Meredith laughed and signed it.
The fact that we were a shifting population and that all the letters were signed by Meredith (or by Sid, imitating Scott's handwriting) sometimes led to anomalies. Jim got into a lengthy correspondence about modern music with one client, then quit, and the client was turned over to Lester del Ray, another Meredith employee. The client, who had been hearing from Jim about Bartòk and Hindemith, now began getting letters about Ravel's _'_ _Bolero'._
My contribution to these letters was the term 'paper dragon plot', meaning the frequent plot in which the ending discloses that there never was a problem. The work was exhausting and challenging, and I liked it. We were certainly exploited, but the training we got was invaluable. A long line of Meredith employees went on to become editors. Meredith encouraged this, on the theory that such people would be inclined to buy from the agency, and in most cases he was right.
The office was in the middle of the entertainment district, and at lunch time, when we had finished eating what we had brought (Jim once complained that Virginia had given him a potato-chip sandwich), we walked down the street to a penny arcade and spent the rest of our lunch hour there. Our favourite was the hockey game, on which I developed a bank shot that exasperated Jim by going through the space between the opposing player's stick and his cast-iron body.
A new office girl called Trudy Werndl joined us, just out of high school, blonde, plump and pretty, and seemed to be impressed with Jim and me because we were writers. We took her out for a beer after work, and I invited her to come and see me that weekend. One thing led to another, and when I asked her to come and live with me she agreed, but her girl-friends were shocked when she told them that, and with some misgivings I married her instead. Just at this point the Blishes had taken a house in Staten Island and asked us to come and share it. Trudy and I were married in the Little Church Around the Corner (chosen by one of the girlfriends) during the worst snow-storm of the decade.
As soon as the novelty wore off it became evident that our marriage was a mistake. We were not well suited, sexually or in any other way. Commuting to work from Staten Island, half an hour on the ferry alone, was exhausting for me and staying home all day was boring for Trudy. I was promoted to the pro desk, which had a huge backlog of work. After a month or so I became ill with cerebrospinal meningitis and was carted off to the Staten Island Hospital, a few blocks away, where in my delirium I read phantom manuscripts. Shortly after I came out, Trudy got appendicitis and went in. Meanwhile we and the Blishes were getting on each other's nerves a little; Trudy and I decided to try to improve our relationship by moving back into Manhattan. This was at the height of the wartime apartment shortage, and we were able to move into a Greenwich Village studio apartment (so called because it had a little skylight in the living room) only by buying the previous tenant's furniture with $700 put up by my mother.
Now began the most miserable and boring time of my life. My relations with Trudy deteriorated. We acquired a large circle of new friends, mostly musicians who met once a week at Julian Goodenough's apartment. He lived alone, in the apartment over his silversmith shop and in his little bedroom, under a pink light, he kept a row of high-heeled patent leather shoes in an assortment of sizes. At his weekly jam sessions, he sometimes played the bass, sometimes thumped on the piano, grinning around his cigar. He could not drink – one highball made his face flush red.
In the Village I met Stewart Kerby, a former sf fan who had published a limited edition of one of David H. Keller's stories. A friend of his, Kenneth Koch, sometimes hunted Stew up and brought him to my apartment to compose tunes to his poems on the piano.
Needing money, I returned to Meredith's, where I found myself in company with Don Fine and James A. Bryans, a slow, gangling man who later became editor in chief of Popular Library. Still later Fine became the head of his own publishing house. I trained myself to pot either of them with a wad of paper, sitting or standing.
When Ejler Jakobsson invited me to go back to Popular as his assistant, I was pleased, particularly since Jake had inherited Al Norton's department, which included the two science fiction magazines. (Norton was now associate publisher.) This was the reason Jake wanted me, anticipating my help in a field unfamiliar to him, but in this we were both disappointed. Jake rejected stories I recommended with enthusiasm, including two early Charles Harness stories, and filled the book with other things that I thought barely publishable. We disagreed about the merits of the pulp-style covers, and he was not amused when I traced one of them, putting football uniforms on the figures instead of spacesuits.
Wollheim married Elsie at last; they moved out to Queens, to an apartment with a sunken living room that featured a photo-mural. Kornbluth married Mary Byers and they went to live in Levittown. Lowndes was living in Westchester, married to a woman whose name suddenly changed. Pohl told me that he had called Lowndes on the phone and had said casually, 'How's Louise?' The following conversation ensued :
LOWNDES: Who?
POHL: Louise.
LOWNDES: _Who?_
POHL : Louise, your wife.
LOWNDES : (with great emphasis) : She who was Louise is now Bar-bar-a.
Pohl married Judy Zissman. They went househunting in Red Bank, New Jersey, and because they were wearing old clothes the house agent assumed they were rich and showed them a huge three-storey house. They bought it and Fred still lives there.
I omit the details of my breakup with Trudy. Because the marriage was less than a year old and there were no children, it turned out to be possible to get an annulment rather than a divorce. She stayed with Julian for about a year, lost a lot of weight, bought new clothes and became svelte and elegant.
I did not have to appear at the annulment hearing; I had appeared earlier, however, in Judy Zissman's divorce action against Danny, in which I testified that Dan and a girl not his wife had spent some time in a bedroom in my apartment. The divorce referee was an old man named, appropriately, Schmuck. He asked me, 'What were you running, a whorehouse?' and muttered frequently, 'There'll be no divorce in this case, no divorce'. He granted it, all the same, and Judy at her request became legally Judith Merril.
I was fascinated by the permutations of Judy's names, and once when we were in a restaurant together wrote a poem about them on a napkin :
Juliet Grossman Zissman Pohl
Hated her name from the bottom of her soul :
Went to court in imminent peril;
Changed her name to Judith Merril.
At a party I had met Lester del Rey's wife Helen and later had learned that their marriage was breaking up. I took her out to a Chinese movie, and again one thing led to another. I turned the studio apartment over to Dick Wilson and moved in with Helen. Later we were married.
Lester del Rey signed his early letters to _Astounding_ R. (for Ramon) Alvarez, and he had four or five other given names; the whole thing went something like Ramon Felippe Maria something something Alvarez-del Rey. He explained that his father was descended from a royalist branch of the Alvarez family, and so on. In conversation he liked to defend unlikely propositions. If he tossed out some assertion that aroused his hearer's incredulity he would immediately repeat it with more emphasis, and even if he had only thought of it a moment before, he would be prepared to defend it all afternoon, quoting sources which might or might not be imaginary: all this with a goblin grin and such evident enjoyment that it was hard to dislike him. I described this aspect of Lester among others in 'A Likely Story', in which he appeared as Ray Alvarez.
In the introduction to one of his stories I once called Lester one of the most contentious men alive. His wife Evelyn later told me that when he read this Lester shouted, 'I am not contentious ! '
I was tired of Popular again, and wished I had my own science fiction magazine to edit. I asked Fred Pohl if he knew of any publisher who might be interested; he suggested I try Alex Hillman of Hillman Publications. I sent Hillman a written proposal and was called in for an interview. Hillman, who looked something like Charles Coburn, hired me in ten minutes. When he asked about salary, I said I was getting $75 at Popular (an exaggeration) but would like to do better than that; we settled on $85 a week, the most I had ever earned in my life. I paid off some debts and bought two new suits for the first time in my life. I had never owned more than one suit, mostly second-hand, before.
I wanted to call the magazine _Science-Fantasy,_ but the firm's lawyers, after a haphazard search, advised against it because both words were in use in the titles of other magazines. We finally settled on _Worlds Beyond,_ swiped from the title of a symposium edited by Lloyd Arthur Eshback, _Of Worlds Beyond._ My handshake agreement with Hillman was so hasty that I discovered afterward I didn't even know if the magazine was to be a monthly. I was too green to ask for a contract guaranteeing a minimum number of issues, or to settle details of production and format. Hillman was leaving on a vacation, and told me to have a cover ready for him when he got back.
Fred, now an agent, laughed with delighted disbelief when I told him I had sold Hillman the magazine. I bought several stories for the first issue from his clients, and one or two others from Meredith. From a young writer named Richard Matheson, then almost unknown, I bought a story called 'Clothes Make the Man', a deft little satire about a suit of clothes that takes over its owner's personality. This was the story I chose to illustrate on the cover. I called in an artist named Herman Bischoff and gave him the commission; he turned out a fine spooky painting of an empty suit of clothes waving its arms at a startled girl. When he came back, Hillman rejected the painting and would not be dissuaded, even though a vice-president took my side. I discovered that I had only thought I had authority to order the painting made; what Hillman had meant was for me to get a sketch made for his approval. Bischoff was never paid. I turned to Paul Callé, who I knew had a painting that had been turned down by Popular, and we bought it for $100.
The atmosphere at Hillman Publications was utterly unlike that at Popular. I had an office to myself for a week or two, then was put in with the staff of Hillman's fact detective magazines, headed by an irascible, popeyed man whose name I have forgotten. Every editor seemed alone at his little desk, even though several of us worked in the same room. There was no camaraderie and no fraternization. Meeting Hillman in the hall was an unnerving experience. Smoking a cigar, he lumbered down the hall staring straight ahead, hands clasped behind his back. When I said good morning, he continued to stare and lumber. (I used him as the Boss of Colorado in my novel A _for Anything.)_
I had the tiniest of budgets, but since I was using about half reprint material I could afford to pay the going rate for new stories. Fred sent me an elegant satire by Phil Klass which I retitled 'Null-P'. I got stories from Poul Anderson, Fred Brown and Mack Reynolds, John Cristopher and others. I wrote a book review department, which I called 'The Dissecting Table'.
The first issue appeared, with a dumb headline sticker contrived by one of Hillman's lieutenants (something about FLYING SAUCER MEN). It was printed on the poorest grade of newsprint I had ever seen, worse even than Lowndes's magazines. When the first sales report came in three weeks later, it was so bad that Hillman cancelled the project at once. Two more issues were in preparation by then and appeared. The cover for the fourth had been painted. The firm did not want to pay the artist for this, either, but this time I stood by him (his sketch had been approved), and he got his money.
In the forties nearly every science fiction magazine had a book review department, but these were mostly of what I later called the shopping guide type; the reviews were about an inch long, and always ended 'A must for every science fiction fan'. Besides the _Worlds Beyond_ reviews, I had already written one long critical essay, 'Destiny's Child', about the works of A. E. van Vogt, which Larry Shaw had published in one of his amateur magazines. When Lester started two new magazines, _Space Science Fiction_ and _Science Fiction Adventures,_ I was able to talk him into letting me do the book department in one of them. He paid me, if I remember, $15 a column.
After a year or so Lowndes also offered to run any reviews I sent him, no matter what the length, and to pay his usual rates, i.e. half a cent a word. At various times I also published reviews in Harlan Ellison's huge sloppy fanzine _Dimensions_ (where my column was called 'Gardy-loo', a call formerly used when throwing the contents of chamber pots out of windows), in Walt Willis's _Hyphen,_ in _Infinity,_ and finally in the _Magazine of Fantasy & Science Fiction._ When I quit, in a dispute over a review _F &SF_ refused to print, I had been reviewing books for nine years.
These reviews were generally successful, even with the authors. (Bob Tucker told me he had been present when Jerry Sohl read my review of _Point Ultimate,_ and that Jerry had laughed and cried at the same time.) There were a few exceptions, however. Someone wrote a virulent letter to _Infinity_ under a pseudonym objecting to my review of a book of stories by Richard Matheson. (He called me, I think, 'one of the great frustrates of our time'.) _Infinity_ published the letter. I noticed that the address was the same as that of Charles McNutt, a one-time fan who had, not quite incomprehensibly, changed his name to Charles Beaumont. I wrote to ask him if he was the author of the letter, and he replied that he was not but that he knew who was. He wouldn't tell me who, and could not explain why his address had been used. I said I would not duel with anybody who fired from ambush, and that was the end of that.
Horace Gold, the editor of the new magazine _Galaxy,_ had bought a story of mine called 'To Serve Man', and I wrote him another story. He bought that, and a third story, and a fourth. I wrote them one after another sitting on the sofabed Lester had given us, with my typewriter on a kitchen chair between my knees. When I sold Horace still another story I realized that, as a successful author, I was no longer tied to New York. Helen and I stored our furniture and bought air tickets to California.
We rented a cottage on the side of a mountain in La Sierra. The view across the valley was magnificent, and there was a little garden. I put my typewriter on a chair under the pepper tree in the yard and finished 'Double Meaning', the novelette I had started before we left New York.
My relationship with Helen was affectionate and companionable rather than romantic. As long as we were poor we got along beautifully and were very happy together. If we were down to a dollar and a half, we spent it on a movie, knowing that something would turn up in a day or two. We invented balloon-ball, played with a string for a net : whoever let the balloon touch the floor on his side lost a point. I worked out a way of simulating computer-written stories, called it 'logogenetics', and we spent hours at that.
Gold rejected 'Double Meaning', my first clue that all was not well in the writers' paradise. (Sam Merwin later bought it and published it in _Startling.)_ I wrote another story and Gold rejected that one too.
Feeling too isolated in La Sierra, we moved to Santa Monica, where we met Richard Matheson and his girlfriend. We lived in a garage apartment owned by a tv actress. Broke, I went to work in an aircraft plant as a file clerk. They fired me after six weeks, to my relief.
We decided we had had enough of southern California, with its eight months of sunshine and four months of rain. We went back to New York and stayed temporarily in Lester's apartment again. (He was living elsewhere.)
I went back to Popular one more time when Mike Tilden needed someone to fill in for a month or so. He said gruffly that I could keep the job if I wanted it, but I didn't. By then it was routine; I could do it without thinking about it, and didn't like it anymore.
Within a year Popular folded all its pulps and let the editors go. Later I found Mike and Ejler Jakobsson working in the same office with Larry Shaw on a series of porn novels published by Universal Publishing & Distributing Co. Mike's wife had died and his son had committed suicide; he died himself a few years later, broke, unlaundered and patient to the end.
Looking at a map, Helen and I saw names we liked in the Poconos and took a bus there. We found a four-room cabin in the woods and rented it from the owner, a bartender named Diebold and his wife. (I used him in _A for Anything.)_ It was about a mile from Canadensis, itself nothing but a crossroads with a post office and a few stores. There was a good-sized lawn which I had to cut with a scythe. Behind us in the woods was a tiny shed, not much bigger than a phone booth, in which a halfwit lived. I found an old desk in a shed and dragged it into the house; it was fragrant with barnyard odours and had great gaps between the boards of its top. We acquired kittens; one of them fell into the well and another was caught in a trap set by the halfwit. In August of that year our first child was born; we named her Valerie. I began writing again, and finished a long story, 'Natural State', which Gold bought.
This was my first editorial collaboration with Horace, and it left me with mixed feelings. Previously I had just written the stories and he had either bought or rejected them; this time I went to him with an idea and we talked it over. The idea was for a story to be called 'Cannon Fodder', which was to take the form of an epic journey by some soldiers and their 'cannon' – a living creature biologically engineered to be a weapon. Gold turned this around and produced the idea of a whole culture based on biological engineering rather than machines; he also contributed some of the most telling details, such as the knife-bushes. Beyond doubt, this was a much better story than the one I had had in mind, but I could not help feeling that I would rather have written my story. I bore this in mind when I next became an editor.
I wrote another long story, 'Rule Golden', but Gold bounced it and I had to get rid of it to _Science Fiction Adventures,_ then edited by Harry Harrison. Even so, we had money again, and bought another car, an imposing green sedan. Helen's father, who was dying, came to live with us; he was patient and quiet in his pain, made no trouble, scarcely disturbed the air.
I wrote 'Special Delivery', a story about an unborn superman, based on Helen's pregnancy and on a remark of hers, 'Give him one for me'.
I wrote a long story intended for _Beyond,_ called 'Be My Guest', finishing it just before Christmas, but Gold bounced that one and it was years before I sold it to Hans Santesson for _Fantastic Universe._ One of the characters in this story was modelled after a disturbed girl whom Chéster had gone to bed with once, and who for months afterward came around and left little tokens for him – poems, Christmas cards, crushed eggshells.
I had first met Horace Gold in 1950, shortly after the first issue of _Galaxy_ appeared. He was a bald, stocky man, restless and energetic, boastful, innovative, brilliant – all the things that _Galaxy_ was. Under all this there was a hard core of despair. Once when he reached for some small object on his desk it toppled and broke. 'Gold touched it,' he said glumly. After the war Gold had developed an extreme case of agoraphobia, and now never left the East Side apartment where he lived with his wife and young son. There were frequent parties there, and he spent hours in telephone conversations. I was always uneasy around Gold because he was the only editor who was buying my stuff with any regularity and because I wanted to like him better than I did. I have since felt the vibrations of similar feelings from writers I have published. It's easier to like someone who is dependent on your favours than it is for him to like you.
Once I called Horace to ask how the magazine was doing, and he asked me as a favour to write to the publishers praising the first issue. I thought this was an odd request, but said I would, and wrote a letter beginning, 'At H. L. Gold's suggestion, I write to tell you what a great job I think he is doing as editor of _Galaxy'._ They showed this letter to Gold, and he told me his wife thought I was knifing him, but he realized I was merely naive. After that, every time something went wrong at _Galaxy,_ Evelyn said, 'Oh, well, Damon can always take over'.
Gold had an incurable habit of overediting stories; as Lester once said, he turned mediocre stories into good ones, and excellent stories into good ones. He bought Edgar Pangborn's beautiful 'Angel's Egg' and showed it to several writers in manuscript, then rewrote some of its best phrases. He changed the description of the 'angel' (a visitor from another planet) riding on the back of a hawk 'with her speaking hands on his terrible head' to 'with her telepathic hands on his predatory head'. According to Ted Sturgeon, when the issue came out and the story was read in the printed version, three pairs of heels hit the floor at that point and three people tried to phone Gold to curse him for a meddler. Sturgeon got in the habit of marking out certain phrases in his manuscripts and writing them in again above the line in ink. Gold asked him why he did that, pointing out that it made it difficult for him to write in corrections. 'That's why I do it,' Sturgeon replied.
Gold was certainly one of the best idea men in the business, and contributed more to the stories published in _Galaxy_ than will ever be known. Blish complained that his invariable response to an author's idea was to turn it on its head, but in fact sometimes he merely turned it sidewise, to its great benefit.
Once Horace called me up in Canadensis and proposed that I become what he called a utility writer for _Galaxy,_ writing stories to order on whatever themes Horace needed at the moment, and under various pseudonyms – 'maybe even under women's names'. I wanted to say no but didn't dare, and agreed with such faint enthusiasm that Horace knew what I meant, and was disappointed twice – once for my refusal and once for my failure to come out with it. I know now that editors are constantly disappointed by authors' unwillingness to fight, and would often rather have a forthright 'no' than a weak-kneed 'yes'.
I had been disappointed in my early ambition to become an _Astounding_ writer; Campbell returned my submissions via Sturgeon with scrawled comments such as 'early 1930' or 'so what?' which hurt my feelings without teaching me anything. I managed to sell him one-half of a story (a collaboration with Blish called 'Tiger Ride'), and that was all until 1952 when I sold him one story of my own, 'The Analogues', which had a touch of Dianetics in it. In 1964 I sold him another, 'Semper Fi', which he retitled 'Satisfaction'. (My title, which I prefer, is Marine slang for 'Fuck you, Jack. I've got mine'.)
I heard about the four-page letters other people got from Campbell, and I felt left out. Eventually I wrote to him asking for more guidance, and he wrote back inviting me to lunch, but I was about to leave for California and had to decline. No doubt I could have got myself invited to lunch long before, but Campbell's lecture-room manner was so unpleasant to me that I was unwilling to face it. Campbell talked a great deal more than he listened, and he liked to say outrageous things; I could not cope with this, and if my patience gave out, my only response was anger.
Now I saw _Galaxy_ as the longed-for ideal science fiction market, and the fact that Horace was buying nearly everything I wrote made it easy to overlook any defects in it. My stories were consistently appearing in first place in the readers' ballots as long as Gold ran them. (Gold insisted, by the way, that Campbell had told him on the phone that he threw away readers' letters and made up the percentages in his 'AnLab' department.)
When Gold began rejecting my stories and I had to look for other markets, I felt betrayed. It's true that these were not the sort of stories he was used to buying from me, but I felt that ought not to matter, and that the whole point of a magazine like _Galaxy_ was that it should buy, if it could, the best work of the best writers no matter what kind it was.
When I came to edit _Orbit_ I tried to live up to this ideal and found that I couldn't. I bought five or six stories in a row from Gardner Dozois and Gene Wolfe and other writers, and then rejected other stories which they must have had every reason to think I would buy. So it goes.
Helen and I wanted a bigger house, and found one for rent in Canadensis, but the owner's eyes narrowed when I said I was a writer. Finding nothing else nearer, we went house - hunting in Milford, with Judy Merril's energetic help.
The first Milford colonists had been the Blishes, who had answered an ad in the _Times_ and had signed an agreement by which they would buy a house in installments and would not get the deed until it was all paid for. This kept them out of the mortgage mill – they had no money for a down payment anyhow – but it made them nervous for years lest they lose the house and all their investment. Their house was a charming two-storey cottage with a sundeck overlooking a long sweep of lawn down to the Sawkill (which later flooded them out).
Judy came next, rented a cold Victorian house on Broad Street, reassembled her family and set out to be a mother. Both children had been living with their fathers and both came to Judy voluntarily, but in taking them she violated custody agreements and that led to trouble later. Fred sued her to regain custody of their daughter Ann, and there was a messy court hearing at which nearly everybody we knew was drawn in to testify on one side or the other.
Milford is a quiet little town on the Delaware. The permanent population then was about a thousand. The streets are lined with old maples, and are beautiful in the fall. Most of the houses are white-painted frame, many of them Victorian houses, with gingerbread, gables and slate roofs. The town has a high society composed of old residents, second and third generation; newcomers are never admitted to this, but anybody who stays one winter will thereafter be treated as human. Tourism keeps Milford alive; to the north of it there are towns like Hawley which are shockingly decayed. The town has always been known for its restaurants, among them the Fauchère, which serves an old-fashioned menu and requires its guests to be decently dressed.
We found a cottage on Ann Street at $35 a month and moved in. The house was painted white inside and out, had sagging wood floors and a bow window with window-seats. The front room was unheated except for a fireplace and in the winter we found that we had to close it off or we couldn't heat the rest of the house. The front room was where we kept the television, however, and there was no convenient place for it in the middle room. Our solution was to tack a blanket across the open doorway and watch the tv over it.
I had been writing longer and longer things, and I thought I was due for a novel, but I still shrank from the idea of doing all that work from scratch. Instead, I thought of a sequel to a story of mine called 'The Analogues'. The sequel, 'Turncoat', was a little over twenty thousand words, and then I had enough to offer with an outline of the rest to Walter Fultz of Lion Books. He gave me a contract, and I finished the book as _Hell's Pavement._ The novel was about the consequences of an invention, and it was more or less legitimate for it to be broken down into a short section (the original story) introducing the invention, then a longer one showing its early development, and a still longer section winding up the plot.
I thought I would try this again with another gadget, and this time I chose the matter duplicator, because I thought previous writers had handled it badly. George O. Smith, in 'Pandora's Box', had preserved civilization by introducing coins made of an unduplicable substance. I thought this was a rabbit out of a hat, and that the thing to do was to let civilization collapse and then see what happened. (Later an Alaskan writer, Ralph Williams, took exception to my version and wrote a lovely story called 'Business as Usual, During Alterations', in which he argued persuasively that civilization would not even shudder.) I wrote the first part and sold it to _F &SF_ as 'A for Anything', then, with that and an outline, got a contract from Fultz for the novel. My thesis was that following the collapse of industrial civilization a new slave-holding society would arise, and that the new masters would necessarily take over the only existing houses big and isolated enough for their purposes, like resort hotels. I put my hero in a real place called Buck Hill, not far from Canadensis; the description of the grounds and the exterior of the house is from observation.
I got to within about ten thousand words of the end of this novel and then hit a block – I knew what was to happen next, but just couldn't write it. Fultz by this time had left Lion, to be replaced by his former secretary; the firm had gone into liquidation and its assets had been acquired by a new corporation doing business as Zenith Books. In order to finish the book I plunged in and wrote it as best I could. The treatment of the rebel leader in the last chapters was perfunctory, but otherwise the ending seemed all right to me. I delivered the ms. to Zenith and asked Fultz's successor to hold off publication for a few months so that I could sell serial rights; she refused, saying that she needed the book right away, and since I was late with it, I gulped and agreed. The book was not published until nearly twelve months later.
Zenith's emblem was a V-shaped thing, and the fact that it pointed downward made me suggest to the editor that the company ought to call itself Nadir Books; but she didn't get it. I was right, though.
In 1955 the partners in a new fan publishing house called Advent approached me with the idea of making a collection of my book reviews. They gave me a contract under which I was to get half the profits after the costs of production had been met. I put the collection together from tearsheets and carbons, although my agent would have no part of it, and said I would never see a nickel. Anthony Boucher contributed an introduction, and I insisted that he get a percentage too. The collection was published in 1956. A revised and enlarged edition was published in 1967, and the book has brought in a few hundred dollars every year since it was first published, for a total of about $2,000.
In 1958 James L. Quinn, the publisher of If, asked me to become the editor of the magazine. Larry Shaw had been the editor in the early fifties, when he published the original novelette version of Blish's 'A Case of Conscience'; but when Larry returned a story of Judith Merril's because he thought she could sell it elsewhere for more money, Quinn took this as disloyalty and fired him. At this time Quinn had been editing the magazine himself for several years, and its circulation had been going down. He was faced with the choice of folding it or trying another editor. He did the layouts for If himself and was good at it, but his tastes in fiction ran to conventional satires about automobiles and computers. I edited three issues of If, and gave it my best, but the circulation did not go up and Quinn sold the magazine to _Galaxy._
Among the stories I inherited when I began editing the magazine was one called 'The Founding of Fishdollar Five' (I shortened this to 'The Fishdollar Affair') by Richard McKenna. Quinn had promised McKenna he would buy this story if he would cut it in half. McKenna has told how he did this and how important it was to him in his 'Journey With a Little Man'. The story was cut to the bone, and Quinn said he had not expected to be taken literally, but he bought it. I was impressed with McKenna and invited him to the Milford Conference. I also invited a writer named Kate Wilhelm, from whom I hadn't bought anything but whose stories had caught my eye. These were fateful decisions.
I had visualized Kate Wilhelm as a middle-aged woman with iron-gray hair and flat heels; instead, she turned out to be young, slender and pretty. That year we had also invited an MIT student called Shag, who was not a professional writer and really should not have been there; he was hopelessly smitten with Katie. We sat up all night in the Blishes' living room, the last night of the Conference, and in the morning A. J. Budrys and I took Kate to the train, where AJ kissed her and she shook hands with me. When we got back to the Blishes', AJ said to Shag, with a twinkle in his eye, 'She was incredibly passionate,' and Shag said, 'You bastard'.
In 1959 I got a copy of the French magazine _Fiction,_ which had translated one of my stories. _Fiction_ was founded as the French edition of _F &SF,_ but almost from the beginning had been using the work of native writers, and at this time the contents were about half and half. It was then an attractive little magazine, with covers by Jean-Claude Forest, the artist who created _Barbarella._
In the forties I had taught myself a little French with the intention of trying to puzzle out the text of sexy French magazines and books. (I had an exaggerated idea of the naughtiness of _La Vie Parisienne,_ from references to that magazine in early science fiction stories.) I was disappointed in the texts I found, but kept at it anyhow and got as far as reading all the way through a novel of Andre Maurois', _Climats._ This was by no means enough to qualify me as a translator, but I got out my French-English dictionary and sat down at the dining room table with the first story in the magazine, 'Au Pilote Aveugle' by Charles Henneberg (really a collaboration between Henneberg and his wife Nathalie, who continued writing stories much like this one after his death). The story went smoothly into English, happened to be very good, and I sold the translation to _F &SF._ Then it was easy to do more. The work of translation, and even more the correspondence with the authors, improved my French enormously, although I still can't understand spoken French well enough to carry on a conversation. The only time I ever had the courage to try it was with José Sanz, the organizer of the film festival in Rio, who wouldn't talk to us because he was ashamed of his English. After two or three shots of my French, he gave in and began talking in perfect and almost accentless English.
In 1960 Robert P. Mills, who had been the editor of _F &SF,_ went into the agency business, first as an associate of Rogers Terrill, then with Ashley Famous Agency, and finally on his own. I was his first client, and the first thing he said to me was, 'I think you ought to be in hard-cover'. He sent a collection of my stories over to Simon & Schuster, and Clayton Rawson bought it. My original title was _Stop the World_ but Clayt, who had never heard the phrase, vetoed it; I proposed _Far Out,_ and the book was published under that title.
Rawson came out to the Milford Conference next year and proposed to me that I should edit a large retrospective collection of science fiction, an idea that had occurred to him because one morning he had two proposals for sf books on his desk, one about old-time science fiction and another from a very young writer; and it struck him that there must be many young people who had never heard of the older sf.
I had always been convinced that I could edit a superior anthology, but had never found out how you convinced an editor of that unless you had already done one. (I still don't know.) I attacked this assignment with enthusiasm, dragged out all my favourite old stories, and Clayt sent several of them back with sounds of pained displeasure. I looked them over again more carefully and realized with dismay that they were junk which had impressed me in my ignorance when I was twelve and thirteen. In spite of this I managed to put together a collection that pleased both Clayt and me (the excerpt from _Twenty Thousand Leagues Under the Sea_ was included at his suggestion), then found that the second anthology was no trouble to sell. As my production of fiction diminished and my responsibilities increased I turned more and more to anthologies as a way of making a living.
Thomas A. Dardis of Berkley Books asked in 1960 if I would be interested in becoming their science fiction consultant. He had approached Groff Conklin first; Groff had suggested me. I served in this capacity for six years, reading manuscripts and writing reports for Dardis, and also did some freelance copy-editing. In 1963 I persuaded Dardis to let me edit four books a year, working directly with the authors and giving contracts on the basis of outlines. In this way I got first novels from Keith Laumer, Thomas M. Disch and others, and brought Gordon R. Dickson and Poul Anderson into the Berkley list.
In 1961, after my mother died and left me some money, my relationship with Helen began to deteriorate, as if wealth had done us in. We owned property and had money in the bank (most of the time), but we no longer enjoyed each other's company. Later Helen explained it as cabin fever: in my presence she would think, 'Ugh, he's breathing'. We tried this and that, but nothing worked, and eventually I went to an upstairs bedroom.
Helen moved out with the children, first to a little house near the river and then to Port Jervis, where she still lives. We were divorced after the degrading, grotesque and cruel preliminaries then required by the Commonwealth of Pennsylvania.
Next year at the Conference, Katie and I approached each other hesitantly; neither of us knew quite how to begin, but we finally managed. We agreed that Katie would get a divorce, bring her two boys to Milford and live there for a year; then if all went well we would be married. She stayed with Judy for a week or two, then rented a little house on the Dingmans road. Kate's sons, Dusty and Dickie, were then thirteen and nine, and they circled me like strange dogs. Dickie, who wore paratrooper's boots, tried to kick me in the kneecap, but I caught his foot and dumped him. After that things went a little better, and eventually very well.
When I told Judy that Kate and I were going to be married, her jaw dropped. I had read about this in fiction, but it was the first time I had ever seen it happen.
We wanted a real wedding but not a minister, and on finding out that Pennsylvania law permits a couple to stand up before witnesses and declare themselves married, we asked Ted Thomas to perform a ceremony which we devised, basing it on a Unitarian service Ted got for us and altering it here and there. Mac McKenna gave the bride away; Aviam Davidson was my best man and Carol Emshwiller was the matron of honour.
In 1963 when I was working on a short novel called _The Other Foot,_ which is still my favourite, and was having difficulty with it, I turned for relaxation to another novel which I made up as I went along; I called it _The Tree of Time._ It was a wild vanVogtian adventure involving an amnesiac superman from the future and a search for a monster which turned out to be the hero in disguise, etc. I enjoyed writing it, especially the sequences that took place in a zero-G satellite of the future (a nasty little scientist I introduced here was modelled partly after J. R. Pierce). All my friends and well-wishers hated it, but I sold it everywhere – _F &SF,_ Doubleday, book club, paperback.
The Tocks Island Dam and Recreation Project was threatening to inundate the Delaware Valley, and the condemnation line ran down the middle of the highway in front of our house. It was evident to us that if we stayed, eventually we would be surrounded by hot-dog stands. Worse, we were getting air pollution from New York for the first time. We made up our minds to sell the Anchorage and move to Florida.
We found a twelve-year-old house for sale on the bay; the waterfront was narrow and at the end of an inlet, and the house was not as spacious as we had wanted, but it would do and we bought it. In our little back yard there is an ear tree (something like a jacaranda), and from this tree we have hung a bird-feeder, and wild budgerigars cluster on this bird-feeder, along with sparrows, doves, and occasional jays and orioles. Herons perch on our dock. The bay is polluted, but not as much as it was before the experimental sewage plant nearby was turned into a pumping station, and I swim there nearly every day. Kate's son Dick and my son Kris, who had been living with their father and mother respectively, are now living with us, and our son Jon, who ignored the ocean when we first brought him down here, has had swimming lessons at the Bath Club and swims like a blond seal now. Pinellas County has the highest growth rate in the country, and we know pollution and crowding will force us to move on in three or four years, but for the present we are all right.
In 1969 the Brazilian government held a film festival in an effort to put Rio de Janeiro on the map, and José Sanz, a science fiction buff, talked them into holding an sf seminar in conjunction with it. Sanz invited sf writers wholesale, and the writers suggested other writers.
Rio is the only beautiful city I have ever seen. From the mountains around the city you can look out over the blue ocean and see only one ship. We were quartered in one of the hotels along the ocean front, and were ferried every day to the French cultural centre to listen to one of our number make a speech. This was idiocy, but we attended faithfully in order to show our gratitude, and refrained from making any speeches ourselves.
Van Vogt said that the universe to him was a tree with golden balls on its branches, and the next day all the Brazilian papers faithfully reported that van Vogt had said the universe was a tree with golden balls on its branches.
Brian was in our hotel and we saw him several times, but most of the others were in three other hotels farther down the avenue. Katie and I moved in a euphoric glow for the ten days we were there. I can't explain it, but there is an ambience of sexuality and romance on the Copacabana – it is in the air, you breathe it.
The beach at Copacabana is for the people, and you see them there sunning themselves, playing volley ball, the children flying tiny homemade kites that continually flutter down, swoop up, flutter. The sand looks and feels like brown sugar. The surf is strong, and the undertow so swift that if you try to swim in water three feet deep, you find yourself stranded as the water goes out.
A Brazilian writer, André Carneiro, was the chairman of the symposium, and several others hung around the lobby of our meeting place, but they had no part in the proceedings. I tried to organize a meeting of American and Brazilian writers, through Carneiro, but instead it was a meeting of American writers and Brazilian publishers.
At a party we met some of the American embassy people, who appeared to think that the world and their relations to it were all fantasy. The servants were Brazilians; the guests, with two or three exceptions, were North Americans. According to Harlan Ellison, the hostess slipped into the bathroom after him and remarked, 'What happens now is up to you.' (Why are people always trying to seduce Jewish men in the bathroom?)
Once, walking down the Copacabana, we came upon John Brunner sitting in his misery. All Latin-American cities were very depressive, he said. Harlan Ellison, who had brought a large good-looking girl and was not getting along with her, made a scene because the symposium did not want to pay for his long-distance phone calls.
We have been drinking dark-roast coffee ever since. I brought home some _cachaça_ (the local firewater), but it was soon gone, and I can't get any more. José Sanz wrote a letter of apology to Harlan Ellison, and said he hoped there would be another symposium; but nothing has come of it. The postman doesn't knock twice.
I was fifty-one in September of this year (1973) and have been half-expecting a menopausal crisis like the ones I had in my early thirties and forties, but it hasn't come.
The Milford Conference is in its seventeenth year; SFWA is eight years old; _Orbit 14_ will be published in the spring.
I learned to swim in the Delaware in my forties, and lost my fear of water for good when I found out how hard it is to stay underwater. Living with the bay in our back yard and the Gulf two blocks away, I have been swimming nearly every day and am smaller in the waist, bigger in chest and shoulders than I was when we came down here.
Today there was only one other person swimming in the Gulf as far as I could see in both directions. About a dozen terns were on the nearest groin, more than I have seen this year, and on the beach farther down there was a crowd of other birds, gulls and little sandpipers. Sky and water were the improbable Mediterranean colours we see every day (remembering how dingy New York looked when we first went back) – sky a luminous dark blue with puffs of white cloud, water a golden green. (Not bad.)
**Frederik Pohl:**
**ragged claws**
How do you describe what constitutes excellence in science fiction over the long-distance phone, when it's one o'clock in the morning, your coffee is cold in the cup, you've been on the phone for two hours and you have to go to the bathroom? He wasn't a bad person. He had his doctorate in English letters, he was earnestly attempting to say everything that could be said of science fiction in 2,000 words for his newspaper syndicate and he was morally determined that every word would be right. And so I prowled around the 18-foot perimeter of the telephone cord, listening to him tell me why a certain story, by a writer whose name I withhold was better than anything by Bradbury or Arthur Clarke and far superior to the overrated novels of Samuel R. Delany. And from time to time I would try to correct some error of fact or suggest a misinterpretation. The connection must have been bad, I never got through.
I understand you, O telephone pal on Mountain Standard Time. I only wish I could make you understand me. 'What breadth of imagination,' you kept saying, 'what a dazzle of conceptualization and thought.' The thing is, you hadn't ever read any science fiction at all, until your managing editor handed you the assignment. You came to it with your pores all open and receptive, and you were overcome by the combined creation of hundreds of intelligent people writing over a period of a century or so . . . all wrapped up in one schlocky fourth-generation retread.
But you didn't know that was what it was, because it was all new and astonishing to you. In your longlashed virgin gaze This was It. I know how you feel, because I was once you, a far younger and more ignorant version of you.
That was a long time ago. 1930. The world was numb under the great depression. I remember the cold city streets, gloveless relief workers shovelling snow outside the secondhand magazine store where I browsed through the old copies of _Amazing stories_ before selecting the one in which I would invest my dime. I remember a hot summer in my uncle's attic, smelling of salt and curing tobacco, where I found treasure trove, twenty back years of _Argosy_ and _Weird Tales._ I opened the pages of an _Amazing Stories Annual_ and Mars smote me, four-armed green warriors riding six-legged steeds across the ochre plains of the evaporated seas.
No one could have suggested to me then that Edgar Rice Burroughs did not far outshine Sir Walter Scott, that David H. Keller had not a wryer view of humanity than Huxley or that Stanton A. Coblentz was not a wittier satirist than Swift or Voltaire. One might have pointed out that my judgment could have been flawed because I had not read Scott, Huxley or Swift, and only _Candide_ of Voltaire. (Which my mother had given me for my eighth birthday, under the impression it was a fairy tale.) No matter. I imprinted when it was imprinting time. A year or two later I discovered I was not alone in the world, when I joined a magazine's circulation-promotion scheme called The Science Fiction League, attended meetings of its Brooklyn chapter and met people like Donald Wollheim, John Michel and others. Fandom was born : we began starting our own clubs, publishing our own mimeographed magazines, writing our bad little stories. As time went by, they got somewhat better and a few of them got published. By the time I was twenty, which was in 1940, I was editor of two ill-paying but professional sf magazines, and so was Wollheim, and so was Robert Lowndes, and we fans who called ourselves Futurians were now collectively a sizeable part of the professional mainstream of science fiction.
Well, that was quite a long time ago, but I would like to dwell on that time for a moment, the decade or a little more from the late twenties to the end of the thirties. What was it that we imprinted to, so adhesively that so many of us elected to devote our lives to it?
It wasn't world esteem. Mostly we carried our copies of _science_ fiction magazines around with the covers folded over. It certainly wasn't literary excellence. The major authors of the sf magazines of the time – the Kellers and Joe W. Skidmores and Ed Earl Repps – were hamhanded and tone deaf to the point of pain.
And yet Keller in particular was acclaimed by persons in all other ways apparently of normal intelligence as a literary master. I suppose his most famous work was a short story called 'The Revolt of the Pedestrians'. I need hardly describe the story. The title tells it all: in the story's future age, humans have become so used to driving everywhere that they have lost the use of their legs, and their legs have consequently atrophied to stubby stumps . . . until along comes a young man, a throwback, with limbs like Nureyev, who preaches the joys of walking and leads the world back to perambulatory purity. Knocks you cold, doesn't it? Would you believe that, not far from half a century ago, it in fact did strike thousands of readers as revelatory and brilliant?
And yet, although the plot is pretty trivial and the style a botch, justice demands a fairer appraisal of 'The Revolt of the Pedestrians'. It was satire. Swift or Voltaire would have done it better, but neither Swift nor Voltaire happened to be commenting on the current American scene at that time, and those who were limited themselves to japes about prohibition and flappers. What Keller did he surely did badly, but he was one of the very few doing it at all.
Another who ploughed those same furrows was Stanton A. Coblentz. He poked furious fun at money-hunger in _The Blue Barbarians,_ at over-specialization in _After 12,000 Years,_ at all of the foibles of early twentieth-century America in a score of stories. If his style was somewhat more gracious than Keller's (Coblentz is a poet as well as an sf author), his comedy was still on the pratfall level.
Nevertheless Keller, Coblentz and others were opening up a new vantage point from which to observe the human race : that dislocation in space, or in time, which permits what Harlow Shapley calls 'the view from a distant star'. How enlightening that view is! How much more clearly we can see the folly and arbitrariness and transience of our own ways when we look at them from Barsoom or from the end of time.
So that was there, in those ragged pulp magazines of four or five decades ago : social satire. It made a whole generation of us both cynics and dreamers : cynics, because we could see the shoddiness of the now, dreamers, because at the same time other writers were offering us Utopias and magnificent challenges.
The greatest challenge of all, of course, was space. Barsoom. Osnome. Persephone, and the planets of Arcturus and Altair. Where the fantasy life of the normal prepubescent male involved strutting into a Texas town, throwing a silver dollar on the bar and shooting it out with a sheriff, ours involved chopping squat, evil Fenachrone into hamburger with shimmering spheres of force. Where the normal fantasy life of the post-pubescent male involved girls, ours also involved girls – but pink-skinned, oviparous little angels like Dejah Thoris, or the doomed, possessed princesses who haunted the waterside bars of ancient Venus in the stories of Northwest Smith.
And all this was going on, remember, in a world that was gradually putting itself together after the cataclysm of the 1929 crash, en route to the catastrophe of World War II. Out of work veterans were being burned out of their shacks on Anacostia Flats because they were demanding a bonus and the government could find no money to give them. Hungry men sold apples in the streets. Not everyone was broke and despairing in those times, but everyone had seen what happened to a brother or a friend, and everyone was afraid. So we suffered from culture shock. Not from encountering Martians or Plutonians, but from looking up from the pages of _Science Wonder Stories_ or _Amazing_ and seeing the insanity outside our windows that we were told was 'real'.
As quickly as we could, we fled back to Barsoom.
Need I make a parenthesis to explain the meaning of the word 'Barsoom'? Probably not; Edgar Rice Burroughs's invented Mars still lives on; just the other day I saw the covers for a whole new reprinting of the saga, nearly a dozen books, going out in hundreds of thousands to an audience that must still somehow find them as enchanting as we did forty years ago. John Carter was the founder of the house. A veteran of the Civil War (on the rebel side) he happened to find himself trapped in a western cave with a horde of howling Apaches just outside, urging him to come out so they could torture him to death. If Zane Grey had been writing the story, Carter would have come out in a burst of unexpected action, seized a chief, stolen a horse and made his escape. Burroughs got his man out of that spot in a more inventive way, if not a perceptibly more believable one : Carter steals out when the Apaches aren't looking, gazes upward longingly at the planet Mars, holds out his hands to it . . . and somehow, Burroughs did not feel obliged to explain exactly how, is drawn to it, and awakens to find himself on Mars itself, with a four-armed green warrior holding a sword to his throat.
Well, so much for that : the rest of the story is swordplay and rescued princesses. I honestly would not care to relate the plot of any of the Mars books to even the friendliest of audiences.
But plot is only a detail. There is more to the Mars books than that. I do not speak of literary style – Burroughs was a pulpster, hitting his typewriter keyboard with clenched fists. I speak of morality, and perception, and thought. Burroughs used Mars to comment on Earth, just as Swift used Lilliput and Laputa. In _The Master Mind of Mars_ he ridiculed religion, an act not without courage. In all the books he jabbed at prudery (the Barsoomians ornamented their bodies instead of concealing them). The 'stately formal dances' of the Barsoomian nobles no doubt reflect his displeasure with the Charleston and the Bunny Hug. And so on; I will not labour the point, but if the man was writing adventure it was not just adventure.
It was even scientifically accurate. That, I admit, is a claim for Burroughs not often heard, but I think it is defensible. True, Burroughs's Barsoom is not much like the Mars of the Mariner photographs. But it is very like the Mars of Percival Lowell, and that was all that science knew of Mars at the time. Low gravity : Carter leaps over a Barsoomian building with a single bound. (I write it that way because it is how Burroughs wrote it; but how, I wonder, do you leap over anything with a double bound?) Thin air: Burroughs fixes that, by building great 'atmosphere factories' which constantly replenish the molecules that dance out into space. It is even possible to suppose that, in some way, he has anticipated the 'atmosphere factories' that a scientist like Krafft Ehricke would now build on the Moon if he could, because the Barsoomian factories use as raw material what Burroughs calls the 'eighth ray', while Ehricke would use what current science calls the 'solar wind', also a flux of charged particles. We cannot fault Burroughs for his ignorance without faulting all of _fin de siècle_ astronomy. When he wrote _A Princess of Mars_ the question of Mars's habitability was about an even bet. Lowell put his money on the side of Martian life; in fact, he and others were less interested in the question of whether life did exist on Mars, than in solutions of the problem of communicating with the putative Martian astronomers to let them know that life also existed on Earth. (The favourite proposal was to dig great trenches in the Sahara, illustrating geometric propositions like the Pythagorean theorem, fill them with oil and set them ablaze so that Martian telescopes could reveal that Terrestrials knew that a2 \+ b2 = c2.) In extrapolating, if some of Burroughs's guesses were bad, a few were extraordinarily good : Ras Thavas, the brilliant and tyrannical surgeon, was doing organ transplants on Barsoom long before anyone transplanted a heart in South Africa or Texas.
Like Burroughs, Edward Elmer Smith Ph.D. wrote stories so big that a single book could not hold them. His Lensman stories run to half a dozen volumes; the Skylark books ran to three, plus a fourth afterthought written decades later, just before his death. Smith wrote _The Skylark of Space_ in 1919, but there was no one to publish it; it languished in a desk drawer until _Amazing Stories_ appeared in 1926 and Smith discovered its existence and submitted his novel to it a couple of years later. Smith did not write the first _Skylark_ by himself. He was unsure of his ability to deal with 'the love part', and so he invited the collaboration of a lady named Mrs Lee Hawkins Garby. In later books he discovered that he could be sufficiently saccharine on his own.
_The Skylark of Space_ is the story of a National Bureau of Standards chemist named Richard Ballinger Seaton who, fiddling with some unidentified chemicals in solution one day, drips some of them onto a piece of copper, puts an electric charge through it and sees it zap out of sight, through a wall, off into space. He has liberated atomic energy.
Corporate baddies in the employ of 'Steel' attempt to rob him of his work so that they can further loot the economy. Seaton foils them, building a spaceship and planning a pleasure cruise, which becomes a rescue mission when the Steel people build a spaceship of their own, kidnap his sweetheart in it and get trapped many light-years from Earth in the gravity field of an enormous dark star. (If only he had thought to call it a 'black hole'!) By the end of the first book Seaton has rescued his sweetheart, encountered a race of disembodied intelligences and discovered a multi-sunned system of green stars where, on a planet named Osnome, he wins a war, accumulates incredible riches and becomes the overlord of two solar systems.
Now, that, I submit, is _exciting._ It certainly excited me. Not only me; a current chief chemist for the National Bureau of Standards admitted in _Scientific American_ a few years ago that _The Skylark of Space_ was what had turned him on to working for the Bureau in the first place : Seaton's job looked so interesting he wanted it for himself, and ultimately got it. (He has not, however, as yet liberated the atomic energy in copper.)
Of course, that is not all there was to _The Skylark of Space._
There was more. Much more. And it pains me, really physically pains me, to have to note that much of that much more was ludicrously bad. Science? Doc Smith did have a doctorate in chemistry, and one would think he would be pretty reliable on fundamental scientific laws; but he gave no evidence of having heard ever of, say, relativistic effects; Seaton gets in the ship and steps on the gas and before you know it he's going ninety light-years a minute, and so what? Style? Oh, not bad, in a functional, adrenal-stimulating way, but not what you'd call literature, maybe. Human relationships? Don't ask. I'm glad you asked, but don't ask. 'The love part' would make a cow blush.
There exists a passage in _The Skylark of Space_ which I must quote to you, at whatever cost to my case for the defence. Seaton's sweetheart, Dorothy Vaneman, has been heard singing, along with another girl named Margaret Spencer who happened to be kidnapped at the same time. Seaton is impressed with Margaret's voice and says so to Dorothy.
'I'll say she can sing ! ' Dorothy exclaimed. 'I didn't know it 'til just now, but she's soprano soloist in the First Episcopal, no less ! '
'Whee !' Seaton whistled.
Unfortunately there is a great deal more in the same vein. In the second volume in the series, _Skylark Three,_ a whole planet of the wisest and most cultured beings in the universe stops still to listen to our intrepid explorers harmonizing on _The Bulldog on the Bank_ . . . _und so weiter._
And yet –
None of this greatly matters. It turns many readers off, and that is a pity; but there are few novels that don't turn a good many readers off for one reason or another, and to close one's mind to Doc Smith because of his conspicuous flaws is to miss his conspicuous virtues. One might as well reject _Moby Dick_ because of Melville's really pathetic inability to write the sounds of Chinese dialect, or because of his gross misstatements of the natural history of cetaceans.
What Smith set out to do he did, and he did it superlatively well, and he taught a hundred other writers how to do it. Richard Ballinger Seaton is not a foil, he is a person. He does not submit to circumstance, he changes it. He is not afraid of science or technology, he uses it; and if it doesn't do what he wants it to do first time around, he works until he finds the way to make it right. All of the things Doc Smith did badly fade in comparison with the one thing he did well. He taught a whole generation how to dream on a cosmic scale.
In the bestiary of science fiction, Doc Smith was a fiddler crab. The male fiddler has one huge claw. It is so big and clumsy that he can't use it to fight, defend or eat, he can only use it to brandish in a sexy, provocative way, impressing the hell out of the dewy-eyed female fiddler crabs.
Smith is not sf's only fiddler crab, they run rampant over the pages of the early _Amazings_ and they are with us today : Harlan Ellison is one, so is A. E. van Vogt, so is Ray Bradbury. They are characterized by such extreme hypertrophy of one aspect of their writing that we forgive them conspicuous lacks in others.
What Doc Smith, and Edgar Rice Burroughs, and a dozen others gave us was a new way of looking at the world, at all the worlds. In the grimy, chill early thirties the vision was revelatory. It is revelatory today. To someone coming to science fiction for the first time the experience can be overwhelming ; O telephone pal, I see myself in thee.
How splendid it would be if we could build a super-crab ! How fine to take the warmth and love of Bradbury and Sturgeon and the boldness of van Vogt and Smith and the word-play of Delany and Zelazny and the scientific insights of Niven, the encyclopedic invention of Heinlein, the unexpected insights of Cordwainer Smith – if we could take all the admirable organs from all the crab herd and assemble one perfect specimen !
As a matter of fact, it happens all the time.
That's where my friend from Mountain Standard Time went wrong; he confused borrowed parts, swiped from a hundred other writers, and reassembled with creation.
Science fiction calls itself a literature of ideas, and in a sense it is; but there is no copyright office to protect ideas, and we all borrowed them from each other. None of us pay royalties to H. G. Wells for using his time machine, or to Murray Leinster for borrowing his notion of alternate worlds. Consider a story about a huge derelict spaceship whose passengers have lost sight of their goal and control over their course : what does it make you think of? Heinlein? Brian Aldiss? The current tv botch, _Starlost?_ It could be any of them.
Not long ago I wrote a story called 'We Purchased People' and showed it to a friend whom I will call 'X'. 'Fine story,' he said, 'but it's _awfully_ like Y, by Z.' 'Bastard,' I replied to him, you are only right insofar as Y is awfully like A _Plague of Pythons,_ which as you know I wrote myself five years before Z wrote Y.' But the more I thought of it, the more I saw antecedents to A _Plague of Pythons,_ too.
It has happened to me a hundred times; it has happened to all of us. It is not unique to science fiction. Poetry grows in the same way.
I don't object to it . . . much . . . but those who are not a part of the process are sometimes confused by it, like my telephone friend, mistaking borrowing for innovation. And some of them are not a bit confused, but use it to do us harm.
These are the creatures who lurk in the entertainment industry. Directors, producers, 'writers'– I remember once visiting an indepedendent Hollywood producer at his studio, viewing his latest Technicolor disaster and chatting with him in his office. 'I used to use writers for my sci-fi flicks,' he said amiably enough. 'I don't any more. I write them myself. If I ever find myself running out of ideas, well – ' he gestured at the walls of his office, lined with _Galaxys_ and _Analogs_ and paper-backs – 'there's always plenty of inspiration there.'
If I were half the man I should be I would have cut his _papier-maché_ throat, or at least gone straight to a lawyer and begun a class-action suit against him for theft and vandalism. I didn't. I let him live.
It is not just a question of monetary harm, it is that these creatures do not even steal well. There are a hundred fine science fiction stories that will never be filmed because some cheapjack operator has skimmed off the surface thrills, neglected the thought behind them and ruled out any proper treatment of the story by a more talented producer later on.
In the pulps of the thirties, we all learned to write by borrowing notions from our betters. Most of us could not do more; we were too young and too raw in the world to have much of our own to say. We needed the practice of putting words on paper and bouncing them off each other, off editors, off readers, to teach us our skills, and only then did most of us stop to reflect on what it was we wanted to use those skills to say.
It was rather easy for an amateur to get published and thus become an instant professional in the thirties. Especially in the latter part of that time, just before World War II, there were dozens of magazines hungry for material, and only a handful of the editors had enough taste and wisdom to distinguish good from bad. Some of us – I was one – found an end-run around even that obstacle by becoming editors ourselves. Of the Futurians – Wollheim, Lowndes, Cyril Kornbluth, Richard Wilson, Isaac Asimov, Damon Knight _et al –_ I was the first to manage that.
I had formed the habit of visiting editors in their offices, trying to sell them stories. I did not often succeed, but for reasons I cannot guess they were pretty hospitable on the whole. Perhaps they thought that from me they might learn what your average teenage reader really liked in an sf magazine. Perhaps they just enjoyed being revered. F. Orlin Tremaine even bought me a lunch two or three times when he was editing _Astounding;_ I don't remember what I talked about, I suppose I mostly listened. When John Campbell replaced him my supply of free lunches disappeared, but John was if anything even more receptive to my visits (and quite as unreceptive to my stories) than Tremaine; he would admit me to his office and sit me down next to his rolltop desk and fit a cigarette into a holder and explain to me why commercial television would never succeed, or how Bell Telephone was the best-run company in the world. After a while I noticed that the subject we talked about in any given month would appear on the stands as his editorial two months later, and realized what he was doing. He started out each month with some polemical notion, tried it on everybody who dropped by, shored up weaknesses and sharpened his arguments on them and by the end of the month had his editorial written. _Amazing_ was then edited by an ancient named T. O'Connor Sloane, Ph.D, with the white beard of a George Bernard Shaw; he never talked to me much, but he did something Campbell never did : he bought something I wrote and published it. It was a poem. I was like seventeen : it was my first sale for actual money – two dollars – and I would have framed the cheque if I hadn't wanted the money more than I wanted the trophy.
Among the editors I favoured with my company was a man named Robert O. Erisman; I don't remember the names of his magazines, but I remember deciding that I knew as much about science fiction as he did and so I asked him if he would like to hire me as his assistant. He politely declined the offer, but instead of throwing me out of his office he sent me across town to see Rogers Terrill, managing editor of the Popular Publications chain. Popular had just started a cut-rate half-cent-a-word line of magazines and seemed to be in the market for additional titles; Terrill offered me a job, and at nineteen I found myself editor of two science fiction magazines. One was _Astonishing Stories._ It sold for a dime, ran to 112 pulp-sized pages, used about 45,000 words of material an issue, and had a budget of $405. That was not just for stories; that was for stories and black-and-white art and an oil painting to be used for a cover.
Of course, at that kind of money I couldn't get the real top writers in the field. I had to be content with Kornbluth and Asimov and Heinlein and Bradbury and Kuttner and . . . well, they weren't bad, for amateurs. Some of them turned out rather well.
For the first six months I was there Popular paid me ten dollars a week. (Comparatively speaking it was not all that bad. The same week I came aboard they hired another editor, and he had to work three months for nothing before they raised him to ten dollars a week.) I should have paid them. I loved every second of it. Not only was it pure pleasure, but I learned the facts of life in that office. I learned that writers are human beings. I learned that it is impossible to publish any story so good that someone won't loathe it, or so bad that someone won't hail it as a masterpiece. I learned that even famous and widely published writers can desperately need a sale, even a tiny sale, to buy the next day's groceries. I learned what applause feels like – and criticism, too; I've had both, and applause is better. I learned that readers, particularly science fiction readers, are often astonishingly perceptive, both for good and for ill : quick to perceive a writer's meaning even when he elects to express it elliptically, quick to detect fraud when the writer tries to get away with shoddy work. I learned that nearly all readers would rather like you than hate you; once they have bought your work it is their intention to enjoy it, otherwise they would not have bought it, and it takes a serious fault on your part to turn them against you. I learned that it is impossible to tell from a writer's conversation or appearance whether his stories are going to be any good. I learned the mechanics of preparing a manuscript for the printed page, the complexities of dealing with artists and the inflexibility of deadlines; and I learned that ten dollars a week didn't go very far. Not even in 1939.
The godlike figures who owned and operated Popular Publications knew that perfectly well. They expected us to supplement our pitiful salaries by writing for ourselves and each other – at our pitiful word rates. Popular published fifty or sixty magazines : nearly half of them Westerns, the rest a mix of detective, horror, sports, air-war, love, general adventure. I never could bring myself to write a Western, but I wrote for all the others. Even the love pulps. I'm glad I did it. I don't suppose I would have missed any important lesson about writing if I had not had the pulp experience . . . but I might have missed some important eating.
And then, along about 1942 or early 1943, I began to notice that something was missing. I was an old hand by then, all of twenty-three, considerably more prosperous; and somehow I began to perceive most of what I had written was crap. Most of what I had published was crap. Some of it was commercially pretty good – I was selling to outside markets by then, usually for better rates than I could pay myself. But, by and large, there was not much justification for pride. In four years I had managed to publish a few good stories in my magazines, shopping for bargains in the reject pile from _Astounding;_ but in little more time than that John Campbell had revolutionized the science fiction field. A reasonable proportion of my own writing had become competent, but very little of it said anything.
So I began playing little games with myself, trying to trick myself into something worth being vain about. I knew how to write for editors. I began writing stories that I didn't think any editor would buy, experimenting with different forms, trying to find ways of using words that were not imitations of someone else. And at the same time, I began typing out manuscript copies of stories that had impressed me, seeking to see what the tactile sensation was that went with putting words like that through the typewriter keyboard. I had trained myself to sell my first drafts, beginning a story with clean white paper and a carbon in the machine; to break that habit, I began to write my first drafts on scratch paper, lined paper, the back of old correspondence – anything that would take typewriting, but that I could not possibly submit without retyping, and thus revision.
I was trying to make a break with bad habits. As it happened, I was helped to make the break. There was a war going on. I was deferred, but I didn't have to stay deferred if I didn't want to, so I went down to the draft board, asked to be reclassified and spent the next couple of years in places like Oklahoma and Colorado and Italy and France as a sergeant in the US Army Air Force.
I will not trouble you with my wartime reminiscences, but there is one event that pertains to what I am talking about. There was a time, when I was with the 456th Bomb Group in a ploughed-under potato patch south of Foggia, when I had some time to spare. As I was also faintly homesick for New York City I thought I would deal with both by writing a novel about life in New York. Because it seemed sexy and exciting to me, I elected to write about life in the advertising business; and one way or another I managed to put together two or three hundred pages of copy on the subject by the time I got back to civilian life.
At this point it occurred to me that as long as I was writing a novel about advertising, it might help if I knew something about it. So I bought a Sunday _Times,_ applied for three advertising copywriter jobs listed in the Help Wanted section, got one of them and spent the next three or four years writing ads.
That novel never did work out. I had a summer home with a big fireplace one year, and I spent one whole night reading that novel manuscript in front of the fire. And page by page, as I read it, I threw it in the fire. It was awful. Not ludicrously awful, or hopelessly unpublishably awful. Just why-are-you-bothering-to-tell-me-all-this awful. But if the novel was a bust, the advertising experience wasn't; a couple of years later I began writing a science fiction novel about advertising, decided I needed help about a third of the way through, showed it to my old friend and former occasional collaborator Cyril Kornbluth, and came out with _The Space Merchants,_ which probably isn't the most successful sf novel ever written but, with somewhere over ten million copies in print in about forty languages, hasn't done really badly either.
Why-am-I-bothering-to-tell-you-all-this?
Because I think one reason _The Space Merchants_ succeeded was that I bloody well knew a lot about what I was writing about, not because I had read it in somebody else's book, but because I had spent several years of my life learning it. I don't think those years in advertising were wasted. I don't think any time I've ever spent that taught me something I didn't know was wasted, not the six months I spent fascinated with the theory of numbers, or the decade as a part-time political party official, or the year I spent playing chess four hours a day. I haven't used all of them yet. I don't know if I ever need to 'use' them, but they're there.
John Campbell once said the same thing to me, when I was maybe seventeen and he was well, J*O*H*N C*A*M*P*B*E*L*L, pushing thirty and secret ruler of the universe. John got out of college in the pit of the depression, no jobs, had to scratch. When he finished prepping that month's editorial with me that day, he began to reminisce. Jobs he had had. I said resolutely it was a waste for a mind like his to be doing things like that, and he put another cigarette in the holder and said I didn't understand, all those things were part of him, without them he would be lesser, they were components of what he thought and did and wrote. I think he was right. Can you find anything in, say, the Don A. Stuart stories that reflects his six months as a used-car dealer? Neither can I, but that doesn't mean that, in some mutated and transmogrified way, it isn't there.
It is not much good copying science out of a textbook onto your manuscript page. The bare bones show. It is not much good transcribing part of your autobiography into your story either, it seldom fits, the ends don't mesh; but what you have read and what you have done and what you have felt and what you know, are, after all, the only things you as a writer have to sell, and in some way they gurgle through the sloshy pipes of the brain, losing an amino acid here and picking up an enzyme there, and what emerges is part of you. Probably the best part of you. The best part of all of us is in what we write.
My grandfather died when I was sixteen and, for one of the few times in my life, I saw my mother cry. I was hurting, too, because I liked the deaf, bent, bald-headed old Irishman, in the aloof and self-interested way a teenager likes any drastically older person, but I discovered something. One part of my mind was trying to respond to my mother's pain and feeling pain of my own. Another part was taking notes : this is how grief is shown, here is how a chair looks when someone who sat in it has died; how can one write down the sound of a sob?
All is grist to a writer's mill. Every experience, even the worst, is in some way a deposit to that bank account we all draw on every time we put words on paper.
One of the things that has helped me write science fiction is that I am a fan of that greatest of spectator sports, science. I take no credit for knowing what goes on in science, it is fun for me; I subscribe to about a dozen journals and other periodicals, read a lot of books, spend a fair amount of time, when the opportunity presents, with people who are working at science. The result is that with not enough formal education to qualify me as a mailman I know about as much about, say, astronomy as any biochemist, as much mathematics as most physicists and so on. I don't do it because I am researching stories. I do it because it is more fun than playing bridge. (Although I also like to play bridge.)
Most of us are like that: we like to learn things. Not always science, for some of the best of us are allergic to science, hate it or fear it or simply don't include it in our lives. But we like to learn, and to share what we have learned. Among us we have produced poems and plays and journals as well as science fiction stories; works about history and biography and politics and travel and model-building and salesmanship and bicycle repair. We are all risk-taking and inquisitive by nature, and if there is a Renaissance man alive I think he is a science fiction writer.
Having come this far, I would like to tell you what science fiction writers _are._
Probably you already have some notions of your own. Perhaps they are the world's notions : the archetypal science fiction writer as a gnomish little creature who sits in his attic and spins dreams about imaginary worlds, as uninvolved in the real world outside his windows as Proust in his cork-lined room; a little odd in his manner; a little strange in his attitudes; more prone than the rest of the population to fall apart and have to be put away.
These notions aren't wrong. There are science fiction writers like that. But they also aren't right, because there are other science fiction writers who are extroverted, gregarious, athletic and even stable. The first general statement that can be made about science fiction writers is that there are few general statements that can be made about all of them.
There are somewhere between 750 and a thousand of us, I think, scattered all over the globe. Perhaps 300 or more are in the United States, another 75 or so in Great Britain, as many as 150 (no outsider really knows for sure) in the Soviet Union. There are colonies in Germany, Japan, France and Italy; small groups or individuals, some of them world-renowned, in Poland, Brazil, Mexico, the Scandinavian countries, Australia and almost everywhere. I don't suppose there is a country in the world (possibly excepting mainland China) literate enough to support a publishing industry at all that has not at least one or two local sf writers. I once received a manuscript from a man who described himself as the second-best science fiction writer in Iran, and for all I know perhaps he was.
There was a time when all the world was islands and a science fiction writer in Paraguay would hardly know of the existence of another in Brazil, although both would probably read English novels and American magazines. The barriers are no longer high. Now we all compete with each other, and learn from each other, and from time to time we meet. How we meet! In Tokyo and Trieste, Montreal and Moscow, London and L.A. The streets outside the hotel windows look different, but inside there are the same familiar faces. If I ever find myself marooned on an Alp or lost in the Sahara I have an infallible system for getting help. I will simply announce the existence of a science fiction symposium, and in half an hour Brian Aldiss and Arthur Clarke will be there.
Most science fiction writers do other things than write sf. Many are scientists : the astronomers (Fred Hoyle, R. S. Richardson), the biochemist (Isaac Asimov), the mathematicians (Chandler Davis, Eric Temple Bell alias 'John Taine'), the medical doctors (David H. Keller, T. J. Bass), the electronics experts (John R. Pierce, Ben Bova). A number (Ray Bradbury, Harlan Ellison, Robert Bloch, Leigh Brackett) double as film and television writers. An even larger number (Jack Williamson, James E. Gunn, Richard Wilson) are in the faculty or administration of universities. Squads are or have been in advertising and public relations (Bob Shaw, Algis Budrys, James Blish, Theodore Sturgeon). L. J. Stecher was captain of the US Navy's guided-missile cruiser _Columbus;_ A. Bertram Chandler commands a cargo vessel in the South Pacific. Keith Laumer was a captain in the US Air Force. Wilson Tucker was a movie projectionist. Roger Zelazny works for the Civil Service. J. F. Bone is a veterinarian. There are even a few (Larry Niven) whose grandparents were kind enough to leave them fortunes so they don't have to work at all – but they do, at writing sf.
The sf writer may be of any age from early literacy to advanced decay. I once invited some of my writers to join me for cocktails at my hotel in Los Angeles; one, whom I hadn't met before, respectfully had to decline. As he was only fourteen, he wasn't allowed to stay out that late. But Wilmar Shiras was a grandmother before she wrote her first story, and E. Everett Evans only began to be successful in his sixties.
Once started, few ever stop. A few years ago _Time_ did a story on science fiction and attempted to write a 'where are they now' sidebar on the great writers of the past. After a great deal of research they found that those who were not dead were all still writing.
What keeps us at it? A few years ago two sf fans named Bill Bowers and Bill Mallardi asked about ninety-four of us that question, among others. Some of the answers :
Damon Knight : ' . . . to tell the truth, I've never been interested in anything else.'
John W. Campbell : 'There's room to think and move ! '
Isaac Asimov : 'Because science fiction, in these times, is the most significant literature one can write.'
Brian W. Aldiss : 'I am a surrealist at heart; that is, I'm none too sure whether the reality of the world agrees with its appearance. Only in sf, or near-sf, can you express this feeling in words.'
Philip K. Dick : '... it's the broadest field of fiction, permitting the most far-ranging and advanced concepts of every possible type; no variety of idea can be excluded from sf ; everything is its property.'
Theodore Sturgeon : 'It gives me almost complete freedom of speech, and absolute freedom of thought.'
Arthur C. Clarke : 'Because most other literature isn't concerned with reality.'*
'Freedom'. 'Significance'. 'Reality'____ Does anyone remember the McCarthy years in the United States? The pipsqueak senator from Wisconsin had a nation terrorized, and while presidents and newspaper publishers were running for the storm cellars there was one surviving area of free speech in the country: the science fiction magazines. I have met ministers who used to put the current issue of _Galaxy_ on sale in their churches in those years, because nowhere else could their parishioners read anything else but the straightest of orthodoxy. I wish Horace Gold and John Campbell could have known that whole congregations were praying for their continued well-being in those days. It would have cheered – and surprised – them both.
Science fiction writers, by and large, are stubborn, smart, unconventional, independent makers-and-shakers. This is an opinion I have formed over forty years of living among them, and is supported by the only scientific study I know of, conducted by psychologists on a representative batch of us. The psychologists, John E. Drevdahl and Raymond B. Cattell, sent out questionnaires to 356 individuals, divided into three categories : general writers, artists, and science fiction writers. From the questionnaires they developed psychological profiles for each subject, averaged them out and published the numbers for a variety of traits for each group.
I'll give you some of the numbers. I should explain what the numbers mean. The average of all Americans is taken as 100% for each trait. The scores published give the departures from the average. If sf writers score 10% higher on something than the average, the score is given as +10. If 6% lower, —6. They scored in tendencies towards one or another of opposing pairs of traits (e.g., 'intelligence vs defective ability'), and plus score indicates movement in the direction of the first trait – in this case, higher intelligence. A minus score, as you no doubt already have deduced, means the direction is towards the opposed one.
Here are some of the results :
| _General_ | | _SF_
---|---|---|---
_Factors_ | _Writers_ | _Artists_ | _Writers_
Intelligence vs defective ability | \+ 10.9 | +8.4 | +14.9
Ego strength vs general emotionality | \+ 4.8 | +4.0 | +5.3
Dominance vs submission | \+ 4.7 | +4.0 | +8.7
Super Ego, identification with group standards vs lack of identification with group standards
| –8.4 | –6.9 | –9.5
Adventurous cyclothymia vs withdrawn .. .. .. ..
| +5.7 | +4.7 | +25.4
(Of course, you know that 'cyclothymia' is a mild manic-depressive state. It doesn't mean that sf writers are manic-depressives. It only means that, to the extent they are, they are highly likely to do something about whatever troubles them, and hardly likely at all passively to endure.)
Science fiction writers also scored significantly above the norm on traits like 'Bohemian unconcern' and 'radicalism' – not any particular brand of radicalism, either left-wing or right. Just unwillingness to accept conventional wisdom. Altogether, the picture the profiles display is of a bunch of stubborn, intelligent individualists. Visiting any science fiction convention gives much the same picture. It may be that persons of other sorts simply could not write science fiction at all. As Drevdahl and Cattell say, we 'appear to possess what Mathew Arnold and other great writers have described as "the divine discontent".'*
And, you know? they're right. We do.
If this 'divine discontent' is real and not some flattery our private mirrors whisper to us, I wonder how it comes about.
I think one thing that happened to most of us, somehow, is that as children we were loners. I know I was. I was an only child, and we moved too often for me to make long-term friends until I was in my teens.
I was born in New York City – 26 November 1919 – but I left the city early. I spent my first Christmas at sea, and by the time I was six weeks old I was living in Gatun, where my father had got work as a machinist for the Panama Canal. I spent the first few years of my life in places like Panama and Texas and New Mexico, following my father's work; and even when we returned to New York City to stay we seldom remained in one house, or one neighbourhood, for more than a few months. My father had his own discontent. He did not much like being a machinist. Wall Street was where his rainbow came to earth, and he desperately wanted part of that action. So through all of my childhood we alternated between wealth and want as his bubbles grew and burst.
Sometimes we lived in luxury hotels. Once or twice we lived nowhere at all, and I found myself thrust on relatives for brief periods (I suppose no more than a week or so, though they seemed terribly long) while my parents tried to locate a landlord trusting enough to take them in. It did not seem an unusual way of life to me. It was the only one I knew, and somehow or other I was always clothed, housed and fed.
My mother had quirks of her own. She took me to school at the usual time, five or six years old, and I promptly came down with whooping cough; she tried again when I was well, and I developed scarlet fever. These were pre-antibiotic days. Scarlet fever, for instance, entailed a Board of Health quarantine sign on the door and everything I owned baked two hours in an oven or thrown away. So after I had kept on doing that for a full year (during which I was present in class maybe a dozen times, few of them consecutive) she opted out of the system (red-haired, half Irish, very sure of her own mind). And for the next years I stayed home, and we sometimes dodged, sometimes debated with, the truant officer. My mother had a teaching degree, though she never taught in a school; she sometimes was able to persuade the authorities that she was tutoring me suitably at home.
Well, I think she was. What she did teach me was to read. I don't know when; I can't remember a time when I couldn't read.
The first book I owned was, grotesquely, Emerson's essays, leather binding with a silk marking ribbon. I cannot remember what prankster gave it to me, or guess at with what motives. I certainly never finished reading it; but I appreciated the thought.
Of course there had been other books long before that – I had to have been eight, at least, before encountering Voltaire – but I do not remember them as individuals, only a general recollection of kid's books by Percy Keese Fitzhugh and Leo Edwards, plus Tom Swift and even the Bobbsey Twins. Fitzhugh did make an impression. His books were about scouting. Reading them, I was desperate to reach the age of twelve so that I could become a fully-fledged Boy Scout and participate in octagonal hikes and tie knots and wear a sleeve full of merit badges. Being a Cub was beneath my dignity. A few years later I was in fact old enough to join, but by then I had long lost the impulse. (Later on I went through the identical experience with the Presbyterian church, and again with the Communist Party after that.)
The effect of it all was that I made few friends and kept them not very long, as a young child. My best friends were books.
I don't mean that I did nothing but read. I was passably good at some sports, particularly on and in the water, and I spent a good deal of time exploring the world I lived in. When I was seven or eight I discovered that I need not do my exploring on foot. It was possible to climb a subway embankment, step gingerly over the charged third rail and emerge on the platform of a BMT station, which meant that by means of carefully learned interchanges I could travel almost anywhere in the city of New York free of charge. Getting back was more problematical unless I happened to have a nickel for the turnstile, but there were ways: an ill-fitting exit I could slip through, or a fence I could climb. I was caught at one of these unauthorized excursions once or twice, when I visited an old neighbourhood and was recognized and reported to my parents. But there was no way they could stop me if I did not choose to be stopped.
For shorter distances it was also possible to hitch rides on moving vehicles. Trolley cars were available. The reel that held the cable to the trolley was just meant to be used as a handhold; the skirt that extended out at the back was sloping and slippery, but if the wind was not too sharp or the rain too intense one could ride for half a mile at a time that way. In practice one didn't, because of police. The proper thing to do was to ride a few blocks to a major intersection, drop off, walk across and pick up another hitch on the far side. Trucks and even private cars were also possible, but more dangerous. I fell off one once, at about twenty miles an hour, and that discouraged me from that particular sport. But all in all there was free transport to cover a whole city, and I used it dedicatedly.
Roaming and reading: I suppose I've spent a major fraction of my waking hours in one or the other. When I was old enough to own a card in the public library the world opened up totally to me. I read everything I could put my hands on, out of the limitless resources of one of the largest library systems in the world. At twelve and thirteen it was mostly fiction : all of H. G. Wells, all of Kipling, enormous doses of O. Henry and Anatole France, of Voltaire and of Mark Twain. Around fifteen it was mostly plays: a little Shakespeare, every word of Shaw, Elmer Rice, Eugene O'Neill. By seventeen it was politics, a nibbling of history, a great deal of biography and non-fiction of one kind or another. The library filed its non-fiction according to the Dewey Decimal System, which was a convenience. Long before I learned to use catalogues I discovered that if I went back to the place on the shelves where I had found a book on an interesting subject, the books nearby were likely to be on related ones. And I read authors : William Beebe's wonderful books of undersea exploration, _Half Mile Down_ and _Beneath Tropic Seas;_ Richard Halliburton's sentimentalized and hyperbolic travel books. It helped that I was a rapid reader. Counting professional stints, I imagine I have averaged something better than a book a day for every day of my life. So it was no great loss if any particular book wasn't much good. I read it anyway. Because the library prissily filed O. Henry under his real name, William Sidney Porter, I read half a dozen novels by Gene Stratton Porter before realizing it wasn't the same man. It didn't much matter.
There was no science fiction to speak of in our library, or apart from Wells and Verne, in any library of that era, so there was a sharp line between my library sources and my private sources of reading. While I was cropping the library I was also reading every word of sf I could find, all the back issues of _Amazing_ and _Wonder_ and _Astounding,_ the Carl H. Claudy juveniles, S. Fowler Wright and W. Olaf Stapledon. _Et al._ And I discovered bridges between sf and mainstream. Somehow I got turned on to James Branch Cabell and read every volume of the 'Biography of Manuel' before I was seventeen, as well as _Shirt-Smith-Smire_ and a wonderful jape called _Special Delivery._ This was a volume of fan letters and the wickedly witty replies he should have made (but didn't make) to them. Great discoveries came from this kind of reading. I learned that writers were human from Cabell, and that connective tissue existed linking _Amazing stories_ to the library shelves. Wells was of course in both; but _Back to Methusaleh_ put Shaw firmly into science fiction for me; _With the Night Mail_ and _As Easy as ABC_ gave us Kipling; Cabell and Wright and others seemed to live happily in both worlds. And also I found myself reading poetry of all kinds : the sonnets, rather than the plays, of Shakespeare; Omar Khayyam (I think I can still recite at least half the Rubaiyat); Coleridge; Eliot. I think that came from my mother. I don't remember her ever telling me a bedtime story, but she used to recite poems for me when I was small.
It was so easy and natural for me to pass from reader into writer that I don't quite know when it happened. The first poem I remember writing was when I was ten. I still remember it, though I will not for anything recite it aloud. The first story I remember completing was when I was twelve, in an idle hour in an English class. I had already read the textbook and I had no interest in whatever the teacher was gabbing about, so I wrote out a six or seven-page story about Atlantis. It was of course a thoroughly hopeless story, and it is a blessing to all of us that it is lost or I might have published it somewhere along the way.
Sometimes I am asked what writers influenced my development. There are two answers. One is, 'I don't know.' The other is, 'All of them.'
I can't think of anything I learned, or borrowed, from the French decadents, for instance, but I spent so much time with Proust and Huysmans that I cannot believe I remained untouched. (There was a time around the age of twenty when I found myself reading so much science fiction professionally – I had begun editing magazines, and had every day's slushpile to contend with – that the only recreational reading I could stand was the previous Frenchmen, and the brooding Russians. I can't really find any traces of Tolstoi or Gogol or Merejkowski in my work either.) I know I owed a lot to Lewis Carroll and the _transition_ stable, mostly James Joyce, at one time, when I pleasured myself with coinages and nonsense rhymes and plays on words. And in science fiction I am sure I owe much to many writers; but we all do, we build on each other as a way of life.
The word that comes to me when I think of my teens is disorder'. I planned nothing. I was palping the world at random. Where it gave, I pushed a little harder; where it resisted, I turned away. I suppose if I had had a proper school background that might have imposed at least some skeletal suggestion of structure, but I gave up on school early on. At the age of twelve I entered high school. I had no real notion of what I wanted to learn but some glimmering that 'science' and 'fiction' were somehow involved. I had never heard that one might study the craft of writing (I still don't wholly believe it; most writing courses are preposterous frauds, and most of the good writers I know have had little formal learning in letters), but 'science' was a possibility. And so I applied to Brooklyn Technical High School. One needed to pass a special examination to get in, and maybe that appealed to my snob sense, too. Once in I was confronted almost at once with the necessity of making a lifetime career decision. At _twelve._ Brooklyn Tech offered only specialized courses. I could specialize in electronics or civil engineering or aeronautics or a dozen others, but I had to make up my mind which. 'Chemical engineering' sounded least unpleasant of them, and so I began to be a chemical engineer. By the age of fourteen I knew that wasn't for me, but it would have been troublesome to replace my steps and anyway I had lost interest. So I quit school with a mixed bag of failures and certificates of excellence as soon as I could – at seventeen. My entire school career was less than a decade, from beginning to end, and I begrudge most of that.
I married at twenty.
I married again at twenty-four, and at twenty-eight, and at thirty-two; for a while there it seemed that in every Leap Year of my life I was destined to get married to someone different.
There are companies in which I find this admission a little embarrassing (although the present company of this volume does not happen to be one of them), but there was no malice involved on anyone's part. All three of my ex-wives were excellent women, who simply never should have married me, or I them. The first time we were both unweaned adolescents. The second and third time we confused attraction and some common interests with an ability to live together permanently, which did not turn out to exist. All of my ex-wives achieved interesting and rewarding lives – we became friends again, after the marriages were over – and so did I, because on the fourth try practice made perfect, and Carol and I have now been married for better than twenty years. I would not have it any other way. We have had so many joys together, and so many tragedies – our first son died before he was a month old, and if there is a worse pain than that I pray to be spared it – that I do not imagine either of us has any very objective view of the other; but I know she is a beautiful and joy-giving person. Further than that I cannot say. What I know of her I cannot convey to you, and what I can convey would not do her justice.
It has not always been interesting and rewarding. When I turned fifty there was a bad time when I thought I was going to die, and actually rather preferred it that way. Everything was bad at once, in more ways than I can count. It seemed to me that, although there was some possibility I could survive, and even surmount most of the lacks and harassments in my life, and go on living, and perhaps even experience considerable joys and triumphs, it was nevertheless unlikely that anything would happen that hadn't happened at least once before. I might earn a great deal of money, or receive some honours, or discover a new city to roam, or publish a story that won praise, or experience any of a hundred sensory pleasures, or learn a thousand exciting new things. All of these were great joy; but I had had all these joys at one time or another. Nothing new seemed likely to happen, and a repeat performance did not seem worth going to any great trouble to achieve. So, for a time, I think it must have lasted two or three months, I waited to die. By and by it occurred to me that I was still living, and as I had nothing better to do I might as well work at it. So I began taking cautious steps towards rebirth. I began writing again, and discovered that I could actually write about as well as I ever had, even write a few stories – _The Gold at the Starbow's End, The Merchants of Venus_ and _Shaffery Among the Immortals_ among them – which (I speak wholly without modesty) seemed to be written about as well as it was possible to write those particular stories. I palped the world again for soft spots and pushed where it gave . . . and most of it gave.
Was this what is sometimes called the male menopause? I don't know. I only know that it seems to me that I had a complete lifetime that ended when I was fifty, and now I am several years into a bonus life which has expanded into places where I had never been before. It is almost as though that other Frederik Pohl had bequeathed me a fair estate, which I can draw on as I choose but which I do not have to earn.
I am not quite sure what I will do with this new life. You see, at this moment I am only about four years old.
One thing I am likely to do with it, as long as I can manage to put words on paper, is write science fiction. And edit it, when convenient. And lecture about it, and about the future in general, and about those tactics for shaping the future we call 'politics' and 'sociology'. There is a great good thing about my life which makes up for almost any loss, and that is that I don't have to do anything to earn a living. I spend my time doing the things I like to do for their own sake – writing and editing and lecturing – and people are kind enough to give me money for that.
. . . It is hard for me to write openly and honestly about myself. It is hard for any writer. We tend to confuse autobiography with hagiography, and we waver between twin terrors. There is Scylla on one hand, which is the peril of untruth, and over there is Charybdis, equally dreadful, whose other name is tedium. There is no easy course between bad morality and bad art, and I doubt that I have found the right one. But one tries to keep one's promises. We have collectively promised to tell not only what we know about science fiction but what we know about ourselves. And there it is.
We do the best we can.
We do not so much to please you the reader – not even so much to please one another – as to please some perfect model in our own minds : to try to capture some thought no one has ever thought before, to try to suggest some new way of looking at the world, to build a temporary world as plausible as the one outside the window, and more interesting. We fail a hundred times for every success; but now and then there is a success. Stubborn, difficult, divinely discontented if you will – we do keep trying.
So I despair of telling my long-distance friend what matters in science fiction. I despair of convincing him that Stapledon and Smith and Weinbaum are great while the novel that turned him on is a multipley plagiarized potboiler. It's no use persuading him to read the originals. He would only look up from them and exclaim, 'Why, these are no great writers, they are only fiddler crabs'.
But when these stubborn and difficult people are doing the things they do best, it is the fiddler crabs among them, not the bland syncretists or the popularizers, who show the others how. They are the innovators. They blaze the trail. After them come the pulp hacks, the tv adapters and the Michael Crichtons. The proper measure of the stature of a science fiction writer isn't the size of his bank balance or his audience, it is the degree to which other writers copy him.
And so we go, divinely discontented, snapping our one gross claw; it is not the most perfect way of life one could imagine but where, in this chill and gritty real world we live in, is there another as good?
* * *
* _The DOUBLE : BILL Symposium,_ D.B. Press, Akron, Ohio, 1969.
* _Personality and Creativity in Artists and Writers,_ John E. Drevdahl and Raymond B. Cattell, _Journal of Clinical Psychology,_ Vol. XIV, No. 2, 107–11, April, 1958.
**Brian Aldiss:**
**magic and bare boards**
The difference between fiction and non-fiction is the difference between magic and bare boards.
Imagine you're going to the theatre. The auditorium fills, the orchestra plays, the lights dim, the curtain goes up____
The stage is bare, or cluttered with old and dusty props. Flats lie shuffled at the rear. A light dimly burns. Something has gone wrong. There will be no play tonight, no magic.
A man appears left, looking awkward and vulnerable, as people do when they walk unrehearsed on a stage. He says, ' "Poets are born not made." Nevertheless, a writer can look back over his life... .'
There's an uneasy hush in the stalls.
Well, that's the difference between fiction and non-fiction. Tonight, the stage is bare and lit by one naked bulb. I come on, looking awkward and vulnerable.
'Poets are born not made.' Nevertheless a writer can look back over his life and calculate the steps he took that made him the kind of writer he is. Many of those important steps are taken in childhood. As F. L. Lucas says, in childhood, fate determines character – after which, character determines fate.
So what I intend to do is give some account of my childhood–'edited highlights', as tv commentators say of the day's sport – and then go on to discuss what may be called my writing career; from which we can move to the subject of writing in general. It is no use talking in a vacuum. If you have some notion of what the man is like, you can make some estimate of the value of his ideas.
My father had a great physical gift : he remained thin and neat however well he ate. To the end of his life he could refer to himself self-deprecatingly as a dapper little man.
My mother's mother was the daughter of a farmer. She taught my mother the art of cooking, so that my childhood recollections include eating well. Norfolk was overrun with rabbits in those days, and rabbit made an excellent dish. Mother would serve them cooked whole in casserole, when the tongue was regarded as a great delicacy; or in pies, with whole small onions and chunks of ham. Her pies were beautiful, crowned with splendid patterned pastry. One of her specialities was salmon pudding, a noble edifice served with parsley sauce and new potatoes. Pickled mackerel were another speciality; they used to stand overnight to soak in a jug of vinegar, their skins shining with beautiful subdued colours.
Mother's pudding courses were also splendid. Summer would bring summer puddings, which disgorged lashings of gooseberries and black and red currants. She was expert at cake-making. Maids of Honour made regular appearances on the tea-table, together with Melting Moments and Surprise Buns (which had jam inside them, though you could not tell from the outside). Our cooking was done in a huge oven heated by three paraffin burners underneath.
Mother made music in that oven. Each Christmas, she excelled herself. What cascades of mince pies, what mountains of sausage rolls, appeared under her hands ! And she would create, on those occasions, exquisite little marzipan titbits in beautiful colours. Young Hoggins, peering over the edge of the kitchen table, would be allowed to pinch a morsel now and then.
In short, my mother had a natural sensibility for British cooking. We were innocent of garlic, yet I never enjoyed food as much as in childhood, often washed down with lemonade made from real lemons, which slopped like skeletal millstones in our huge bone china jug. No doubt of it, all that generous eating stood the family in fine stead for the war years which were to follow. If you learn to eat well young enough, you will have a strong appetite for other pleasures.
While mother was upstairs doing the cooking, father was working downstairs in the section of the shop over which he held sway. I shall have to say something about the shop, it was so much a part of my childhood.
We lived over a part of H. H. Aldiss and Sons in what seemed to me an enormous flat. It had a long corridor, along which I could pedal like mad on my three-wheeler; once I broke the speed record (Sir Malcolm Campbell was my idol, even from an early age) and pedalled right down the stairs.
This was in East Dereham, a prosperous little market town in the rustic heart of Norfolk, where I was born in August, 1925. My mother loved giving my sister and me treats; one of the great yearly treats was when she took us to the fair. Fairs in Dereham in the thirties were really something. The whole of the market square was taken up with cake-walks, dodge'ems, coconut-shies, boxing booths, big and small roundabouts, and all the other stalls. The big roundabout had splendid cockerels with high heads, and dragons that seated three, as well as galloping horses. Mother was always lucky at the stalls; perhaps the gipsies recognized the psychic element in her. Most years, we would come away triumphantly with a goldfish or a Norwich canary, yellow and trim. One canary lived for many years, laying an egg every month. It was given brandy on a feather when it was ill.
What a pleasure it was to go to bed at night and listen to the high piping music of the organ on the big roundabout. We were near enough to the market square for it to be clearly audible. The other excitement was to go to school through the fair booths. At that time in the morning, men and women would be coming out of their travelling wagons and caravans which filled the back streets, washing themselves, singing, arguing, perhaps blowing their noses into the gutter with two fingers squeezing the bridge of the nose. How I admired that gesture – and could soon copy it accurately.
When the fair arrived in Dereham, trade in our shop was brisk. The fair people would all come to H. H. Aldiss for new clothes. So would the countryfolk, who arrived from out of town to enjoy the traditional jollifications of harvest-time, the season at which the fair was held.
When the excitement was over and the market square again deserted, my sister and I would be stood in the bath and our clothes removed, garment by garment, as mother searched us for fleas; they too rode into East Dereham with the fair.
Gypsies were never far from Dereham. On the outskirts of the town lies a tract of wild heathland called the Netherd, on which travelling wagons were often to be seen. The Netherd was ever-mysterious, ever attractive, and seemingly boundless. The gorse bushes were so tall to a boy that one could stalk and be stalked, like a small animal. In the winter, the Netherd pond used to freeze, whereupon we would skate and slide on it. Once I went to the pond with my cousins during a summer drought, to see the fish dying in the mud. We 'saved' one or two fish in jam jars full of water, but they died after we got them home.
The Netherd was a reminder that George Borrow was born in Dereham, at Dumpling Green. Borrow was a strange elusive man, and a tremendous writer. How little his works are read now! It is oddly appropriate that the only quote from him in familiar usage is 'There's a wind on the heath; life is very sweet, brother; who would wish to die?' Though the _Oxford Dictionary of Quotations_ also grants him 'every dog has his day': 'Youth will be served, every dog has his day, and mine has been a fine one'.
Borrow's _The Bible in Spain,_ despite its off-putting title, is one of the best and most living of travel books, to be set on the honoured shelf beside Kinglake's _Eothen._
The other writer associated with Dereham is the poet, William Cowper. Borrow was born there, Cowper died there. The Aldiss family, devout or pretence-devout Congregationalists – they defended their faith to the last child – went to worship every Sunday at the Cowper Memorial Church. And moreover were forbidden to take even a glance at the stills outside the Exchange Cinema on their way home to the Sunday joint. Time was when I used to read all of Cowper's poems (I actually liked 'The Sofa'), and his letters too; these days I haven't the patience. But the letters show a good English style. He went gently mad. His poem which begins 'I was a stricken deer that left the herd .. .' is moving in its eighteenth-century way.
My mother's relations lived in Peterborough, a city now ruined by industrialization. There, I had three splendid uncles, two of whom were architects, and a grandmother. Grandmother was the farmer's daughter. Her house was run on purely Victorian lines with, as I first remember it, two daily maids, a washerwoman who appeared on Mondays and wielded a ferocious dolly-tub, and a boot-boy. Every room in her house astonished me. The windows sported venetian blinds with wooden slats. The doorknobs and light-switches were enormous brass affairs. A stuffed fox snarled over the drawing-room door. All the furniture was huge and intricate, forever breaking out in animals, foliage, and faces. The breakfast-room had a vast grate which glowed angrily behind its bars and had to be black-leaded regularly for the sake of its complexion. The drawing-room held lace antimacassars and spindly glass cases containing collections of china; one was forbidden to play ball in there.
In grandmother's delicious dank cellar, which always comes to mind when I read Edgar Allan Poe, next year's Christmas puddings dangled from hooks overhead. They had to mature for eighteen months before being ripe for eating. In that house, huge hams were cured and huge pork-pies made. Huge trifles were consumed. For breakfast, one ate porridge or those gritty little Grape Nuts; the silver sugar sprinkler was magnificent; and one looked at the adventures of Pip, Squeak, and Wilfred in the _Daily Graphic,_ then a _respectable_ paper. Although the strip was aimed at children, it must have had some political significance, for one of the minor characters, I seem to recall, was a terrible old anarchist-bolshevist called Boskofski, who dashed about with a smouldering bomb in his pocket.
My grandmother was a tiny woman with white hair and ivory-coloured skin who outlived her husband by twenty years. She always wore black for mourning with a little white lace frill of collar. Although she died in 1945, when I was in Sumatra I recall her with absolute clarity. Yet how I _felt_ about her remains a mystery. She was never other than extremely kind to me, and I never remember when she was not smiling, even when a canine friend and I were busy wrecking all that scrupulously kept house; but some reserve in her kept me ever from loving her.
On the other hand, I unreservedly loved that monster, my grandfather. That was Granpa Aldiss, H. H. Aldiss of Dereham, J.P. My father went in awe of him until his last days, and always called him 'the Guv'ner'.
The Guv'ner not only governed, but could be seen to govern. He dominated what would now be called a department store, employing some fifty staff. There were three main departments, the ladies' drapery and millinery, ruled directly by the Guv'ner, the gents' outfitting, and the household goods and furnishing, ruled by the Guv'ner through my father and my uncle. Behind these departments were 'factories' and warehouses where carpets, beds, rolls of linoleum, and many other items were stored, and workshops where men sat cross-legged and made up suits or women chattered endlessly and created dresses or hats. H. H. Aldiss also did removals and funerals. I can just recollect the glass hearse pulled by two enormous black horses wearing black plumes, and driven by Nelson Monument in his black silk topper. Then a motor-hearse was bought and the old hearse stood in one of the back yards to rot. Soon the stables were empty, although kittens – and once a brood of startingly ugly barn-owl chicks – were raised by their feral parents in lofts above the stables.
The whole complex of buildings was a playground for me and my cousins. The various departments were connected by long, often dark, corridors; below them were 'stoke-holes', where the terrifying business of keeping the departments warm went on. Established in various nooks throughout this loose assemblage of buildings were little knots of people going about their different trades. A small boy's relationship with adults is always precarious. In some nooks, one would be accepted as a welcome diversion, in some rejected as a tiresome nuisance; one might be made much of or teased; and the reception could change as the predominant character of the workroom changed.
The millinery department was established upstairs, in what had once been a dwelling house. Behind the scenes, old bedrooms with striped bedroomy wallpaper did duty as boxrooms, scruffy corridors retained the dignity of ancient gas-brackets. To this department I would advance with a fluttering heart, provoked by hope and trepidation. Yes, and love! For one of the young girls in the millinery was very beautiful. She had dark hair, blue Norfolk eyes, a pale complexion, some freckles, and a lovely red mouth. She was known to get me in one of the old bedrooms, to take me on her knee and cuddle me, to kiss me. But she also teased me. That teasing I could not bear; nevertheless, I did bear it for the sake of those wonderful kisses.
Long after we had left Dereham, I would dream of that face, and those lips, and scheme to go back and give a more grown-up account of myself.
Among the rambling buildings were nooks numberless in which to hide and ambush assistants – who sometimes retaliated by hauntings and ghostly noises from places of concealment. There were many opportunities to annoy old Monument, who would come rushing out of his harness-room swearing as his chimney was blocked from outside by a sack, and smoke poured into his little reeking room. There were endless chances to go ratting with the terriers, or to climb over and along walls, endless chances for disappearing.
Occasionally, the Guv'ner would appear in his wrath and give us the yard-stick about the legs or bottom. Or he would send father to do the same. Father would do as he was told, but he had his own fun, even in shop hours. The outfitting department was frequently convulsed with laughter at some prank of which my father was prime mover. He sometimes eluded reps ('commercial travellers' they were in those days) by dressing up in _one_ of his own fitting-rooms and sweeping past the man with a polite lifting of his top hat, sporting a false moustache the while.
In his youth, father took part in amateur theatricals. We had a photo of him in pierrot costume, looking droll. East Dereham inflicted respectability on him, but he still enjoyed an absurd prank; he would enforce silence on us at mealtimes, then sit eating with his napkin-ring screwed into his eye like a monocle. Wars of thrown dusters were covertly conducted among the serious business of his shop. He could do conjuring tricks, tell jokes, and sketch beautifully. We regularly went to the swimming pool to see him win tortoiseshell clocks and other prizes in local events.
Everyone worked tremendously hard in father's shop, enduring long hours and low pay. On late nights, a tray of tea would be brought down to father in his office. Sometimes it would stand untouched for an hour, until the tea stewed and I snaffled the buns.
Over all, the Guv'ner presided. He was short and stocky. He had small neat jowls, and a mysterious little lump on one cheek. The Guv'ner was always well-dressed. He had a sharp tongue. Everyone respected him. He took his annual holidays in Torquay or Gibraltar, turn and turn about. Once, he brought me back a wood-and-metal object which imitated bird-calls when twisted. His leisure pursuits were few. He gardened and went to chapel twice on Sundays. He was teetotal. He read his Bible but few other books; I know he enjoyed anything about Scott of the Antarctic, whom he admired.
His wife, my grandmother, spent many years confined to bed with one of those mysterious illnesses which have gone out along with gutta percha, Birkett Foster, and chemist's sealing wax. I used to climb upstairs to see her in her cool room, where long windows looked out on garden and meadow. What sort of woman was she, I wonder? I rather enjoyed being in that room, among the bunches of grapes, flowers, and ornaments; but it was a relief to slip downstairs again and be free.
The Guv'ner's habitat hardly seemed to be in that house. We saw him there only at Christmas. The shop was his proper environment. In the middle of it all he had his office – so much in the middle of it all that the office had no windows. One might look instead into a large safe with a green door. The Guv'ner sat in a swivel chair, with ladies about him on high stools, industriously working at ledgers as if dear Disraeli had never died. But he always had time for me; I knew he loved me in his gruff and inarticulate way, and I would always go to kiss him respectfully good-night on Saturday evenings, when he would press a sixpence into my crafty little palm.
My grandmother died at last. Late in life, the Guv'ner married again. He married one of his cashiers, a nice young lady who looked after the ledgers. It caused no end of a stink in the family. Nobody could talk about it, even decades later, without getting heated. But the old boy did right. His new wife was a marvellous woman in every way, and always very kind to my sister and me into the bargain. She made the Guv'ner happy for the last few years of his life.
All this must sound worlds away to my American readers. It is worlds away to me. In all but date, our little corner of Norfolk in the nineteen-thirties was Victorian in its modes and its thought. Nor had the sun dreamed of setting on the Empire.
When I was thirteen, we went to live at the seaside, at Gorleston-on-Sea. There, father would go down and fish by the old Dutch pier, which was swept away some years later by a heavy sea. He would bring back a bucket full of fish for us and our neighbours. Sometimes we would have crabs. You could buy large crabs, dressed, from the fishmonger in the High Street, for ninepence; they cost almost two pounds now, undressed. And we could buy boxes of soft and hard herring roe – a teatime treat. Mother saw to it that we still ate well. I must owe my health and physique, not to mention my appetite for any exotic dish, to those years of good eating which, compared with the miserable diet of boarding school, were like Heaven to Hell.
My father was a keen fisherman. He liked nothing better than to spend all day in some quiet backwater, preferably on one of the Norfolk broads, watching his line. He took me to Hickling Broad one day. We arrived very early. Nobody was about. Birds were busy over the vast silences of the water. I would have loved it if I had not had to fish.
Only after he died in the mid-fifties did I realize that my father thought of himself as a failure. A failure ! That handsome, witty man who worked so hard for us! Impossible! But time brings deeper insights. I can see now how painful it was for him when he sold up his share of the business after the Guv'ner's death and we left Norfolk for good. Nothing was ever the same for him again. He was a humorous man, but his jokes grew more bitter and turned against his family, who must have seemed increasingly a burden to him. He died when my first son, Clive, was less than a year old. Now I have three more children who know nothing of him, Stanley Aldiss.
When I told my father that I was thinking of throwing up my job and becoming a writer, he was horrified. His life experience had taught him to cling to what was. Whereas my life experience had taught me that better things were round the corner. Already, before we left Dereham, I could feel the prison shades closing in on me; well do I know how H. G. Wells felt in his draper's shop, for I was destined for the same fate. I escaped. My father's supreme misfortune was my good luck.
There I pause, looking back at that lad of thirteen. Some things he had already decided. He had decided that he was no more interested in his father's business than was Kafka in similar circumstances. He had decided that he was not interested in the religious rituals droning on round him. But, more deeply, he had decided that he had a contempt for the safe bourgeois way of life in which he found himself.
I was an omnivorous reader even then, although it was mainly trash I read; but I had got hold of a book that firmed up my ideas on the subject. A cell of French culture had penetrated to East Dereham (perfect Bovary country, come to think of it), and I lit on a translation of what could have been Murger's _Vie de Boheme_ but was more probably Zola, the mere sight of whose four-letter name always had an hypnotic effect on me. I don't recall what it was, although I can hear a clock ticking as I read it. The novel gave me a longing for bohemian life which, because I have never much gratified it, I have never lost, and a feeling for being socialist as part of life, although I have never been a Socialist with a capital S. Those feelings seemed to be lacking in the circles in which I was brought up.
Along with that went an interest in science. I have mentioned my hero, Sir Malcolm Campbell, the death-defying man who frequently broke the world's speed record. He set up a new world record when I was ten, driving on Daytona Beach, a name I used to recite to myself under my breath. Later, he broke the water speed record. Both his cars and his boats were called 'Bluebell'. I had a little flicker-book, advertising Castrol or another oil, which, when thumbed rapidly, showed the mighty 'Bluebell' surging across Coniston water !
The boys' magazine I took every week was _Modern Boy,_ published by Amalgamated Press. I loved _Modern Boy_ dearly. Sir Malcolm Campbell used to contribute to it – and Flying Officer W. E. Johns, for his Biggles stories were being published even then. What I was absolutely addicted to was Captain Justice. Justice was an elegant adventurer, much given to wearing white ducks and a naval cap and smoking a cigar. He had various bases round the world, chief of which was Titanic Tower, significantly in mid-Atlantic.
From Titanic Tower, Justice sorted out the troubles of the Anglo-American world in story after story. The stories were written by 'Murray Roberts', the pen name of Robert Murray Graydon. Justice and Co ventured into Africa to find an empire of slaves ruled by strange forces, confronted giant insects, battled with enormous robots, overcame alarming flying machines, survived a world plunged into darkness (the most enthralling of all his adventures !) and also paid regular visits to any runaway planets which happened to be passing through the solar system at the time.
So science fiction entered into and began warping my life from an early age. My hero-worship changed from one Campbell to another, from Sir Malcolm to John W.
Science also took my fancy. Looking back, I can recall no interest in art or science among the members of our family. But my parents gave me a microscope at the right age; peering down its barrel became one of my favourite occupations for a while. Anything I could get hold of went on those flimsy slides, and what I saw I would draw and colour with watercolour in a special notebook. There was a compelling aesthetic attraction in that microscopic world.
Other landmarks of my journey towards realizing that sf was my main dish I have already catalogued in _Shape Of Further Things_ or _Billion Year Spree._ The discovery of _Marvel, Amazing,_ and _Astounding_ on Woolworth's counter. The purchase for a shilling of Alun Llewellyn's remarkable novel _The Strange Invaders._ The continuous reading of H. G. Wells's novels, not only the sf stories but _Tono-Bungay_ and _The New Machiavelli,_ and so on, which I enjoyed for their socialism. But it was _The Strange Invaders_ which persuaded me that I had discovered something worthwhile. Although I enjoyed the magazines – and was particularly mad about Kuttner's 'Time Warp' in _Marvel,_ because of its erotic element – I thought they were appallingly written and was a little ashamed of reading them.
Where I got that critical idea from, I cannot tell. By the age of eight, I had begun to appreciate style, the vehicle of fiction, as well as content.
What happened to me at eight was a terrible thing. I have so far painted a cheerful picture. At the age of eight, I was sent away to boarding school, and at boarding school and public school I remained until I was seventeen and old enough to go into the army – whereupon I was promptly whisked abroad to the Far East for four years. So my severance from home and parents began early in life, far too early.
I was sent, like Pip from Joe Gargery's forge, to be educated and to become a little gentleman. The treatment, as with Pip – Dickens was always the perceptive reporter– created a chilled and conventional creature, cut off from his roots. Thousands of English boys endured and still endure the public school system; most of them survive in some way or other. But it seems to me a pernicious system, deadening to a wide sector of life, and it perhaps accounts for much of the legendary coldness attributed by foreigners to the English. Many years of adult life passed before I shook off that cold shadow of exile.
Which is not to say that there is necessarily a great deal wrong with the schools per se. My last school, West Buckland in Devon, was a fine one and I was happy there. Nevertheless, the possibilities for torment in an authoritarian community, within which one is confined for twenty-four hours a day for many weeks at a stretch, are many.
At least school provided some chance for writing and reading. When did I make my first book? I cannot remember. Certainly I was forever making books at prep school, my microscope book among them. By then, I was an authority on prehistoric life and used to give lessons on dinosaurs to other lads at a penny a time (with a cunning mixture of scholarship and commerce which I have never entirely abandoned). Later came my Victorian melodrama period, my epic drama period, my ghost story period, my space story period, my pornography period, my space-pornography period (girls raped by huge vegetables on Jupiter and quite enjoying it), my horror-and-blood period, and so on, until, in the army, I wrote 'Her Dear Dead Body', a sort of erotic detective story. I also kept a voluminous diary.
The diary went on for years and years, growing annually larger and more pretentious. It evolved into a gigantic freestyle journal containing many millions of words. I still have it, in something like two dozen volumes. It contains the ramblings of a stranger and is without interest; only a mixture of shame and pride makes me preserve it; after all, getting rid of all those countless days and sentences must have been of some value !
All these sorts of writing, slowly becoming more ambitious, stood me in good stead when it came to attempting anything professionally. Writing by then was an integral part of life.
When I left the army, I had no ambition except to write. To be propelled back into civilian life in 1948 was extremely disorienting; I had no knowledge of society, except the out-of-date rules I had hazily gathered as a child – though I was fortunate in finding myself in Oxford, where mores were a little fluid. Even while I was still on repatriation leave, I began writing a novel about a soldier's experiences in India, Burma, Sumatra, Singapore, and Hong Kong; it was to be entitled _Hunter Leaves the Herd,_ and five chapters were written before I bogged down. The impulse behind it was as much nostalgic as creative. In truth, I longed for the sunlight and the whole ambience of the East. That region, that experience, has remained with me continually. People have less in the East; but they do not seem to suffer from envy as much as we do.
I wanted to be a poet, although I was not familiar with the work of any poet later than Thomas Hardy. I got a job in a bookshop in Oxford, where I came into contact with the physical material of our culture. Bound sets of Macaulay, Gibbon in calf, Richard Burton in folio, Hogarth in elephant folio, were things of romance to me. And of course we were always throwing out books – a useful reminder that, however successful one is, every dog has his day, in Borrow's phrase.
In those dim cloisters, I encountered such splendid eighteenth-century heroes as Candide, Tom Jones, Rasselas the Prince of Abyssinia, Vathek, and the time-haunted Tristram Shandy, while meeting for the first time in complete versions with those redoubtable voyagers Robinson Crusoe and Lemuel Gulliver. Formidable creations, all of them, and all created by authors who project strong _personnae_ into the works concerned. Each work, in its way, is a grand entertainment which also constitutes an enquiry into the world.
Fielding, Johnson, and Sterne, in particular, aroused the sedulous ape in me. For some months, my desk was cluttered with Sterne-like entanglements and Fielding-esque disquisitions, written in Johnsonese. Then I moved digs and burnt them all. I began to read that neglected contemporary writer, Eric Linklater, and imitated him instead.
After a while, I took to contributing to the trade paper. The editor commissioned a series called 'The Brightfount Diaries', about life in a fictitious and pleasant bookshop. The rest, as they say, is history, although obscure enough history to bear repeating here. One day, I received a letter from Faber & Faber, from Charles Monteith, saying that I had my fans at Faber's and would I ever think of turning my series into book-form.
I did know the series was successful and, as it happened, I had been thinking of nothing else. So I made the book, my first publishable one; Pearl Falconer – then a fashionable illustrator – drew some delightful pictures for it; and Faber published it, with a pleasing modest success, in November 1955. The edition stayed in print for years and Faber, bless them, never remaindered. They sold me the last thirty copies cheap in the late sixties.
For me, there were few rejected manuscripts, few rejection slips, no starving in garrets. I should feel guilty but don't. There were plenty of years of hardship, when I was too skint to buy myself a morning paper for fear of wrecking the budget, beers were far apart, and I ate at places called British Restaurants.
My regular association with Faber continued through twenty books and seventeen years; a very good partnership. Happily, Charles Monteith was also an sf reader – in the fifties, they were few and far between in England, particularly in any position of power. Charles knew Kingsley Amis and Robert. Conquest and Bruce Montgomery (Edmund Crispin), all of whom had been up at Oxford and read _Astounding._ It was a fortifying experience to meet them.
There was no difficulty in following up _Brightfount's_ with a collection of sf short stories, _Space Time and Nathaniel._ By this time, I was relinquishing the idea of being a poet – persuaded by reading Eliot, Auden, MacNiece, and John Donne, who showed me what poetry really was. My memorial to that ambition lies in the contents page of _STAN_ (we always knew the book by its acronym), where the titles of the three types of story are laid out in octet and sextet form, as in a sonnet. There are fourteen titles.
For Faber to publish this volume in 1957 was an act of faith on their part. I had had only thirteen stories published; a fourteenth had to be hurriedly written to make up the number. Paperback rights never sold until 1966. With my next book, _Nonstop_ (retitled _Starship_ in the States), I had the same paperback problem. Eventually, rights were sold to Digit for £75 and I was glad to get the money. _Nonstop_ has been translated into thirteen foreign tongues so far.
By 1957, I was earning as much money from writing in my spare time as from working from 9 till 5.30 in a bookshop. So I left the bookshop. From that day to this, I have never done an honest day's work, and have lived happily ever after !
Not that the financial position improved greatly at once. I have just come across a note in an old diary for 5 April 1958 which says, 'This is the end of the financial year. I've got £110 less in the bank than I had a year ago. And £20 less in the PO Bank. In fact, we've got £60 left, sum total of our wealth. Ye gods, outlook's black ! ' Outlook improved during that year, and has never been too perilous since.
It was early in 1957 that the national Sunday paper, the _Observer,_ announced the results of its competition for a short story set in the year 2500 AD. My story, 'Not For An Age' tied first and was published in the paper, complete with an illustration by Leonard Rosoman. Since all the British science fiction writers had entered the competition, the ensuing kudos for an unknown was great. The late Arthur Sellings had a story in the top twenty prize-winners. Then it was I learnt that such a thing as fandom existed; I received a letter from Helen Winnick asking me up to the Globe, the pub where London fans met regularly. There I met two young writers, John Brunner and Sam Youd (John Christopher). Sam's famous _Death of Grass_ must already have been out by then.
By this time, I had been appointed literary editor on the _Oxford Mail_ under W. Harford Thomas. I had been reviewing science fiction and ordinary novels, as well as non-fiction of all kinds, for the paper. My sf column ran from 1954 until 1967, making it the longest regular sf review column ever to appear in a newspaper. It must have been read by thousands of undergraduates (who are called students now). Later, I reviewed sf for the _TLS,_ but abandoned it when the standard of work fell so low. I hate to think how many hundreds of books I devoured for the _Mail._ Fortunately, I always had a good appetite. Now I'm doing a stint for Ian Hamilton's _New Review._
Criticism and creation always went hand-in-hand as far as I was concerned.
My intention was partly to write social novels. John Osborne's play, _Look Back_ in _Anger,_ and Kingsley Amis's novel, _Lucky Jim,_ embodied for me then much of the experience of my generation – after all, neither author is much more than half-a-dozen years older than I. _Lucky Jim,_ over which I howled with laughter, altered something in my approach to life. Laughter is very persuasive.
But there was a bigger persuader: the atomic bomb. When the Bomb was dropped, my division was in India, resting after our bout of involuntary heroism in Burma – and training to be launched against the Japanese in a seaborne assault on Singapore. So I had good reason to rejoice in the flattening of Hiroshima and Nagasaki. My bacon was saved. My admiration for that fine president, Harry Truman, dates from that time.
The new-born nuclear power was something greater than social life, greater than almost all the people (not greater than all people : not greater than Truman : for it was he who decided that the bomb remained in the hands of politicians instead of being passed to the generals; that decision forms one of the nodal points of modern history). The Bomb dramatized starkly the overwhelming workings of science and technology, applied science, in our lives.
So I perceived, and have been trying to perceive more fully ever since, that my fiction should be social, should have all the laughter and other elements we associate with prosaic life, yet at the same time be shot through with a sense that our existences have been overpowered (not always for the worse) by certain gigantic forces born of the Renaissance and achieving ferocious adolescence with the Industrial Revolution.
But such matters are best reserved for discussion in fiction, rather than in a chatty memoir.
Since this piece is addressed in part to an American audience, I will seize on the chance to examine my dealings with the States. It is a curious thing for a writer to think of his natural audience as being in another country, although not too strange for me.
One reason was what I have called the shadow of exile which loomed over me. Another was the whole American dream under which I was brought up – or under which I brought myself up, to be more accurate. I was a movie fan from an early age, encouraged by my cousin John. All that I admired in the cinema was American. Then again, the sort of music that moved me most deeply was American, the blues and jazz. The same with all the music we called 'hot' in my adolescence.
The best ideas, too, seemed to be American. The States stood then as a bastion of freedom and continued revolution. It was a sad thing to see that image crumble after the war; at the same time, the American view of Britain as a gallant little fighting island was collapsing. Particularly before the second world war, when more and more of Europe seemed to turn into an armed enemy camp, the lights of America burned bright, viewed from English shores.
Then of course there was sf. Not that it ever could be said to paint a cheerful picture of the States; very far from it; but it radiated an image of a land of dynamism and change. This I apprehended as very different from Britain, with its oppressions, unemployment, class structures, and trading difficulties (all of which existed in the States also, as I failed to realize when a lad). I did get the message that the most exciting sf came from New York.
The war. France fell. The boy of fourteen watched with mingled fear and excitement as his father brought his two guns out of the cupboard, handing the lighter one over and saying, 'You'll have to use this when the invasion comes, so we'd better get some practise in.' The invasion never came. Britain miraculously survived, right on the edge of black Europe. Our great ally was the USA, across the Atlantic. Americans were very much treated as heroes. Liberators of Europe and all that. Times change; it ill becomes an sf writer to complain of the fact. Change is the material from which his cloak is cut.
All of which made it natural that I should write my science fiction with America in mind. That first novel Non-stop, was part of American fiction. I had read and been fascinated by Robert Heinlein's _Common Sense,_ about the interstellar ship on which a catastrophe has occurred, although I was struck at the same time with the poverty of characterization and feeling in it. My novel was designed at least in part as a response, an antidote.
Another thing. At the time I began to write for publication, there were really only two going sf magazines over here, _Authentic_ and _Nebula._ I disliked the _Authentic_ format, which had long boring stories in it bearing titles such as Immortal's Playthings (it improved later under E. C. Tubb's editorship). _Nebula,_ edited by Peter Hamilton in Glasgow, was more exciting. I found an issue (No. 3) in Freshwater, Isle of Wight, read it, and decided that I could do almost as well as the authors performing there. Hamilton took a lot of trouble trying to make the stories I submitted publishable, but without much success – for one thing, I've always disliked rewriting at editors' behest.
Then Ted Carnell's magazines got going on a regular basis. Ted can never be enough commended for exercising that principle of punctuality to which he adhered, for his sane good humour, or for his scrupulous honesty in professional and financial matters. But it was always apparent that most of his authors were little better than illiterate, and were short on imagination too. Jimmy Ballard has said of them that they were 'the most dingy and pathetic bunch of third-rate ex-journalists' he had ever met; so they were. In fact, they were feebly copying an ageing American tradition. Many of their stories were rejects from American magazines.
It was a poor thing to talk with an assumed American accent in a British magazine. I have no doubt that it was because of this kind of down-at-heel tradition that the axe fell and revolution, signalled by Moorcock and his feathered hordes, had to come.
Even before Moorcock's day, there were only two authors I could read with interest in Ted's magazines. One was me. An author should always read with care his writings when they appear in print, when sufficient distancing has set in for him to be able to perceive his errors and victories. The other was J. G. Ballard. His record for not writing American retreads was always unblemished.
Before us, Arthur C. Clarke had always used an English idiom instead of the bogus Yankee employed by some of our chums.
At first, events moved more happily for me in the States than in my own country. I was lucky to have Truman 'Mac' Talley as editor at Signet; he really seemed to care, and I appreciated his advice. When he received the manuscript of _Hothouse_ (which he insisted on retitling _The Long Afternoon of Earth_ so that it did not fall among the horticultural section), he altered it throughout very extensively in pencil. Every page had something changed. He sent it to me to look at. I was horrified. I wrote back and said I would prefer that the novel should not be published rather than that it should be published in such mutilated form. He wrote back, 'Okay', and printed my text as written.
Under Mac, the best of my early titles appeared in the States, neatly packaged with Powers covers, and I was proud of them. I was also getting a response from Anthony Boucher and Fred Pohl, men whose work I respected. The atmosphere was sharper and more intellectual in New York. And I was grateful to Don Wollheim at Ace for publishing my more shaky early attempts.
It was a stimulating time in which to enter the lists. Not a great deal seemed to be happening in England then, apart from John Wyndham and John Christopher. I admired and still do admire the latter's _Death of Grass_ (called _No Blade of Grass_ in the States), and his very funny and clever novel, _In the Year of the Comet._ There was also Arthur, whose writings I greatly enjoyed; but his distinctively English style was generally exported to New York.
In the States, much more was going on; all the other contributors to this volume were already making a considerable mark on the world. Pohl's _The Space Merchants,_ in collaboration with Cyril Kornbluth, was already on the way to being accepted as a classic. Bester had exploded his two great fireworks, _The Demolished Man_ and _Tiger! Tiger! (The Stars My Destination),_ leaving considerable retinal damage in my case. Silverberg was already filling all the available magazines; I had the pleasure of meeting him in 1957. Knight was still writing fiction, often with a humorous emphasis much to my taste, whilst his book of reviews, _In Search Of Wonder,_ had already appeared. (I must have read that volume more often than any sf novel, with the possible exception of _Earth Abides_ and _Out of the Silent Planet.)_ Harrison's _Stainless Steel Rat_ and _Deathworld_ had already appeared, paying enough for him to bring his family to Europe at the start of their long peregrinations.
Kingsley Amis's _New Maps of Hell_ appeared in 1960. In many ways, it must be regarded as a special event. It did put sf on many maps. Amis was the first man to lecture on sf in a university, taking it as a serious subject for discussion, and his book remains exemplary in its wit and literacy, even if its scholarship is a little hasty. _Hell's Cartographers_ is a title that reflects back some of Amis's heat and light.
It was as part of the hospitality I enjoyed from US sf readers that I received a shield from the 17th World SF Convention nominating me 'Most Promising New Author'; there were a Hugo and a Nebula to come later.
I still gain encouragement from that 'Most Promising New Author' plaque. It was a sign that someone cared, and I took it as such. I was incensed when a Mr Devore of Chicago later suggested the plaque be withdrawn.
(It's true that, later still, the Australians voted me 'World's Best Contemporary SF Writer', but I suspect that was a friendly conspiracy by Lee Harding, John Bangsund, and Bruce Gillespie. Still, the title looks good on dust jackets.)
An author's career is an haphazard matter, subject to chance like most things. I suffered some misfortune in the States, changes at Signet and trouble with agents; meanwhile, success started coming my way in my own country.
Writing from England in 1974, I could list many troubles under which the country labours, and many vexations from which the individual suffers. But the sixties were in some ways a halcyon period. It is difficult to encapsulate what happened but, in my view, the Romans were busy becoming the Italians. The British had lost or relinquished most of their empire and, what's more, the Goth was within the gates of the capital. The Goths were civilized people, as were the waves of Italians, Indians, Pakistanis, Chinese, and Americans who, for their different reasons, began to invade England. They still help to make London one of the pleasantest and most cosmopolitan capitals in the world.
The sixties was the time of 'Swinging England', the Beatles, Mary Quant, Flower Power, permissiveness, and all that. More deeply, personal relationships opened out, people became much more frank, would make friends more easily, and would more readily tell those they disliked to go and find another pad. This was a general phenomenon, by no means confined to the very young – although that particular revolution had its storm-warning in 1961, when a schoolgirl called Helen Shapiro bounded into the national consciousness singing 'Don't Treat Me Like a Child' and 'Walking Back to Happiness'. It was a great time. I don't recall the country feeling so pleasant.
For me personally the weather also set fair. I too had my freedoms. In 1965, I remarried. The adorable Margaret Manson became my wife. That long shadow of exile of which I have spoken had at last dissolved. It proved to have been an exile from my true self.
The effect of this beneficent change upon my writing was slow but undeniable. One can see it by comparing the stories in one of my early volumes, such as _Canopy of Time_ (which appeared in the States in different format as _Galaxies Like Grains of Sand),_ with those in my latest collection, _The Moment of Eclipse._ A Cambridge reviewer said of the earlier volume that the stories were perfect, classically perfect, almost too perfect; whereas the later stories are never content with a static form, and shape and content form a living whole, varying as needed. The dynamism is redirected.
It was in the mid-sixties that Mike Moorcock took over _New Worlds._ There one was allowed to cut one's capers – to the advantage of all, and especially to the advantage of oneself. I must say something about _New Worlds,_ although that subject is entered into more fully in _Billion Year Spree._
We can identify two streams in science fiction, two streams which have now (though uneasily) become one : what I will call the 'once-for-once-only' stream and the pulp stream.
The 'once-for-once-only' stream is a direct literary response to a new factor or a change in society, generally brought about by a technological development, as are most societal changes. The first good example is Mary Shelley's _Frankenstein_ (so much better a book than all the horror movies allow), provoked by new sciences, new discoveries (the opening up of the South Seas for instance), and new philosophies, such as the evolutionary speculations of Erasmus Darwin. The line of inheritance then goes on through – shall we say – Verne, de l'Isle Adam, Butler, Bellamy, Wells, Kafka, Huxley, Skinner, Orwell, _et al._ The books of these men are once-for-once-only responses to or examinations of some new aspect of the world.
The other stream is the pulp stream. When Gernsback started his magazines in the late twenties, he invented sf as a genre, a category. No chance of the 'once-for-once-only' approach here! By their very nature, the pulps appeared regularly, with a regular amount of space to fill. So modern sf was established, wherein a lot of underpaid authors rather frantically cribbed or sparked ideas off each other – at its rare best, the process did become some sort of a meaningful dialogue (for more on that subject, I must ask you to consult the intros and commentaries in Harry's and my _Astounding/Analog Reader,_ 2 vols.).
The pulp stream also witnessed the triumph of the editor over the writer. It was something we missed in England, whether because writers are more individualistic or editors are poorer, I don't know.
But the great distinction to be made between the two streams in their heyday is that whereas the 'once-for-once-only' stream was, by its nature, a critical literature, even in the case of its most noted practitioner, H. G. Wells, the pulp stream quickly turned to power-fantasy and escapism. Here, the great practitioner is Edgar Rice Burroughs, one of the best-selling, if not the most bestest-sellingest, sf writers of all time. Burroughs' influence has been pervasive and often detrimental; among Tarzan's descendants is certainly Heinlein's Michael Valentine Smith of _Stranger in a Strange Land._
It is up to everyone to pick the bones out of their own reading preferences; I've already declared my preferences in my history. Sf is the best-equipped of all literatures to indulge in power-fantasy; it encompasses the universe, and it can give us the infinite policing powers of the human mind, as well as the gross material conquests that lie beyond the atom.
But for a writer to indulge himself and his readers in these ways is ultimately to ruin his credibility as a writer. As a real writer. The disgrace that the sf community still (perpetually?) thinks it is in is precisely the shame it shares with pornography, of transforming a man into an organ of conquest in a knocking-shop of wish-fulfillment.
Mike Moorcock, in kicking the old gang out, allowed in the more traditional kind of sf, responsive to the current situation. There were excesses, but excesses are part of every revolution. The four-letter words, the sex, the indulgence in style for its own sake – all these were in defiance of the old pulp tradition; so was the attention paid to the other arts. This is what opponents of the new wave, like Sam Moskowitz, Isaac Asimov, or, more literately, Robert Conquest, never seem to have grasped.
It is disappointing that someone of Conquest's critical standing refuses to comprehend that this new wave had much in common with many previous revolutions, major and minor; it was reacting against the decadence it superseded. Kingsley Amis shares Conquest's general view, of literature as of politics, but as a popular novelist Amis is more aware of the writerly necessity for breaking new ground, as his continuing exploration of genres shows.
Moorcock's revolution was always part-powered by American writers like Thomas Disch, John Sladek, Norman Spinrad, Samuel Delany, Pamela Zoline, Kit Reed (all of them, incidentally, people of great charm), and other exiles who visited London from time to time, as well as English writers like Mike, Ballard, David Masson, Charles Platt, Lang Jones, and me. It found sympathy with a whole lot of people who were out there reading sf occasionally but unhappy about its limitations. And when I sought a grant for _New Worlds_ from the Arts Council, we could rely on immediate support from a wide range of people in the arts and journalism who enjoyed sf, Angus Wilson among them. From then on, sf has prospered widely in England, and not just in a narrow commercial sense, cut off from other arts. To the arts sf can contribute whenever it, or any author who wishes to write it, ceases indulging in arabesques of power-fantasy. Of course, power-fantasy always sells. That's not what I'm arguing.
It may be that early promises were not entirely fulfilled. Such is the way of early promises. Good writers are always too few. But Spinrad remains a considerable talent who will forge further yet. Disch will recover from his present doldrums. While Kit Reed, in such books as _The Better Part_ and _Tiger Rag,_ has proved herself a considerable psychological novelist. That elusive man, Samuel Delany, already has a large fan following.
While this excitement was going on, I was involved with other matters. Margaret and I bought an old Land-Rover in 1964 and headed for Jugoslavia, where we travelled all of the six republics that make up the state. A wonderful country, a great experience, tremendous to be able to snatch six months out of life. We've never managed to fit in a similar expedition since. Out of the Jugoslav trip came my one travel book, _Cities and Stones._
Margaret and I _recognized_ each other from our first meeting, which meant that we could stand a lot of mutual nonsense, and do so even with an amount of pleasure.
It was important that she recognized me clearly as I was. From then on, I became able to realize myself; my writing changed in accord with the mysterious gear-changes which carry individual evolutions forward. With my first wife, no such recognition took place. Indeed, as that marriage was approaching the last stages of destruction, I said to a certain person whose business it was to know us both that one of my anguishes was that, while I felt I had a clear picture of Her, she had no picture at all of Me. This was confirmed. It is impossible to live day after day with a shattered mirror. Rapport is the sun of existence.
The early sense of formlessness with which I had been afflicted was a search for identity. When I found it, it was startling and protean. The years of exile brought their recompense. Through a new understanding of myself, I was better able to understand others. I have a quick empathic sense; now I try to use its findings to warm the essentially cold medium of modern sf.
One amazing incident on the Jugoslav trip. We had a letter from Harry Harrison in May, c/o the British Embassy in Belgrade, saying that he and Joan were going to drive from Denmark, where they were living, to Hungary in July – a distance of about a thousand miles; they would look us up in Jugoslavia if we named a time and place. It sounded a bit remote, but we wrote back saying we'd be in Makarska on the afternoon of 24 July, in the local camp (we saw from the map that there was a camp, although we had never been there). The mail went astray then. We got no answer. But we rolled into Makarska camp a couple of months later – and the Harrisons turned up ten minutes after we did.
Harry and I had already begun collaborating. With the aid of Tom Boardman, we put out _SF Horizons,_ a little review of sf. We issued only two numbers, but it seems to have been influential, not least on us. We went ahead with other collaborations. The most successful collaboration has been Harry's idea for an annual _Best SF,_ although my role there is merely as a talent scout. Our recent _Astounding/Analog Reader_ in two volumes is doing well. But our partnership has been most fruitful for the insights into writing we have been able to give each other. The fact that we never cheat or let each other down has helped a good deal, too.
It is generally assumed that my main contribution to the Moorcock era lies in _Report on Probability A_ and _Barefoot in the Head._ Both certainly appeared in _New Worlds (Barefoot_ in chunks as 'The Acid Head War' stories). In fact _Report_ was written some years earlier, in 1962. After a while, a writer grows too firm in his own ways to be actively influenced by anything new but, in 1960, I was much persuaded by the French _nouveau roman,_ the anti-novel, as practised by Michel Butor and Alain Robbe-Grillet (Marguerite Duras was less to my taste).
I admired their scrapping of many literary clichés. I was attracted by the way that Robbe-Grillet and Duras translated readily into cinematic terms. In particular, I was stunned by the Robbe-Grillet-Resnais film, 'L'Année Dernière à Marienbad'. with its temporal confusions, mysterious agonies, and alien perspectives. It still embodies for me many of the things I set most store by in sf (while many other valuables are to be found in Luis Bunuel's recent film, 'Le Charme Discret de la Bourgeoisie').
Robbe-Grillet's novels are not exactly works one wants to read through many times. But Michel Butor's _Passing Time_ is less austere, and to be recommended as a permanent book.
From these exemplars I took courage. I would cleanse my prose of its antiquities. So I developed the central situation in _Report,_ a situation charged with a drama which is never resolved. Moreover, I withhold the emotion involved, so that a reader must put in emotion for himself.
Later, I met Anna Kavan, author of _Ice,_ and found that she too had been haunted by 'L'Année Dernière'. That brave woman took her life before she could learn of her new reputation, to my lasting regret. She was marvellous, and it is entirely fitting that a cult is now growing about her name.
After Moorcock had published _Report,_ I was able to sell it as a book; Faber published it in hardcover and Sphere in paperback, where it still reprints merrily. Unfortunately, I have not yet been able to secure a French translation. It is a novel that owes much to the cool Paris of the intellect.
In the States, _Report_ seems to have secured none of the supporters which a book of its nature requires in order to gain ground. For it does need close attention, particularly if one has been nourished on a diet of pulp. My staunch friend Larry Ashmead published it in hardcover from Doubleday. None of my regular paperback publishers such as Signet would support it. Eventually it appeared from Lancer, but clearly it has not met with the same response as in England. A pity. It is the novel in which I came nearest to fulfilling my intentions, and its essence remains with me in a way I could claim for few of my other novels.
_Barefoot_ is more ambitious. I believe it has its triumphs. There again, it has met with a mixed reception in the States. Larry again backed it. The paperback publishers again shied like startled virgins. Ace eventually produced rather a pleasing edition. From the start, the novel had its supporters. My particular gratitude went to Jannick Storm, who translated it into Danish – a great accomplishment.
Doubleday finally pulped their hardcover edition. It sold fewer copies than had the Danish translation (in a country of only four million people!). But eccentrics like Harlan Ellison, Norman Spinrad, Theodore Sturgeon, and James Blish have aired some of the novel's virtues to a reluctant public, and there is a chance that it will survive and flourish in the States as it does in England.
The growing band of academics should espouse the cause of novels which venture to be unpopular by breaking new ground. As far as I know, only Charles Platt has used _Barefoot_ in an sf course so far.
The form that _Barefoot_ takes is entirely dictated by its content. If you write about a Europe overtaken by drugs, how do you best convey the experience to the reader? I had no doubt about the answer : you plunge him in to that world as deep as you can. And you do that by the chief means at your disposal – the use of language. By the deployment of suitable phraseology, you make him feel what it is like to belong in an entire culture gone hippie and yippie. The logic cannot be faulted, however you judge the end result.
The book's not just style. It is full of things, ideas, images. It took me almost three years to write and, when I'd finished it, I felt I had written myself out of sf. I wrote ordinary novels instead – _Hand-Reared Boy_ and _Soldier Erect,_ both of which went straight to the top of the best-seller lists in England, although they did less excitingly in the States. They embodied too British an experience, maybe. _Soldier Erect_ is probably the best of all my novels, shot through with pain and humour.
As a result of the Moorcockian revolution, sf has achieved a widely based reputation and readership. There were years when it looked as if Ballard and I were the last of the breed and no more sf writers were coming along. Now the situation has changed, with many younger authors such as Christopher Priest, Mark Adlard, and Ian Watson coming up. Bob Shaw has already arrived.
Unfortunately, sf writers, like everyone else, tend to grow conservative as they age, which is why new talents, techniques, and topics are continually necessary. A writer should set his own house on fire if he finds too much dogma lurking there. Try something new. In Bester's phrase, go for broke.
Some of the greatest sf writers have been subversive and gone for broke. Olaf Stapledon is the supreme example. What courage and imagination in _Star-maker!_ Plagiarised often, never rivalled! – although Stanislav Lem's _Solaris_ possesses something of Stapledon's quality.
Most of the science fiction being written is disappointing, and not merely on literary grounds; so many of its basic assumptions are fossils of thought. The philosophy and politics behind the average sf novel are naive; the writer takes for granted that technology is unqualifiedly good, that the Western way of life is unqualifiedly good, that both can sustain themselves for ever, out into galaxy beyond galaxy. This is mere power-fantasy. As I have often argued, we are at the end of the Reinaissance period. New and darker ages are coming. We have used up most of our resources and most of our time. Now nemesis must overtake hubris, for this is the last act of our particular play.
The knowledge should be a challenge. We should not be dispirited. 'May you live in interesting times' was an ancient Chinese curse; whatever we die of, it will not be boredom. The human spirit has to be continually tested. By whom? Why, by man himself. There is no one else.
One thing is certain, as Orwell said : when the world-state dawns, it will be neither Christian, white, nor democratic. But there is a long way to go before that.
Meanwhile, the Western way of life has the termites under it. The extravagance of Europe, of Japan, of the USA, is being curbed. Our foundations were built not upon stone, not even upon sand, but upon oil, and cheap oil has come to an end. These our memoirs are being written during the Great Power Crisis of the Seventies, when the Twentieth Century skidded to a halt. Readers must forgive our concentration on minutiae – to which I now return.
Philip Dick is one living sf author whose writings I admire and enjoy consistently. Dick is a natural subversive. He has a quality often found in great writers : humility. He seems to keep himself open and vulnerable, just as he writes about vulnerable little people. Dick made a brave speech at the 1972 Vancouver Convention, where he spoke of an ill-used girl he knew and then said about the future, 'I can only imagine it as populated by modest, unnoticed persons like her'. This loving quality in Dick balances nicely with all his seedily glittering wonders. Silverberg has that quality too, markedly in _Dying Inside._ Dick does not have to make his events plausible; the internal plausibility is always there.
One of the traditions of the novel which has been most staunchly defended over the decades is that fiction should be more plausible than life. Not too many coincidences, no over-exaggerated caricatures, as few flat characters as possible, no preposterousness. Anthony Trollope is one of the great practitioners here; whereas Charles Dickens fails on all these counts, although the scope and power of Dickens's mind is such that he has far more to offer the reader than Trollope. While it is true that the descendants of Dickens are many and often unexpected – Kafka for example – nevertheless, the other tradition of 'nothing implausible' still rules.
Science fiction cuts across this dividing line. Sf that stirs my imagination often boasts a major implausibility (maybe with attendant minor implausibilities hanging from it like subordinate clauses), which the writer then gradually makes plausible. That is, he integrates it with the world picture we already accept, so that our view of the world is thus changed. One good example of this process is in Jack Williamson's _Darker Than You Think,_ where we are slowly forced to take a darker view of our own ancestry than normal. More sophisticated examples lie in Kafka's two great novels, wherein almost nothing happens that could not happen in real life, until we realize that it is precisely the _banality_ of life that defeats us.
One element in the contributions to this book, and in its very inception, is phenomenal. It will be self-evident to sf fans and a cause of amazement to those who know little about science fiction. I mean the loyalty we all show to our chosen medium. Bob Silverberg confesses he felt that loyalty even in the days when he was exploiting it to the tune of several million words a year. Other writers who have gone beyond it – financially, like Arthur Clarke, popularly, like Kurt Vonnegut, or artistically, like J. G. Ballard – still recognize the tremendous power of that abstract idea of sf, an attempt, however crude, to build some sort of philosophical and metaphysical framework round the immense changes of our times brought about by technological development – a development which has largely obliterated the ramshackle old frameworks of medieval thought and organized religion.
Apart from writers of the comic-apocalyptic, such as Terry Southern, Thomas Pynchon, Joseph Heller, Philip Roth, and so on – all of whom apparently owe a debt to sf – the mainstream novel singularly fails in exactly those areas where sf is strong; it fails to interest itself in that abundant wilderness existing outside the narrow zone of activity covered by an average newspaper.
It is infuriating the dull way in which sf writers cling to the same topics over and over. The number of writers who actually _invent_ are few. All the same, I feel the loyalty. Did I not, I should never have written that complex volume, _Billion Year Spree._ The brief lives recorded here have been devoted to an obscure and unpopular genre (as it was until a few years back, when we had made our reputations). Julius Kagarlitski calls sf 'the intellectual novel for lowbrows'. If it is so, then it is a new genre, and we are among the first to succeed in it.
Unfortunately, genre materials wear out. I long ago swore off FTL travel and ESP in my writing (unless for comedy effects) because (a) I felt both subjects were refuges for tired minds, (b) they had been overdone to death, (c) while I could not say that I absolutely disbelieved in either concept, it was apparent that a writer who rambles on about concepts he only pretends to understand or believe in _is_ going to become a purveyor of drivel, and (d) that it is often an advantage for a writer to accept limitation of subject-matter, particularly in a field where he can get away with almost anything after two sentences of double-talk.
This voluntary restriction of subject matter has prevented me from turning out a regular ten novels, or five novels, or two novels, or even one novel, a year, unlike many of my competitors. It has made me think harder and care more about what I do write.
Not only do genre materials wear out, but the form itself wears out, and the form of sf is still essentially that of the Gothic or post-Gothic (Leslie Fiedler has much to say on this topic), with a mystery posed, hints of danger, a thread of suspense throughout, and a final revelation of some ghastliness. It is hard to resist the idea that to continue to use this form, patented at least as long ago as the turn of last century, is to put oneself in danger of obsolescence; although, in view of the threatened obsolescence of the Western way of life, this could be merely in accord with the times – but the most interesting writers are never content to be merely in harmony with their world.
I can see the problem. I'm still looking for a solution.
There are a number of stop-gap solutions. Two of the contributors to _Hell's Cartographers_ are noted practitioners of such solutions. Harrison has his own patent brand of comic-apocalyptic which uses as material not so much reality as sf itself; I'm thinking of _Bill, the Galactic Hero, The Technicolor Time-Machine, A Transatlantic Tunnel, Hurrah !_ (which his US publishers plonkingly retitled _Tunnel Through the Deeps),_ and the _Stainless Steel Rat_ books. In _Bill_ one can detect among the strata of Harrison's comedy the fossil bones of Heinlein's _Starship Troopers_ and, deeper yet, the crumbling foundations of Asimov's planet Trantor.
Bester's novel, _The Stars my Destination,_ is an earlier example of cannibalism, wherein all the elements of gaudy science-fantasy are seized up and kneeded into one tasty cake. Bester really ended that dynasty by doing the job definitively. Of course, there are minute intellects around who have not noticed the fact, and continue to do it over yet again.
These, as I say, are stop-gap solutions to the problems of a dying art (not that I don't think that all arts are not perpetually dying and almost-perpetually being renewed). I have another solution in my recent novel, _Frankenstein Unbound._
Barry Malzberg's _Beyond Apollo_ is a novel which uses the traditional elements of a space journey but, by setting it in retrospect and within the mind of an astronaut, equates the journey with a man's exploration of his own life and motives, and shows something of the inter-relation of the two. Incidentally, it is often a very funny novel.
Malzberg's format may appear complex at first, but he is not wooing complexity for its own sake; his complexity is the measure of Evan's, the astronaut's, mind. We should respect those writers who go out on a limb; they bring home the triumphs and the failures that the rest will be using casually, or casually avoiding, in a few years' time. Best-selling authors don't need our respect to anything like the same degree; they're working with the discoveries of the day before yesterday, often without knowing it. And they are generally richer than the pioneers.
Talking about solutions in the abstract is one thing. It's quite another thing when the individual writer shuts his study door, sits in front of his typewriter, and stares infinity in the face. Every writer has to work out his own solution in specific terms. Most writers are interested in little beyond a financial solution, although they may seek fame or notoriety as well. Whilst admiring the sort of writer who struggles to keep beer in his fridge and his mistress in dresses, whilst indeed admiring the whole idea of brave defeat and going-down-fighting, I am addicted to the idea of success. I naturally mean success in my own terms – any other kind of success is a defeat.
The problems of the serious science fiction writer cannot differ greatly from those of the serious writer. If the novel itself sinks, then sf itself will soon follow down to the ocean bed. I cannot think the novel will founder yet; there is still so much to be said for it, both for the understanding of life that it brings, and for the sheer convenience of its format. One of the qualities it does not now enjoy which it used to do is that very quality after which it was named: novelty. But the sf novel possesses novelty still.
Novelty may be the elixir which keeps an sf reader reading. It is less attractive to the general reader. In consequence, a good science fiction writer may be more neglected than an average writer or (certainly) an average playwright. An extreme instance of this is that great experimental writer Olaf Stapledon.
Such blind attitudes may change. This volume is itself a hopeful token of change. Meanwhile, we have our own successes, of which we are the best judges. I have written a number of books which I believe contain something like a creative vision, no matter in what other ways they may be flawed. Although I see my true strengths to lie in the short story field, I have novels for which I cannot but feel some warmth; most of them are involved with the portrayal of landscape, such as A _Soldier Erect, Report on Probability A, Barefoot in the Head,_ and _Greybeard,_ all of which depict figures in a landscape. _Non-Stop_ and _Frankenstein Unbound_ show figures swallowed by their landscapes. So, I suppose, does _Hothouse,_ a novel from which I always feel distanced, perhaps recalling the miserable circumstances under which it was written. _Cryptozoic (An Age)_ has landscape as surrealism, _Male Response_ landscape as comedy. _Eighty-Minute Hour_ has an exploded landscape.
Being a self-conscious man, I am well aware of how – in England if nowhere else – one can be both famous and remarkably obscure. It is curious to have arrived at a point where one's standing in this way so precisely matches character.
Damon Knight, who has read what I write here, complains that I begin my piece interestingly with concrete details of childhood, only to fade into the twilight of theory. It may be so. But the theories now mean more to me than childhood.
Lest I have been too abstract, let me add a portrait of the author as a middle-aged scribe. I am many people. Most of my opinions and emotions come in cycles, as does the weather – I am strongly influenced by the temperature, like an old allosaurus. As a youngster, I mistrusted this apparent shifting sand of character: how disgraceful that one's opinions should change with the company. Now I have learned to live with and profit from the phenomenon; it is of tremendous value to an author to be able to play the chameleon, to have an empathic sense. My empathic sense carries me away, and what can be disastrous in real life becomes a triumph on the page, where one is one's characters.
I am cynical and sentimental, foolish and wise, wanton and puritanical, courageous and cowardly, religious and atheistic; not all at once but in series. I hate exhibitionism, perhaps because there is a streak of it in me.
Some things are permanent, though. Permanent is the belief that the human species does not improve or deteriorate rapidly from one generation to another. But the emphasis changes. The America of Ford and Nixon suffers largely from values derived from advertising; there is too much eagerness to sell the product, irrespective of the worth of the product. One sees this among the self-advertisers of sf. And one sees their success. If you keep telling the world you're good, it will read your books and believe they're good. Very saddening. The false values Fred Pohl warned against in _The Space Merchants_ are taking over.
The England of today suffers from national exhaustion. I mentioned my three uncles. All joined the Army at the beginning of the First World War, all were captured and made prisoners by the Germans, an ordeal from which they were slow to recover. My father was gassed and involved in the debacle of Gallipoli in that same war. All my nearest and dearest male relations were scarred by that ferocious struggle; and many of their generation emerged into a world bereft of values.
The shame of Munich occurred when I was a schoolboy. The Second World War broke out when I was twelve. The map of Europe turned black. I used to have nightmares in which I was pursued and shot by the Gestapo. Murder and destruction were the commonplaces of our youth. Our minds were peopled with warlords : Stalin, Churchill, Eisenhower, Chiang Kai-Chek, Hitler . . . .
Britain emerged from that second great war broken and apathetic. To my mind, it will need another generation before the trauma is played out. If we have that long.
Responses to the havoc and grandeur of this mental desolation are many; I feel a sort of wonder at the years gone by – which actually seem to include the First World War for, although I was not born until long afterwards, I listened avidly to the stories my uncles and parents told me. Since then, we have come through the attritions of the Cold War and Vietnam. The Renaissance period has been ended by our own innate violence.
Inevitably, acts of kindness, deeds of bravery, works of art, are dwarfed by the landscape of destruction in which they take place.
Inevitably, we are of our times; writers can do well to move beyond those times, as well as merely depicting them.
I will end on a note of gratitude for the fortune that has been mine, much of which has come to me through writing. Sf fandom, with its essential kindness and relish for the sort of nonsense I relish, has been the only society I ever felt I belonged to; for once I left Norfolk, I never belonged anywhere again – not even at my own hearth for many years. I have had the pleasure of meeting many of sf's leading writers; there was that moment in Rio de Janeiro when Harry was in his finest form, lining up two gentlemen to greet me and saying, 'Brian, I want you to meet A. E. van Vogt and Robert A. Heinlein'.
I have visited many exotic and ludicrous spots, and have good friends abroad, particularly in Scandinavia, which to me is the best slice of the world – except for the fact that its climate is just slightly chillier, darker, and cooler than England's. At home, my luck is as good, with an early Victorian house which is not unlike my grandfather's old home in Dereham, a jolly family to fill the house, and a wife I adore.
Lest I make the picture too rosy for general credence, let me add that I regret having no faith – for belief in Catastrophe is no faith – more especially since I have lost hope in the idea of Reason as a guiding light. Even a loving family does not entirely compensate for a sense of isolation; nor do beautiful women and good friends erase the knowledge that life and success are mere temporary accidents. The days come and go; the enemy is never forgotten.
Yet I finish this memoir (if memoir it is) on a mild January afternoon with a fire burning and a tranquil country view from my window. Margaret prepares dinner while the children play with a friend. Tonight there is a party. Tomorrow offers new excitements. In the circumstances, a display of stigmata would be inappropriate.
**how we work**
**Robert Silverberg**
It was all much simpler long ago, in the old days when I was a high-volume producer of fiction. I worked a rigid schedule, five days a week, throughout the entire year, with six or seven weeks blocked off for holidays. I kept a sheet on which I listed, day by day, everything I was committed to writing over the next three or four months, and I simply did my daily stint, knowing I'd always be able to complete my allottment. The schedule looked something like this :
March 20 Ace novel pp 140–165
March 21 Ace novel pp 165–190
March 22 AM finish Ace novel; PM short story 3000 wds
March 23 Campbell novelet pp 1–25
March 24 Campbell novelet pp 25–50
And so on, incredibly, month in month out. Like a machine. Getting the ideas was no problem; they arrived mysteriously out of the air, like radio broadcasts, while I was reading or strolling or listening to music. I jotted them down on handy scraps of paper – often nothing but a title (titles usually came before plots for me, and still do) and a brief summary of theme ('Earthman falls in love with bizarre alien woman'). The morning I was due to begin work on a story, I would elaborate the summary into a two-paragraph outline, pick some characters, give them their names and physical descriptions, and start work. Into the typewriter would go a sheet of white paper, a piece of carbon paper, and a yellow second sheet; and, hour after hour, the first and only draft of the story would emerge. I worked from nine in the morning to half past eleven, took a little more than an hour for lunch, and put in another stint from quarter to one or so until about half past two. Somehow this regime produced twenty to thirty pages a day of publishable copy, and everything I turned out this way sold. _Everything._
But, as readers of my early stories know, the prose was simple, functional stuff, the plot showed signs of improvisation, the narrative flow was often congested with padding : when no inspiration came I nevertheless kept my agile fingers moving and the action, however inconsequential or irrelevant, spinning. As I matured this method of work no longer sufficed for me, and by the time of my first 'new' fiction – in the mid-sixties – I was very much more careful, doing detailed outlines before beginning, pausing to rewrite scenes that didn't seem acceptable as they came from the typewriter, going back to put in inserts, in short doing a great deal of planning and revising. Still, most of what I was publishing through 1967 or so was basically first-draft stuff – written more slowly, conceived with greater care, but even so not subjected to any systematic process of overall rewriting. My years as a high-volume producer had given me skills of expression and improvisation that allowed me to say what I wanted to say clearly and effectively in a single try.
What I wanted to say, though, became ever more complex and difficult to express, and during the late sixties my working methods evolved toward what they are today, so very different from my habits of fifteen years ago. I still work a faithful five-day-a-week routine, although my hours now run only from nine to noon, and the holiday weeks are more frequent. I no longer try to set a fixed schedule of pages per day, however, being content to work at whatever pace is necessary for each day's task; and so, whereas in 1957 I could tell you on 3 October what I would be writing on the 12th of December (and almost certainly be right), I now have no idea how long it will take me to finish any project. I used to do short stories of up to 7,500 words in a single day; now they require, sometimes, six to eight weeks, and writing novels, once a job of two or three weeks, has become an endless procedure.
I still begin with a title and a brief statement of theme. Then, on the backs of old envelopes, come structural notations having to do with overall form and texture of the work, lists of characters, bits of background data, suggested sequences of chapters. I usually have the beginning and the end of any story fairly clear before I start; the middle is subject to development once the story acquires life of its own, and so I constantly write memoranda to myself as I go along. (The final paragraphs of _Born With the Dead_ presented themselves to me, unbidden, while I was watching 'Last Tango in Paris', and quite distracted me from the movie; the moment the film was over I grabbed pen and paper and wrote everything down, feverishly, before it vanished. Some day I must go back and see the second half of 'Last Tango' again.)
I never try to get away with one-draft writing any more. Using old letters, advertising circulars, bits of manuscript, or any other sheets of paper with one blank side, I write (single-spaced) a paragraph or two, go back and do it again, do it three or four times if necessary, then try another passage, and so on until I have a thousand words or so of work that I consider acceptable. At this point I usually type a fair copy, double-spaced, and put it aside; then I continue with more rough work, and so I proceed through the story, now doing new material, now revising, now retyping, the final manuscript constantly growing as I plough onward into new territory. Very frequently I discover that I have been premature about committing the early pages of a story to final-copy form : it has become almost customary for me to halt, thirty or forty pages along, realize I've made a false start, consult frantically with my wife (who has the unique privilege of seeing works in progress, and only in their earliest phases, when I'm least sure of myself) and go back to page one. This creates a mound of blank-on-one-side waste paper out of what for a while had been my 'final' draft, but I recycle it into the drafts to come.
Here are some samples of the rough-draft process :
I pondered for a couple of days. Break my word to Carvajal and save Qu
But could I let Quinn mess himself up?
Half an hour after I left Carvajal I could see no reason for not telling Quinn
Half an hour after I left Carvajal I was ready to warn Quinn to skip the Kuwait dedication. I didn't call him.
I didn't put through an immediate call to Quinn, but it was close. Once Carvajal was out of sight I found myself wondering why I had decided to remain silent. Carvajal's inflexible
I didn't put through an immediate call to Quinn, but I came close. As soon as Carvajal was out of sight I found myself wondering why I would hesitate at all. Carvajal's insights into things to come
And so on. All of these are possible openings for a scene; in the end I chose a variant of the last one. Choices I once made automatically and unconsciously I now must work out on paper, and there seems no way around it. At first I was restless and bothered by the inescapable need to do all this fussy sentence-by-sentence tinkering; now I accept the reality that my work will no longer flow as it once did. My output now is less than a tenth of what it was in my most prolific days. I suppose by some standards I'm still a terribly productive writer, but it doesn't seem that way to me, not with a net output of 800 words or so per day. There are times when I miss the old ease of production; but I do prefer today's results.
**Alfred Bester**
'Writeing is a nag – ' is a line from one of Ring Lardner's superb stories, 'A Caddy's Diary'. It sums up my attitude toward my work; it isn't a knack, it's a nag, a damned compulsion, and this is how I handle it.
I've often said, whilst living abroad, that it was difficult for me to learn another language because I was too concentrated on learning my own. This has more meaning than a mere cop-out. In my opinion it's essential for a writer to think, speak and write the identical language. This takes a tremendous amount of self-discipline and training.
Next, preparation and dedication. It's tough explaining preparation. It means that every item of life, no matter how minute, must be observed and noted. You never know when it may become useful, perhaps never, but it must be there waiting. I don't know how many times I've combined notes made years apart into one story. This means that the author must split his personality. Half of him is participating in the scene; the other half is watching himself and the other members of the cast keenly. It's rotten. It's hell. It's a price an author must pay.
Dedication: I don't write two hours a day, or four or eight; I write twenty-four hours a day. I can't help myself. I keep thinking story, even in my sleep, not as an observer but as a participant. I'm inside the story. I'm all the characters. Of course it must be understood that many of these stories never come to fruition, let alone writing. I pass them off as fantasies and let them go. A few of them still haunt me, however, and I often wonder whether my unconscious is making its own notes.
My writing, naturally, is the sum total of my education and experience. I studied music at college and am extremely sensitive to tempo and timing. It was pointed out to me once that I usually write in a presto three-quarter beat. No argument, although I try very hard to relieve it occasionally with a four-four andante. I was deeply affected by the energy of Dickens, Reade and Dostoievsky, and have sometimes been criticized for my razzle-dazzle style. I can't help that. I'm compelled to try to match their dash because I regard writing as an entertainment for myself and the reader, raree show which will amaze and amuse.
I'm sometimes condemned for being an elitist in my lifestyle and my writing. I don't think that's true but I don't defend myself. All I can say is that I am what I write and write what I am and it's up to the reader to keep up with me. I never try to put the reader down, but I'll be damned before I write down. I have too much respect for people to do that.
I return to the questions of discipline and split personality. After many years I learned to edit myself. This means that I'm both writer and editor. I recall one of my publishers giving me the manuscript of a science fiction writer and asking my opinion. 'It's dull,' I reported. 'Well,' he said, 'he's a rather dull writer.' 'No,' I said, 'he makes a silly mistake. He accepts the first idea that comes to his mind.' This is something I cannot permit myself to do. The editor in me nods and says, 'Yes, but that's obvious. Surely you can come up with something better, my boy.'
And I try. God knows, I try! With what success I don't know, but I try. I was once rebuked for being a perfectionist. So I am. What destroys me is the conviction that I've fallen so far short of perfection. This is what nags me into writeing and writeing and writeing.
**Harry Harrison**
Writing about writing can be a very tricky business. It seems very pretentious to talk about how one goes about the physical act of writing – yet who can deny the hideous attraction of the topic. The _Paris Review_ series of interviews were supposed to get the reader inside the heads of the great authors, but about all I can recall of them is Hemingway saying that every good writer had an inborn automatic shit detector, and Simenon writing 5,000 words a day in a scruffy hotel and X-ing out each day boldly on the calendar until the end of the book was reached within two weeks.
With this love-hate aspect of writing in mind the writerly act has been tucked into this corner of the book. Read it if it interests you, pass it up and I shall not mind.
Writing is a fragile act. When one is pulling language, thoughts, ideas out of thin air, or the turgid subconscious, any disturbance disturbs. Harlan Ellison makes it a point to write in the middle of booming, drunken parties but, no drinker himself, I feel that what he. writes there reads as though it were written in some place like a booming, drunken party. I'm sure he does his best work in the quiet of his study because I know of no serious writer who does not need his solitude, his sitting and thinking time, in addition to his writing time. Even the familiar, no matter how peaceful, can intrude. Brian, I know, goes to a cottage on the Thames when he needs that sort of quiet. Alan Nourse has an A-frame deep in the Washington woods, totally deserted. I take the camper to the beach in Mexico where I can neither talk nor be talked to. Not that any of us lack peaceful studies at home, we all have fine ones. But there are times when a bit of Walden Pond is needed by all of us.
Not that non-writing others always understand. In Mexico Joan literally beat a 'friend' away from my study door with a broom because he knew old Harry didn't mind talking to him at the time. Joan's mother, a paragon of virtues in all other ways, does not realize the basic needs of a writer or she would not have opened the door when I was writing, as she did once years ago when we were staying in her home, and say, 'Harry, since you aren't doing anything, would you go to the store for me.' A writer's family understands; my daughter knows when I have that glassy look in the eye and am staring into space that I am not to be disturbed because I am 'working'.
That is part of the working time. What is needed next is the sitting-at-typewriter time which is when the penny finally drops. During my freelance art days I found I worked best from noon to about four the following morning but, thankfully, I have changed my schedule since then. I know some who still work these kind of hours. I find that a regular daylight schedule is more productive.
In the morning, anywhere between eight and ten, I go to the studio (called that instead of study or office by reflex from the art days) and put in a day's work. When I am writing I emerge at cocktail time in need of strong drink. When I am editing I emerge at the same time in greater need since most editing is such drudgery. When I am working at a piece of fiction I stay with it for as many days as it takes. Early on, in honour of the christian work ethic, I used to work the six day week and take Sunday off. I found this broke the motion of a book so I began to work straight through on the first draft. Since I average about 2,000 words a day this means at least a month of continuous writing. That's fine. It also means weeks off at a time when others are in their offices. There are really only three advantages to the freelance life. (1) You can live wherever you wish. (2) You can work the hours you want. (3) You can wear comfortable work clothes and old shirts while on the job and save a fortune in suits and white shirts.
I work from an outline always, more or less detailed, but always an outline. Many pages for a book, just a firm idea in mind for a short story. I know writers who start a story with no idea how it will end; I would rather die first. I am a firm back-plotter and must know the ending before I begin, then expend writing energy disguising the fact that I always know what is coming next.
My study contains a long desk, formerly from a drafting room, cut down so the whole thing is at typewriter height. On this are in and out baskets for correspondence, a holder for paper and stationery, the telephone – with a switch to turn it off so it won't ring – and a calculator so I can figure how much money I am owed at any given time.
Some few years ago Joan rubbed in an awareness of a basic financial question. I had always bought cheap second-hand typewriters, and the one I had decaying at the time had cost £20 second-hand, had been made in East Germany, which meant that American repair men would not work on it (it was a commy machine), and I had written seven books on it which amortized out at $7.1428 a book (the calculator, remember?) not even counting the short stories. I was looking around for another cheap wreck when she told me that everything the family ate and drank, everything physical in our lives, came from the typewriter – so why was I being so chintzy? Buy the best. It was a good argument and I did. An IBM selectric, the writer's friend. I have never regretted this action for a moment since.
Books and books ago I found it hard to start work each day, a variation of the writer's block we have heard so much about. I discovered that if I did my correspondence first I could get the. fingers flicking about and the typewriter humming. Then I would trick myself and slide in paper and try to write. It usually worked. Things are better now. Most of the time I can approach work, begin and end.
To begin I read the pages done the previous day. I give them a quick proofing but no elaborate rewriting – though I do make marginal notes like BAD! OUT! REAL CRAP! to cue myself on the second rewrite. Having reread the one day's work, but no more, I slip white paper and carbon-set into the machine, take a deep breath – then turn the machine off and think a bit. Then I write.
The carbon-set is a must. Early in life I found I needed copies of letters uncopied, carbons of manuscripts lost in the mail and such. Now a carbon goes in with anything, other than labels or envelopes, that I put in the machine.
I labour this way until the first draft is done. Less than a week for the usual short story, the mentioned month at least for the novel. If the work to hand is a book I take an extra large drink after I type those fine words THE END and lay the whole thing aside for a bit. A week usually. Joan and I take a weekend in Mexico, or some such place, and sun, water, food and drink cleanse the mind. Then I begin the rewrite, something that gets progressively harder each time through. I do try to get through the book at least five times. I write very tightly and rarely do more than change words right on the page, punctuation, grammar, the usual thing. When this is finally done I emit what is called an intense sigh of relief. Some writers retype their mss, thereby finding a chance for more rewrite, but since I am the world's worst typist I bundle the entire thing off to my typist, the pearl-beyond-price, Mrs Fitzhamon. (First making a xerox of the rewritten ms. which differs a good deal from the carbon.) She lives near Brighton and makes no errors and finds all of mine, and if you have ever had a bad or indifferent typist you will understand just how good a superlative one can be.
That is it. By the time the typing is done I am well into the next piece of work and planning the one, two or three pieces ahead of that. At the same time earlier stories or books will be at the publisher or being published. When people ask me 'how is the book coming' I can respond only by blinking a glassy eye and muttering 'which one?' This is no act because at any given time I may have at least four or five books somewhere in the area between idea and on sale.
It may be art, but it is business too. That is why I have an agent who earns, ten times over, the ten percent for his labours.
**Damon Knight**
I work at a large wooden desk made for me by a carpenter years ago at a cost of $40; it has a well for the typewriter and open shelves on either side for paper, letterheads and envelopes. As far as I can see there is only one thing wrong with it, and that is that the desk is so big that it tempts me to pile all sorts of things on it, and I am forever trying to clear it off. The desk faces a wall lined with bookshelves. I found out years ago that I could not work facing a door or a window. Someone might come in through the door, and there is usually a tree outside a window; then I watch the tree and forget to write.
I use blue second sheets. It is absurd that the colour should make any difference, but it does – I need something to distinguish the work from final copy, otherwise I get self-conscious and freeze up. When I began writing I used yellow second sheets, the cheapest kind, but now I dislike them.
I once had a Frieden Flexowriter on trial – a marvellous machine, with a tape cutter and reader built onto the chassis of an IBM Model A electric typewriter. It cuts an eight-channel tape, and you can feed the tape back in, make any corrections you like while the machine cuts a second tape, then feed in the corrected tape and watch it chunter away by itself turning out final copy. If I were very rich I would own one of these machines, but they are expensive and delicate. Ideally, every professional writer would have one of these now, and would furnish a tape to be used in typesetting along with each manuscript. Later we all ought to have computer terminals which would do the same thing more efficiently, and would also provide instant information, correct our misspellings and typographical errors, etc. With sophisticated programming they might do a lot of the routine work of writing as well; in some kinds of commercial fiction there is no reason why they shouldn't do most of it.
By the time I feel ready to start a short story I usually have a pretty clear idea of its form, and have written down a list of scenes. A novel is different, and I always write reams of background material first. Then I throw out most of it and write more. This is a terribly inefficient way of making a novel, and it partly accounts for the fact that I have written so few.
I am particular about the names of characters, and at one time kept a list of good ones, crossing them off as I used them. Names in stories should have the variety and absurdity of real names. They can be used to suggest what part of the country a character comes from, what his parents were like, etc; they can also be used, as Hammett used them, to give subliminal cues about people in a story.
I write one draft and correct it heavily for style, etc. If part of it gets too bad, I cross it out and write in a new version; I often throw pages away and start over, but I try to keep the work up to a satisfactory level as I go, rather than running out a complete first draft and then correcting the whole thing. This is not efficient, either.
I think I can tell by the look of a page whether it is well proportioned or not. If I know where I'm going in a scene it will come out in approximately the right form and length, but if I am just noodling around hoping for inspiration to strike, it will go on drifting forever.
I always throw away beginnings. If the beginning is wrong, the rest of the story will be wrong, but if it's right the rest of the story will spin itself out of the first sentence or two.
I model characters after people I have known, but they always take on their own personalities, and I never know what they're going to say until they open their mouths.
I once tried to write a series of stories with titles derived from the running heads of Donald Days _Index to the Science Fiction Magazines._ 'Stranger Station' was one of these.
**Frederik Pohl**
How do I write? Any way I can. There is a disciplined way, an inspired way and a way of desperation. The inspired way is most pleasurable; it is joyous to conceive a story, sit down at the typewriter and bring it to birth in a single sitting. Unfortunately inspiration hardly ever lasts past the first ten pages, so that only works for short stories, and not many of them. The way of desperation is surest, the times when I have so boxed myself in that I have no choice but to write. I try to avoid the post-deadline, bills-due, editors-screaming sort of desperation that makes it hard work because it is uncomfortable, and also because often enough it produces bad work. But sometimes it produces good work, too – I cannot say why sometimes it is the one and sometimes the other – and anyway, without at least a little desperation I am not sure I would finish many stories.
So year in and year out I mostly rely on the disciplined way. For me the procedure is simple. Once a day I sit down at my typewriter, roll a sheet of paper in and do not get up until I have produced four pages of copy. If I am going well, maybe I'll produce more; but however badly things are going, I will produce those four or die for it. It's as simple as that. Sometimes it takes forty-five minutes to write the four pages, sometimes twelve hours. But when I am on my quota system editors smile at me, I meet deadlines, I produce satisfying amounts of material and all the world is mine.
Is what I write in this soulless and stultifying way any good? Sometimes yes, sometimes no – like what I write in any other way.
What do I do if it's bad? I put it away and work on something else, till the ageing process has had a chance to work. Then I take it out and look at it, and most times I can see what is wrong and what I need to do to fix it. (I never submit first drafts. As I have mentioned elsewhere, I make it impossible to submit first drafts by typing them on the backs of old circulars and fund appeals.)
Provided I have not boxed myself into a deadline, that almost always works; I put aside what is going badly, and work instead on something that goes well – until it too begins to go badly, and then it gets put aside in turn. It's a slow way to write. Most of my novels are three or more years in the works for that reason. Some have been ten. Short stories have been five years from first page to publication. In fact, I have short stories started much longer than five years ago that haven't been published, or even finished, and some of them never will be. A good thing about being a successful writer is that everything you write gets published. A bad thing is that it gets published even if it shouldn't be. So I try to censor myself and not send out a story until I am pretty sure it's as good as I can make it – sometimes I've later come to think I was very wrong – which means that I have quite a few stories in one stage or another which I don't offer for publication. I suppose that, counting everything, there must be well over a million words: five books, three of them actually completed, twenty or thirty shorter pieces, and God's own quantity of (at least temporarily) abandoned fragments. Several times I have decided against publication not only after finishing a work but even after signing a contract and receiving an advance on it; three of the books were in fact contracted, and I bought the contracts back, because I didn't like the books well enough to want to publish them.
If imposing the four-page-a-day rule on myself works all that well, why don't I do it all the time? Human frailty, that's why. Partly it is rebellious unruliness in my head, but partly it is a matter of losing quality after a while; I go stale and need fresh inputs.
What are my inputs?
Science is one of them; as I have said, I am a fan of science, it is the greatest of spectator sports for me. Because many scientists were, or are, science fiction readers I have the opportunity quite often to spend time with men who are doing fascinating kinds of research; and as they like to talk about what they are doing, and I like to listen, there is a lot of good conversation to add to the magazines and journals and books and papers and symposia and speeches. That is probably my favourite perquisite, but surely next to it is the oportunity to do a good deal of travelling. I usually get out of my own country two or three times a year at least, and inside it I have managed to visit forty-eight of the fifty United States. (If anyone cares to offer me a lecture engagement in either Alaska or North Dakota I'm available any time.) And then – well, I read a lot, and I permit my curiosity to lead me into out of the way places. I've taken formal courses, in the last six months, in things like guitar-playing, ballroom dancing and transcendental meditation. I'm a trustee of a Unitarian church, and of a private school for retarded children. From time to time I am involved in politics; and within the last year or two I've made my debut as folk-singer, night-club comedian and actor – all very amateur, to be sure. But interesting, informative – and inputs. I do not pretend that I do these things because I want to write about them, but some of them do show up in what I write, and I don't see how I could write well without them.
Where do I work? On the third floor of our home, in a four-room suite – three are mine, the fourth (since she began doing anthologies and painting covers on her own) my wife's. It is a wickedly profligate consumption of space, and I love it. Everything is here: three desks, four typewriters, two kinds of recording equipment, file cabinets full of manuscripts, correspondence, documents and reference materials. We are plagued with books – thirteen of our fourteen rooms have bookshelves in them, and so do two of the halls. But the books on the third floor are my working library: scientific books, six shelves of them in the northwest room; political books in the northeast; histories, language guides, encyclopedias and so on in a passage between two of the rooms; my own published work spread out wherever there is room for it, mostly in the southwest room and the hall. God knows what we'll do if we ever move.
But really I don't need it all, not for more than one or two sidelights in an average work, which I could of course as readily pick up in a public library if I had to. It is convenient, but not essential. Some of the best writing I've ever done has been with a portable typewriter perched in my lap, in a hotel room or a bus station, on a plane, wherever. A few years ago while my wife was taking a nap in our California hotel room I sat on the sunny balcony outside our tenth-floor window, looking out over the old 20th Century-Fox lot, and in about an hour and a half I had written one of the short stories I'm most pleased to have done, 'I Remember a Winter'. It's a fragile sort of story, and if I hadn't written it right then when it formed itself in my mind I don't suppose I would ever have written it at all.
When do I work? Optimally, at night – starting at midnight, and continuing until I have done my quota for that day, whereafter I read a little, eat something, listen to a little music and go to sleep, generally about sunrise. (My old friend, Cyril Kornbluth used to say, 'If God had meant man to be awake by day He wouldn't have given us the electric light.') This makes problems when I interface with the real world. Publishers and university people in particular have a foul habit of trying to telephone me in the morning, when I am generally asleep. But I won't change; from midnight to six there are no phone calls, the household is asleep, no one comes to the door and that is how I like it. I am all too easily distracted. Years ago I used to try to get my family to tell callers I was out in the morning. Now they just say I am asleep.
Being a writer is difficult; you not only need the talent and technique to write, you also need the discipline to make yourself do it, and the critical judgment to know when you are done. If you are a bricklayer, say, people tell you what to do – 'twelve courses of glazed yellow, ten yards long, staggered and faced with stone at the gate – ' and when the wall is up you are through. If you are a writer you have to set yourself a task, make yourself do it, and evaluate the result when you are done, with little or no real help from anyone. To be sure people will try to tell you what you should do, and give you all the criticism you want, maybe more than you want, once it is irretrievably in print; but the only opinion that really matters is your own.
The way I write is the way every other writer writes : the best I can, in whatever way I can; and that is the Whole of the Law.
**Brian Aldiss**
The most important thing first. I never submit outlines or synopses of novels to publishers. For better or worse, generally worse, my novels are entirely mine. Only when they are finished do agents and publishers get a look. That way, art or whatever-you-call-it and commerce do not get too mixed up. Nine out of ten writers, I'm told, must have an advance before they begin writing; but that's not my way.
I've never written for radio or tv or the films. I have often written for free – but no longer, with four children and several publishers to support.
I'm one of those fools who likes to think he is writing for posterity as well as the present. I know, I know. But my older children, Clive and Wendy, read my books voluntarily; it would be nice to think that their children could enjoy them too.
Science fiction used to be regarded as ephemeral; ephemeral is a relative term, but experience indicates otherwise. All the sf that I ever wrote is now in print and continually being reprinted. Several of my collections of short stories, from _Space, Time, and Nathaniel_ onwards, remain in print in various editions, and at least three of them have sold towards 200,000 copies each – a figure that would be phenomenal outside the sf field but must have been exceeded many times over within the field. Which suggests that it is worth expending time (i.e. love and care) over even the shortest pieces.
Much of my time – Harry says the same thing – is spent doing nothing. I suppose I work about nine months of the year and, during that nine months, I do a six-day week every week. I'm in my study by 9.30 or 10, remaining there till about 4.30, with an hour or two break for lunch, when I may, or more likely may not, take a stroll. For half of those days, my time is spent answering correspondence from all over the world – I must be mad to do it. Half of the rest of the time, I do nothing, either sitting and gazing relaxedly out of the window at the twentieth century unfolding, or walking about twitchily with ideas half-formed, occasionally picking out a book from the shelves and reading an odd page. Only in the rest of that time do I actually manage a little fiction.
My study is at the top of the house, a spacious and comfortable area with cacti, sofas, an easy chair, three desks, and thousands of books. Phone, radio, tv, cassette-player, electronic calculator, photo-copier, intercom; but no booze or cigars. I never smoke or drink when I'm writing, except on occasions at the end of the day when I'm getting a bit ragged.
When really absorbed, I like to work in the evening, but there has been less of that in recent years. Small children impose regular hours on authors; there is nothing so bourgeois as children. A verse of Clarence Day's comes to mind at this juncture :
Who drags the fiery artist down?
Who keeps the pioneer in town?
Who hates to let the seaman roam?
It is the wife, it is the home.
An understanding wife is one of a writer's great assets. Not only is Margaret my first and shrewdest reader, she makes it easy for me to slip off for the odd week to a cottage in the depths of the country, where I work in absolute peace, sometimes writing for twelve hours a day (but I never go without being clear about what I wish to write).
The other thing a writer needs is a good agent. My agent, Hilary Rubinstein, owns the cottage I use as refuge; the loudest noise you ever hear there is the dropping of apples in the autumn into the long grass. Several recent novels have been finished in Willow Cottage.
Generally, I am serene when writing. Between times, I am occasionally depressed to reflect on how much of my art I have never learned and probably never can learn; I can't bear to think of Leo Tolstoi. (Even thinking of his wife is bad enough; she wrote out _War And Peace_ for him five times in longhand.)
People always want to know how long novels take to write. Novels impose different schedules, just as they need different approaches. The fastest novel I ever wrote was _Dark Light Years,_ because I did it in a fit of anger. It was all complete in a month – but I did nothing else in that month except write or do nothing. _Barefoot In The Head_ occupied me for over two years. The average time is about a year. _Billion Year Spree_ took the best part of three years to write. Now that I'm moderately successful, I want to spend more time over novels – instead, I seem to spend more time over correspondence.
I have a nice secretary, Pam Woodward, who comes in two mornings a week and will do extra when needed. I also have a typist, Jill Watt, who lives near Bristol and is good on sf as well as typing. And Margaret does a thousand jobs, including looking after all the financial side and managing SF Horizons Ltd. I have never typed out the final draft of any of my own novels or stories; that surely is a job for a professional.
Writers' blocks are unknown to me (he said nervously), perhaps because I have never flogged out copy for cash and still actively enjoy the building of sentences and paragraphs. Of course I get stuck occasionally, but one develops remedies for sticking as for hangovers. There are one or two critical books which can always get my associative and creative juices flowing again – John Livingstone Lowes' _The Road_ to _Xanadu,_ Caroline Spurgeon's _Shakespeare's Imagery,_ and – unfailingly – Mario Praz' _The Romantic Agony_ (a title I have been known to repeat in incantatory monotone while drunk). I rely increasingly on free association for the series of short stories at present under way.
Apart from this series of short stories (about the Zodiacal Planets), I also have on the stocks an ambitious novel about a utopian city-state, Malacia, where change is prohibited by law and time is elongated. Four stories about Malacia appeared in one of Damon Knight's _Orbit_ anthologies; they form the basis of what I hope may prove my most satisfyingly complex fiction, _The Malacia Tapestry._ If I am pleased with results, more on Malacia may follow.
Meanwhile, other novels exist in embryo, leading shadowy existences. The third Horatio Stubbs novel, successor to A _Soldier Erect,_ is largely visualised in my head, and should be the next thing I write after editing this book. It will probably be called A _Rude Awakening._
When I work on a novel, I compose on to a slow typewriter – it is some years since I forced myself to graduate from fountain pen and loose-leaf book, thank heaven! As the pages come out of the machine, they are placed in a pile face downwards, so that I cannot see what I have just written. This is to prevent disillusion. A creative glow is the great necessity for that first draft; by resisting reading back, I can sustain that glow, thinking how wonderful was my vision. Only when the draft is complete and finished dare I go back and read what is written. How much of the vision escaped between head and paper! Disappointment is always great; but hope still abounds, and creative hope mixed with critical discontent carries me through the re-write and/or second draft. Then there comes a sort of lull, where second thoughts creep out of the basement of the subconscious like swine-things in William Hope Hodgson's _The House On The Borderland._ They can be incorporated in the final draft correction, which is almost purely critical in feel. Then the raddled and ratty old typescript goes off to Jill Watt.
It looks better when properly presented. Hopes rise again, and you send the typescript copies off to your agent in good heart. The publisher's proofs bring you to the nadir of hope : the material is stale, you no longer laugh at your own jokes, weep at your own tragedies, blench at your own truths. But, with luck, the whole thing looks much more imposing when you get your six bound complimentary copies. Thus encouraged, you turn like a stag at bay to face the baying of the reviewers____
**selected bibliographies**
ROBERT SILVERBERG
_Novels and Short Stories_
Revolt on Alpha C, US 1955
Needle in a Timestack, US 1966, UK 1970
To Open the Sky, US 1967, UK 1970
The Time Hoppers, US 1967, UK 1968
Thorns, US 1967, UK 1969
The Masks of Time, US 1968, as Vornan–19, UK 1970
Nightwings, US 1969, UK 1972
The Man in The Maze, US & UK 1969
To Live Again, US 1969, UK 1974
Up the Line, US 1969, UK 1975
Tower of Glass, US 1970
The World Inside, US 1971
Son of Man, US 1971
A Time of Changes, US 1971, UK 1974
Moonferns and Starsongs, US 1971
The Book of Skulls, US 1972
Dying Inside, US 1972, UK 1974
Unfamiliar Territory, US 1973, UK 1975
Born with the Dead, US 1974
The Stochastic Man, US 1975
_Non-Fiction_
Lost Cities and Vanished Civilisations, US 1962
The Old Ones : Indians of the American Southwest, US 1965
Scientists and Scoundrels: A Book of Hoaxes, US 1965
The Auk, the Dodo, and the Oryx, US 1967, UK 1969
Mound Builders of Ancient America, US 1968
Mammoths, Mastodons and Man, US 1970, UK 1972
The Realm of Prester John, US 1972
_Editor_
New Dimensions, Vols 1–6, 1971–6
Alpha, Vols 1–6, 1970–5
The Mirror of Infinity, US 1970, UK 1972
Science Fiction Hall of Fame, US 1970, UK 1971
_Awards_
Hugo : Most Promising New Author, 1956
Hugo: Nightwings, 1968
Nebula: Passengers, 1969
Nebula : Good News from the Vatican, 1971
Nebula : A Time of Changes, 1971
ALFRED BESTER
I'd be delighted to fill your request for a bibliography but the hell of it is that I can't. I've sometimes wondered whether it's Freudian; I always dismiss my past and concentrate on the present and future. As a result I don't remember three-quarters of the things I've written, and certainly not the dates of anything. Would these paragraphs serve?
Began professional writing in 1938. Wrote science fiction, mystery and adventure stories, then comics _(Green Lantern_ – _Captain Marvel –_ etc.), then radio _(Charlie Chan_ – _The Shadow – Nick Carter_ – etc.), then TV _(Fireside Theatre_ – _The Winchell Show – NBC Showcase – Fred Astaire –_ etc.). Switched to magazine feature writing for _Holiday_ and _Rogue_ magazines regularly; others occasionally. Four novels, three short story collections, one popular science book.* Only award : the very first _Hugo._ Am not at all glamorous; merely a working stiff.
_*Novels and Short Stories_
The Demolished Man, US 1953
The Rat Race, US 1955
The Stars My Destination, US 1956, as Tiger! Tiger! in UK
Starburst, US 1958
The Dark Side of the Earth, US 1964
_Non-Fiction_
The Life and Death of a Satellite, US 1966
HARRY HARRISON
_Novels and Short Stories_
Deathworld, US 1960
The Stainless Steel Rat, US 1961
War with the Robots, US 1962
Deathworld 2, US 1964
Bill, the Galactic Hero, US 1965
Plague from Space, US 1965
Two Tales and 8 Tomorrows, UK 1965
Make Room! Make Room! (filmed as Soylent Green), US 1966
The Technicolor Time Machine, US 1967
Deathworld 3, US 1968
Captive Universe, US 1969
The Daleth Effect, US 1970
One Step from Earth, US 1970
Planet of the Damned, US 1970
Prime Number, US 1970
The Stainless Steel Rat's Revenge, US 1970
Stonehenge (with L. E. Stover), US 1971
Montezuma's Revenge, US 1972
The Stainless Steel Rat Saves the World, US 1972
A Transatlantic Tunnel, Hurrah!, US 1972
Star Smashers of the Galaxy Rangers, US 1973
Queen Victoria's Revenge, US 1974
The Lifeboat (with Gordon R. Dickson), US 1975
_Juveniles_
The Man from P.I.G., US 1968 Worlds of Wonder (editor), US 1969
Spaceship Medic, US 1970
The California Iceberg, US 1974
_Editor_
Apeman, Spaceman (with L. E. Stover), US 1966
John W.Campbell: Collected Editorials from Analog, US 1966
Nebula Award Stories 2 (with Brian W. Aldiss), US 1967
Four for the Future, UK 1969
The Year 2000, US 1970
The Light Fantastic, US 1971
The Astounding-Analog Reader (with Brian W. Aldiss), US 1972–3
Astounding: John W. Campbell Memorial Anthology, US 1973
Best SF, US 1967–74 (with Brian W. Aldiss)
SF: Author's Choice, Vols 1–4, US 1968, 70, 71, 74
Nova, Vols 1–4, US 1970, 72, 73&74
_Awards_
Nebula: Make Room! Make Room!, 1974
DAMON KNIGHT
_Novels and Short Stories_
Hell's Pavement, US 1955
A for Anything, US 1959
Far Out, US 1961
In Deep, US 1963
Beyond the Barrier, US 1964
Off Centre, US 1965
The Other Foot, US 1965
The Rithian Terror, US 1965
Turning On, US 1966
Three Novels, US 1967
World Without Children and The Earth Quarter, US 1970
_Biography and Criticism_
In Search of Wonder,US 1956; rev. ed. US 1967
Charles Fort, Prophet of the Unexplained, US 1970
_Translations_
13 French Science Fiction Stories, US 1965
Ashes, Ashes by René Barjavel, US 1967
_Editor_
Orbit Vols 1–15 (US 1966–74) and many other anthologies
_Awards_
Hugo : Best Book Reviewer, 1956
FREDERIK POHL
_Novels and Short Stories_
Alternating Currents, US 1956
The Case Against Tomorrow, US 1957
Tomorrow Times Seven, US 1959
Drunkard's Walk, US 1960
The Man Who Ate The World, US 1960
Turn Left At Thursday, US 1961
A Plague of Pythons, US 1965 Digits and Dastards, US 1966 The Age of the Pussyfoot, US 1968
Day Million, US 1970
The Gold At The Starbow's End, US 1972
The Abominable Earthman, US 1973
_with C._ M. _Kornbluth_
The Space Merchants, US 1953
Search The Sky, US 1954
Gladiator-At-Law, US 1955
Wolfbane, US 1959
The Wonder Effect, US 1962
_with Jack Williamson_
Undersea Quest, US 1954
Undersea City, US 1958
The Reefs of Space, US 1964
Starchild, US 1965
Rogue Star, US 1969
_Awards_
Invisible Little Man Award, 1963
Edward E. Smith Memorial Award, 1964
Hugo : Best Editor, 1966–8
Hugo : Best Short Story (with C. M. Kornbluth) The Meeting, 1973
President, Science Fiction Writers of America, 1974
BRIAN W. ALDISS
_Novels and Short Stories_
The Brightfount Diaries, UK 1955
Space Time and Nathaniel, UK 1957
Non-Stop, UK 1958, as Starship, US 1958
Canopy of Time, UK 1959
Male Response, US 1959, UK 1961
Hothouse, UK 1962, as Long Afternoon of Earth, US 1962
Airs of Earth, UK 1963
Dark Light Years, UK 1964
Greybeard, UK 1964
Best, SF Stories of Brian W. Aldiss, UK 1965, as Who can Replace a Man?' US 1965, Rev. ed. 1971
An Age, UK 1967, as Cryptozoic !, US 1968
Report on Probability A, UK 1968, US 1969
Barefoot in the Head, UK 1969, US 1970
The Hand-Reared Boy, UK 1970, US 1970
A Soldier Erect, UK 1971, US 1971
The Moment of Eclipse, UK 1971, US 1972
Frankenstein Unbound, UK 1973, US 1974
The Eighty-Minute Hour, UK 1974, US 1974
_Non-Fiction_
Cities and Stones: A Traveller's Jugoslavia, UK 1966
The Shape of Further Things, UK & US 1970
Billion Year Spree : The History of Science Fiction, UK & US 1973
_Editor_
Penguin Science Fiction, UK 1961
More Penguin Science Fiction, UK 1962
Yet More Penguin Science Fiction, UK 1964
Penguin Science Fiction Omnibus, UK 1973
Best Fantasy Stories, UK 1962 Farewell Fantastic Venus, UK 1968
Best SF 1967–74 (with Harry Harrison), US & UK 1968–73
The Astounding-Analog Reader, Vols 1 & 2 (with Harry Harrison) US 1972–3, UK 1973
Space Opera, UK 1974
_Awards_
Hugo Special Citation : Most Promising New Author, 1958
Hugo : Hothouse, 1962
Nebula: The Saliva Tree, 1965
BSFA Vote : Britain's Most Popular SF Writer, 1969
Ditmar: World's Best Contemporary Science Fiction Author, 1970
BSFA : Moment of Eclipse, 1972
www.doverpublications.com
| {
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} | 7,937 |
Rosenstand is a Danish surname. Notable people with the surname include:
Peder Rosenstand-Goiske (1752–1803), Danish playwright and lawyer
Vilhelm Rosenstand (1838–1915), Danish painter and illustrator
Surnames of Danish origin | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,344 |
{"url":"https:\/\/gasturbinespower.asmedigitalcollection.asme.org\/article.aspx?articleid=1424483","text":"0\nTECHNICAL PAPERS: Gas Turbines: Combustion and Fuels\n\nCatalytic Combustion Systems for Microscale Gas Turbine Engines\n\n[+] Author and Article Information\n\nCenter for Micro and Nano Technology,\u00a0Lawrence Livermore National Laboratory, Livermore, CA 94551\n\nJ. Peck, I. A. Waitz\n\nGas Turbine Laboratory, Department of Aeronautics and Astronautics,\u00a0Massachusetts Institute of Technology, Cambridge, MA 02139\n\nJ. Eng. Gas Turbines Power 129(1), 49-60 (Sep 28, 2005) (12 pages) doi:10.1115\/1.2204980 History: Received August 25, 2005; Revised September 28, 2005\n\nAbstract\n\nAs part of an ongoing effort to develop a microscale gas turbine engine for power generation and micropropulsion applications, this paper presents the design, modeling, and experimental assessment of a catalytic combustion system. Previous work has indicated that homogenous gas-phase microcombustors are severely limited by chemical reaction timescales. Storable hydrocarbon fuels, such as propane, have been shown to blow out well below the desired mass flow rate per unit volume. Heterogeneous catalytic combustion has been identified as a possible improvement. Surface catalysis can increase hydrocarbon-air reaction rates, improve ignition characteristics, and broaden stability limits. Several radial inflow combustors were micromachined from silicon wafers using deep reactive ion etching and aligned fusion wafer bonding. The $191mm3$ combustion chambers were filled with platinum-coated foam materials of various porosity and surface area. For near stoichiometric propane-air mixtures, exit gas temperatures of $1100K$ were achieved at mass flow rates in excess of $0.35g\u2215s$. This corresponds to a power density of $\u223c1200MW\u2215m3$; an 8.5-fold increase over the maximum power density achieved for gas-phase propane-air combustion in a similar geometry. Low-order models, including time-scale analyses and a one-dimensional steady-state plug-flow reactor model, were developed to elucidate the underlying physics and to identify important design parameters. High power density catalytic microcombustors were found to be limited by the diffusion of fuel species to the active surface, while substrate porosity and surface area-to-volume ratio were the dominant design variables.\n\nFigures\n\nFigure 1\n\nBaseline engine schematic\n\nFigure 2\n\nSchematic (a) and SEM (b) of six-wafer microcombustor\n\nFigure 3\n\nComparison of hydrogen and propane gas-phase performance\n\nFigure 4\n\nSchematic of microcombustor showing location of catalyst material\n\nFigure 5\n\nPhotograph (a) and SEM (b) of 95% porous nickel foam substrate\n\nFigure 6\n\nPhotograph (a) and SEM (b) of 88.5% porous FeCrAlY foam substrate\n\nFigure 7\n\nPhotograph (a) and SEM (b) of 78% porous Inconel-625 foam substrate\n\nFigure 8\n\nFully packaged microcombustor\n\nFigure 9\n\nIgnition characteristics for catalytic microcombustors\n\nFigure 10\n\nExit gas temperature for microcombustor with noncatalytic foam\n\nFigure 11\n\nExit gas temperature plot comparing Ni-Pt and FeCrAlY-Pt devices for \u03d5=1\n\nFigure 12\n\nOverall combustor efficiency plot comparing Ni-Pt and FeCrAlY-Pt devices for \u03d5=1\n\nFigure 13\n\nEfficiency breakdown for catalytic microcombustor with Ni-Pt, \u03d5=1.0\n\nFigure 14\n\nTotal pressure loss plot comparing Ni-Pt and FeCrAlY devices for \u03d5=1\n\nFigure 15\n\nPressure loss versus mass flow rate for porous media, comparing estimates from Eq. 13 and experimental data\n\nFigure 16\n\nPeclet number versus diameter\n\nFigure 17\n\nAxial temperature profile in porous media plug flow reactor\n\nFigure 18\n\nAxial fuel concentration profile in porous media plug flow reactor\n\nFigure 19\n\nComparison of model to experiment\n\nFigure 20\n\nFuel conversion profiles for various surface area-to-volume ratios\n\nFigure 21\n\nOperating space for catalytic microcombustor; lines of constant power density\n\nFigure 22\n\nNondimensional operating space; Peclet number versus thermal efficiency\n\nErrata\n\nSome tools below are only available to our subscribers or users with an online account.\n\nRelated Content\n\nCustomize your page view by dragging and repositioning the boxes below.\n\nRelated Journal Articles\nRelated Proceedings Articles\nRelated eBook Content\nTopic Collections","date":"2019-04-25 20:45:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 4, \"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\": 0.24946817755699158, \"perplexity\": 14439.542703522737}, \"config\": {\"markdown_headings\": false, \"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-2019-18\/segments\/1555578733077.68\/warc\/CC-MAIN-20190425193912-20190425215912-00248.warc.gz\"}"} | null | null |
Theophil Johann Noack (* 20. September 1840 in Wusterbarth; † 31. März 1918 in Braunschweig) war ein deutscher Zoologe und Lehrer.
Leben und Wirken
Ab 1858 studierte Noack Naturgeschichte und Geographie an der Universität Halle und anschließend an der Universität Leipzig, wo er 1865 zum Doktor graduierte. Zwischen 1860 und 1874 war er Lehrer in einer Oberschule in Stettin und von 1874 bis 1911 war er Professor am Herzoglichen Real-Gymnasium in Braunschweig. Noacks Forschungsschwerpunkt war die afrikanische Säugetierfauna. 1906 beschrieb er den mysteriösen Afrikanischen Zwergelefanten (Loxodonta pumilio) auf der Basis eines Exemplars, das Carl Hagenbeck vom New Yorker Bronx Zoo erwarb. In der Folgezeit stellte sich jedoch heraus, dass Loxodonta pumilio lediglich ein kleinwüchsiges Exemplar des Waldelefanten (Loxodonta cyclotis) darstellt und somit kein valides Taxon ist. Weitere von Noack beschriebene Taxa sind Noacks Blattnasenfledermaus (Hipposideros ruber), der Kleinohr-Bilch (Graphiurus microtis), die Böhm-Nacktsohlenrennmaus (Gerbilliscus boehmi), die Kaiser-Buschratte (Aethomys kaiseri) und der Somali-Wildesel (Equus asinus somaliensis). 1894 stellte er die Gattung Dorcatragus für die im selben Jahr von Josef Menges als Oreatragus megalotis beschriebene Beira auf.
Viele von Noacks Artikeln wurden in den Reihen Zoologischer Anzeiger und Zoologische Jahrbücher veröffentlicht, die von der Deutschen Zoologischen Gesellschaft herausgegeben wurden.
Noack war mit Alzira geb. Fölzer, die am 8. Oktober 1857 in Porto Alegre in Brasilien geboren wurde, verheiratet. Aus dieser Ehe gingen die Kinder Ricarda und Hans hervor.
Artikel (Auswahl)
Pindari Carmen Nemeacum interpretatus est (Eine Interpretation über Pindars erste nemeische Ode), Cöstlin, 1867, S. 1–30 (Online)
Neues aus der Tierhandlung von Karl Hagenbeck, sowie aus dem Zoologischen Garten in Hamburg In: Der Zoologische Garten Nr. 3, 1884, S. 75
Über das zottelige Nashorn (Rhinoceros lasiotis) In: Der Zoologische Garten Nr. 3, 1884, S. 138–144
Beiträge zur Kenntniss der Säugetier-Fauna von Ost- und Central-Afrika In: Zoologische Jahrbücher, Band 2, Verlag von Gustav Fischer, 1887, S. 193–302 (Online)
Beiträge zur Kenntniss der Säugethierfauna von Süd- und Südwest-Afrika In: Zoologische Jahrbücher, Band 4, Verlag von Gustav Fischer, 1887, S. 94–96 (Online)
Beiträge zur Kenntnis der Säugetier-Fauna von Ostafrika, 1891 (Online)
Neue Beiträge zur Kenntniss der Säugethier-Fauna von Ostafrika, 1893 (Online)
Über die neue von Herrn J. Menges beschriebene Antilope des Somali-Landes, 1894 (Online)
Equus Przewalskii, 1902 (Online)
Ein neuer Hirsch aus der Dschungarei, 1902 (Online)
Die Entwicklung des Schädels von Equus Przewalskii, 1902 (Online)
Das Zebra vom Kilimandscharo, 1902 (Online)
Centralasiatische Steinböcke, 1902 (Online)
Zur Entwicklung von Equus Przewalskii, 1903 (Online)
Der Schädel von Capra Mengesi, 1903 (Online)
Zur Säugetierfauna des Tian-Schan, 1903 (Online)
Die Steinböcke des Altaigebietes, 1903 (Online)
Asiatische Bären der Arctos- und Tibetanus-Reihe, 1903 (Online)
Ein neuer Cephalophus, 1904 (Online)
Analyse der Herberstainschen Abbildungen des Ur und des Wisent, 1905 (Online)
Bären aus der Mongolei, 1905 (Online)
Eine Zwergform des afrikanischen Elefanten, 1906 (Online)
Literatur
Bo Beolens, Michael Watkins, Michael Grayson: The Eponym Dictionary of Mammals. JHU Press, 2009, ISBN 978-0-8018-9304-9, S. 294
Zoologe
Mammaloge
Deutscher
Geboren 1840
Gestorben 1918
Mann
Gymnasiallehrer | {
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Christchurch shooting video triggers lawsuit against Facebook and YouTube by French Muslim group
By Julie Carriat Reuters
2:11 Role of social media in Christchurch shootings
March 15: The mass murder in New Zealand was broadcast on a Facebook live stream. As Seán O'Shea reports, video of the shooting was available to millions worldwide to the horror of critics – Mar 15, 2019
One of the main groups representing Muslims in France said on Monday it was suing Facebook and YouTube, accusing them of inciting violence by allowing the streaming of footage of the Christchurch massacre on their platforms.
The French Council of the Muslim Faith (CFCM) said the companies had disseminated material that encouraged terrorism and harmed the dignity of human beings. There was no immediate comment from either company.
WATCH: March 16 — New Zealand shooting — PM says they've attempted to remove video of mosque shootings
2:25 New Zealand shooting: PM says they've attempted to remove video of mosque shootings
New Zealand shooting: PM says they've attempted to remove video of mosque shootings – Mar 16, 2019
The shooting at two mosques in New Zealand on March 15, which killed 50 people, was livestreamed on Facebook for 17 minutes and then copied and shared on social media sites across the internet.
Facebook said it raced to remove hundreds of thousands of copies.
However, footage could still be found on Facebook, Twitter and Alphabet Inc's YouTube a few hours after the attack, as well as Facebook-owned Instagram and Whatsapp.
READ MORE: New Zealand's inquiry into mosque shootings will look at social media, spy agencies and guns
Abdallah Zekri, president of the CFCM's Islamophobia monitoring unit, said the organization had launched a formal legal complaint against Facebook and YouTube in France.
Both companies have faced widespread criticism over the footage.
The chair of the U.S. House Committee on Homeland Security wrote a letter to top executives of four major technology companies last week urging them to do a better job of removing violent political content.
WATCH: March 24 — New Zealand PM announces Inquiry into Christchurch shootings
2:41 New Zealand PM announces Inquiry into Christchurch shootings
New Zealand PM announces Inquiry into Christchurch shootings – Mar 24, 2019
A spokesman for the Federation of Islamic Associations of New Zealand (FIANZ) welcomed the French group's action. He said his organization had been planning to contact Facebook to register their complaint but had been busy dealing with the aftermath of the attacks.
"They have failed big time, this was a person who was looking for an audience and … you were the platform he chose to advertise himself and his heinous crime," said FIANZ spokesman Anwar Ghani, referring to Facebook.
READ MORE: New Zealand shooting raises free speech debate after manifesto banned
"We haven't been in touch with the (French) group … but certainly something which can deter the social media space in terms of these types of crimes, we would be supportive of that," he said.
A man has been charged with one count of murder over the Christchurch shootings and will next appear in court on April 5.
© 2019 Thomson Reuters
Christchurch ShootingChristchurchchristchurch livestream lawsuitchristchurch shooting lawsuit facebookchristchurch shooting lawsuit youtubechristchurch shooting videochristchurch shooting video lawsuitfacebook christchurch shooting lawsuit
Iran urges Biden to rejoin nuclear deal, says Trump's policies 'completely failed'
'Very proud': India celebrates Kamala Harris' inauguration as U.S. vice president | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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> copy it to revert base.
better for some (many?) mailreaders.
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\section{Introduction\label{s-intro}}
\hskip\parindent
The main purpose of this article is to introduce and to investigate the
variable weak Hardy spaces on $\rn$. It is well known that
the classical weak Hardy spaces appear naturally in critical cases
of the study on the boundedness of operators.
Indeed, the classical
weak Hardy space $W\!H^1(\rn)$ was originally introduced by
Fefferman and Soria \cite{fs86} when they tried to find out
the biggest space from which the Hilbert transform
is bounded to the weak Lebesgue space $W\!L^1(\rn)$. Via
establishing the $\fz$-atomic characterization of $W\!H^1(\rn)$, they obtained
the boundedness of some Calder\'on-Zygmund
operators from $W\!H^1(\rn)$ to $W\!L^1(\rn)$.
Moreover, it is also well known that, when studying the boundedness of some
singular integral operators, $H^p(\rn)$ is a good substitute of
the Lebesgue space $L^p(\rn)$ with $p\in(0,1]$; while
when studying the boundedness of operators
in the critical case, the Hardy spaces $H^p(\rn)$ are usually further replaced
by the weak Hardy space $W\!H^p(\rn)$. For example, if $\delta\in(0,1]$ and $T$
is a convolutional $\delta$-type Calder\'on-Zygmund operator with $T^*(1)=0$, where
$T^*$ denotes the \emph{adjoint operator} of $T$, then $T$ is
bounded on $H^p(\rn)$ for all $p\in({n}/{(n+\delta)},1]$ (see \cite{am86}),
but may not be bounded on $H^{{n}/{(n+\delta)}}(\rn)$. For such an endpoint case,
Liu \cite{liu91} proved that $T$ is bounded from
$H^{{n}/{(n+\delta)}}(\rn)$ to $W\!H^{{n}/{(n+\delta)}}(\rn)$
via establishing the $\fz$-atomic characterization of the weak Hardy space $W\!H^p(\rn)$.
Furthermore, when studying the real interpolation between the Hardy space
$H^p(\rn)$ and the space $L^\fz(\rn)$, Fefferman et al. \cite{frs74} proved
that the weak Hardy spaces $W\!H^p(\rn)$ also naturally appear as the
intermediate spaces, which is another main motivation to
develop a real-variable theory of $W\!H^p(\rn)$. Recently, He \cite{h13}
and Grafakos and He \cite{gh14} further investigated
{vector-valued weak Hardy spaces} $H^{p,\fz}(\rn,\ell^2)$ with
$p\in(0,\fz)$.
Very recently, Liang et al. \cite{lyj} introduced a kind of generalized
weak Hardy spaces of Musielak-Orlicz type $W\!H^\varphi(\rn)$,
which covers both weak Hardy spaces
$W\!H^p(\rn)$ and weighted weak Hardy spaces $W\!H_w^p(\rn)$ from \cite{qy00}.
Various equivalent characterizations
of $W\!H^\varphi(\rn)$ by means of maximal functions, atoms, molecules
and Littlewood-Paley functions, and
the boundedness of Calder\'on-Zygmund operators in the critical case were obtained in \cite{lyj}.
For more related history and properties about $W\!H^p(\rn)$,
we refer the reader to \cite{at07,frs74,fs86,gh14,h13,liu91,lu95,qy00}
and their references.
On the other hand, based on the variable Lebesgue space,
several variable function spaces are developed
rapidly in recent years (see, for example, \cite{aa16,ah10,cw14,dhr09,ns12,
Xu081,yzy15,yzy151}).
Recall that the variable Lebesgue space $\lv$,
with a variable exponent function $p(\cdot):\ \rn\to(0,\fz)$,
is a generalization of the classical Lebesgue space
$L^p(\rn)$, via replacing the constant exponent $p$ by the exponent function
$p(\cdot)$, which consists of all functions $f$
such that $\int_{\rn}|f(x)|^{p(x)}\,dx<\fz$.
The study of variable Lebesgue spaces can be traced back to
Orlicz \cite{or31}, however, they
have been the subject of more intensive study since the early 1990s because
of their intrinsic interest for applications into harmonic analysis, partial
differential equations and variational integrals with nonstandard
growth conditions (see \cite{cfbook,dhr11,ins14} and their references).
Particularly, Nakai and Sawano \cite{ns12} introduced the variable
Hardy spaces $\hv$ and established their atomic
characterizations which were further applied to consider
their dual spaces and the boundedness of singular integral operators on $\hv$.
Later, in \cite{s10}, Sawano extended the atomic characterization
of $\hv$, which also improves the corresponding result in
\cite{ns12}, and gave out more applications of $\hv$, including the boundedness
of several operators on $\hv$. Moreover, Zhuo et al. \cite{zyl}
established equivalent characterizations of $\hv$ via
intrinsic square functions including the intrinsic
Lusin-area function, the intrinsic Littlewood-Paley $g$-function
or $g_\lz^*$-function.
Independently, Cruz-Uribe and Wang \cite{cw14}
also investigated the variable Hardy space $\hv$ with $p(\cdot)$
satisfying some conditions slightly weaker
than those used in \cite{ns12}. In \cite{cw14},
equivalent characterizations of $\hv$ by means of radial or non-tangential
maximal functions or atoms were established.
Very recently, in \cite{yzn15}, Yang et al.
characterized $\hv$ via Riesz transforms with $p(\cdot)$
satisfying the same conditions as in \cite{cw14}.
In this article, via combining some ideas from the theories
of the aforementioned classical weak Hardy spaces
and the variable Hardy spaces from \cite{ns12, cw14}, with the same assumptions
on $p(\cdot)$ as in Nakai and Sawano \cite{ns12}, we introduce and investigate
the variable weak Hardy spaces $\whv$. These spaces are first defined
via the radial grand maximal function
and then characterized by means of radial or non-tangential maximal
functions. Various equivalent characterizations of $\whv$ by means of atoms,
molecules and square functions, including the Lusin area function, the
Littlewood-Paley $g$-function and $g_\lz^*$-function, are also obtained.
As an application, we establish the boundedness of
convolutional $\delta$-type and non-convolutional
$\gz$-order Calder\'on-Zygmund operators from $\hv$ to $\whv$ including the critical
case when $p_-={n}/{(n+\delta)}$ or when $p_-=n/(n+\gamma)$, with $p_-$ as in \eqref{2.1x} below, which is of
special interest. These results further complete the theory
of variable Hardy-type spaces developed by Nakai and Sawano \cite{ns12} (see also
Cruz-Uribe and Wang \cite{cw14}).
To be precise, this article is organized as follows.
In Section \ref{s-pre}, we first recall some notation and notions, and state
some basic properties about variable Lebesgue spaces.
The variable weak Hardy space $\whv$ is also defined in this section
via the radial grand maximal function.
Section \ref{s-max} is devoted to characterizing the space $\whv$
by means of the radial maximal function corresponding to certain
Schwartz function or the non-tangential maximal function
corresponding to Poisson kernels (see Theorem \ref{mthm1} below).
To this end, we first establish a vector-valued inequality of
the Hardy-Littlewood maximal operator $\cm$ on the variable weak
Lebesgue space $\wlv$ in Proposition \ref{mlmveq} below.
Then, by borrowing some ideas from those used in the proofs of
\cite[p.\,91, Theorem 1]{stein93} and \cite[Theorem 2.1.4]{Gra14},
we give out the proof of Theorem \ref{mthm1}.
To prove Proposition \ref{mlmveq}, an interpolation theorem of
sublinear operators on the space $\wlv$ is obtained
(see Theorem \ref{mp1} below), which further induces the boundedness of
$\cm$ on $\wlv$ and may be of independent interest.
We point out that Proposition \ref{mlmveq} also plays an important
role in Section \ref{s-palay} when establishing
the Littlewood-Paley function characterizations of $\whv$.
In Section \ref{s-atom}, by borrowing some ideas from \cite{c77} and
a modified technic based on \cite{lyj},
we establish the atomic characterization of $\whv$.
Indeed, we first introduce the
variable weak atomic Hardy space
$\wha$ in Definition \ref{atd2} below and then
prove $\whv=\wha$ with equivalent quasi-norms (see Theorem \ref{atthm1} below).
To prove that $\wha$ is continuous embedded into $\whv$,
we mainly use a key lemma obtained by Sawano in
\cite[Lemma 4.1]{s10} (also restated as in Lemma 4.5 below), which reduces
some estimates related to $L^{p(\cdot)}(\rn)$ norms for some
series of functions into dealing with the $L^q(\rn)$ norms
of the corresponding functions,
and also the Fefferman-Stein vector-valued inequality of the Hardy-Littlewood
maximal operator $\cm$ on $\lv$
from \cite[Corollary 2.1]{cf06} (also restated as in Lemma 2.4 below).
The proof for the converse embedding is different from that used in the proof for
the corresponding embedding of variable Hardy spaces $H^{p(\cdot)}(\rn)$.
Recall that $L_\loc^1(\rn)\cap H^{p(\cdot)}(\rn)$
is dense in $H^{p(\cdot)}(\rn)$. Hence, to obtain an atomic
decomposition of any distribution $f\in H^{p(\cdot)}(\rn)$,
by a dense argument, it suffices to assume that
$f$ is a function in $L_\loc^1(\rn)$, which makes it
convenient to construct the desired atomic decomposition
(see \cite[Theorem 4.6]{ns12} and \cite[Theorem 7.1]{cw14}).
However, this standard procedure is invalid for the space $\whv$ due to
its lack of a dense function subspace.
To overcome this difficulty, we adopt a strategy used in \cite{lyj},
originated from Calder\'on \cite{c77},
to directly obtain an atomic decomposition of distributions in $\whv$
instead of some dense function subspace.
In Section \ref{s-mole}, we characterize the space $\whv$ via molecules
in Theorem \ref{mothm1} below.
Since each atom of $\whv$ is also a molecule of $\whv$,
due to Theorem \ref{atthm1}, to prove Theorem \ref{mothm1},
it suffices to show that the
variable weak molecular Hardy space
$W\!H_{\rm mol}^{p(\cdot),q,s,\epsilon}(\rn)$ is
continuously embedded into $\whv$.
To this end, the main step is to prove that a
$(p(\cdot),q,s,\epsilon)$-molecule
can be divided into an infinite linear combination of
$(p(\cdot),q,s)$-atoms. Here we use some ideas similar to those used in the proof of
\cite[Theorem 4.13]{hyy} (see also \cite{tg80}).
Section \ref{s-palay} is devoted to establishing some square function
characterizations of the space $\whv$, including characterizations via
the Lusin area function,
the Littlewood-Paley $g$-function or $g_\lz^\ast$-function,
respectively, in Theorems \ref{lpthm1}, \ref{lpthm2} and \ref{10.10.x} below.
We first prove Theorem \ref{lpthm1}, the Lusin area function
characterization of $\whv$, by borrowing some ideas from those
used in the proof of \cite[Theorem 4.5]{lyj} in which
Liang et al. established the Lusin area function characterization of
$W\!H^p(\rn)$ with $p\in(0,1]$ as a special case.
To obtain the Littlewood-Paley $g$-function characterization of $\whv$,
we make full use of an approach initiated by Ullrich \cite{u12}
and further developed by Liang et al. \cite{lsuyy},
which, via a key and technical lemma (see Lemma \ref{lm-12.7} below) and
an auxiliary function $g_{a,\ast}(f)$ (see \eqref{1.21-x} below), gives one
way to control the Littlewood-Paley $g$-function by the Lusin area function.
As an application of the space $\whv$, in Section \ref{s-bou},
we establish the boundedness of the convolutional $\delta$-type
and the non-convolutional $\gamma$-order Calder\'on-Zygmund operators
from $\hv$ to $\whv$ in the critical case when $p_-=n/{(n+\delta)}$ or when $p_-=n/(n+\gamma)$
(see Theorems \ref{bdnthm2}, respectively, \ref{bdnthm3} below).
In this case, any convolutional $\delta$-type or any non-convolution $\gamma$-order
Calder\'on-Zygmund operator may not be bounded
on $H^{p(\cdot)}(\rn)$ even when $p(\cdot)\equiv {\rm constant}\in(0,1]$.
In this sense, the space $\whv$ is a proper substitution of
$H^{p(\cdot)}(\rn)$ in the critical case for the study on the boundedness
of some operators,
which is one of the main motivations to study the variable weak Hardy space $\whv$.
We point out that the approach used in the proofs of Theorems \ref{bdnthm2} and \ref{bdnthm3}
is different from that of \cite[Theorem 1]{liu91}
(see also \cite[p.\,110, Theorem 4.2]{lu95})
in which the boundedness of the operator $T$ from the classical
Hardy space $H^{n/{(n+\delta)}}(\rn)$ to the weak
Hardy space $W\!H^{n/{(n+\delta)}}(\rn)$ was obtained,
and that of \cite[Theorem 5.2]{lyj} in which Liang et al.
established the boundedness of $T$ from the Musielak-Orlicz
Hardy space $H^{\vz}(\rn)$ to the weak Musielak-Orlicz
Hardy space $W\!H^{\vz}(\rn)$ in the critical case, where
$\vz:\ \rn\times [0,\fz)\to[0,\fz)$ is a Musielak-Orlicz growth
function satisfying that, for any given $x\in\rn$,
$\vz(x,\cdot)$ is an Orlicz function and $\vz(\cdot,t)$
is a Muckenhoupt $A_{1}(\rn)$ weight uniformly in $t\in(0,\fz)$
(see \cite[Definition 2.2]{lyj}).
Indeed, when $\vz(x,t):=t^p$ with $p\equiv {\rm constant}\in(0,\fz)$ for
all $x\in\rn$ and $t\in[0,\fz)$, or $\vz$ is as in \cite[Theorem 5.2]{lyj},
the fact that there exists a positive constant $C$ such that,
for any $\bz\in[1,\fz)$, $t\in(0,\fz)$ and any ball $B\st\rn$,
$$\int_{\bz B}\vz(x,t)\,dx\le C \bz^n \int_{ B}\vz(x,t)\,dx$$
plays a crucial role in the proofs of \cite[Theorem 5.2]{lyj} and also
\cite[p.\,110, Theorem 4.2]{lu95}. However, it may not be true
when $\vz(x,t):=t^{p(x)}$ for all $x\in\rn$ and $t\in(0,\fz)$ with
$p(\cdot)$ being a general variable exponent (see \cite[Remark 2.23]{yyz13}),
and hence the methods used in the proofs of \cite[Theorem 5.2]{lyj}
or \cite[p.\,110, Theorem 4.2]{lu95} are invalid in the present article.
To overcome this difficulty, we establish a weak-type
vector-valued inequality of the Hardy-Littlewood
maximal operator $\cm$ on $\lv$ for $p_-=1$ in
Proposition \ref{1.14.x3} below, via an extrapolation theorem
obtained in \cite[Theorem 5.24]{cfbook}.
These variable weak Hardy spaces $\whv$ might also be useful in the study
on the real interpolation between the variable Hardy spaces $\hv$, which is
the main subject of another forthcoming article, to limit the length of this article.
More applications of these variable weak Hardy spaces $\whv$ (for example, in the study
on the endpoint boundedness of operators) are expectable.
Finally, we make some conventions on notation. Let $\nn:=\{1,2,\dots\}$ and
$\zz_+:=\nn\cup\{0\}$. We denote by $C$ a \emph{positive constant}
which is independent of the main parameters,
but may vary from line to line. We use $C_{(\az,\dots)}$
to denote a positive constant depending on the indicated
parameters $\az,\, \dots$. The \emph{symbol}
$A\ls B$ means $A\le CB$. If $A\ls B$ and $B\ls A$, we then write $A\sim B$.
If $E$ is a subset of $\rn$, we denote by $\chi_E$ its
\emph{characteristic function} and by $E^\complement$
the set $\rn\backslash E$.
For all $r\in(0,\fz)$ and $x\in\rn$, denote by $B(x,r)$ the ball
centered at $x$ with the radius $r$, namely,
$B(x,r):=\{y\in\rn:\ |x-y|<r\}.$
For any ball $B$, we use $x_B$ to denote its center and $r_B$ its radius,
and denote by $\lz B$ for any $\lz\in(0,\fz)$ the ball concentric with
$B$ having the radius $\lz r_B$.
\section{Preliminaries\label{s-pre}}
\hskip\parindent
In this section, we aim to introduce the variable weak Hardy space via
the radial grand maximal function.
To this end, we first recall some notation and notions on
variable Lebesgue spaces and then state some of their basic conclusions
to be used in this article.
For an exposition of these concepts, we refer the reader to the monographs
\cite{cfbook,dhr11}.
\subsection{Variable Lebesgue spaces\label{s2.1}}
\hskip\parindent
A measurable function $p(\cdot):\ \rn\to(0,\fz)$ is called a
\emph{variable exponent}.
Denote by $\cp(\rn)$ the \emph{collection of all variable exponents}
$p(\cdot)$ satisfying
\begin{align}\label{2.1x}
0<p_-:=\mathop\mathrm{ess\,inf}_{x\in \rn}p(x)\le
\mathop\mathrm{ess\,sup}_{x\in \rn}p(x)=:p_+<\fz.
\end{align}
In what follows, for any $p(\cdot)\in\cp(\rn)$, we use $p^*(\cdot)$
to denote its \emph{conjugate variable exponent}, namely, for all $x\in\rn$,
$\frac{1}{p(x)}+\frac{1}{p^*(x)}=1$.
For a measurable function $f$ on $\rn$ and $p(\cdot)\in\cp(\rn)$,
the \emph{modular functional} (or, simply, the \emph{modular})
$\varrho_{p(\cdot)}$, associated with $p(\cdot)$, is defined by setting
$$\varrho_{p(\cdot)}(f):=\int_\rn|f(x)|^{p(x)}\,dx$$ and the
\emph{Luxemburg} (also known as the \emph{Luxemburg-Nakano})
\emph{quasi-norm} is given by setting
\begin{equation*}
\|f\|_{\lv}:=\inf\lf\{\lz\in(0,\fz):\ \varrho_{p(\cdot)}(f/\lz)\le1\r\}.
\end{equation*}
Then the \emph{variable Lebesgue space} $\lv$ is defined to be the
set of all measurable functions $f$ such that $\varrho_{p(\cdot)}(f)<\fz$,
equipped with the quasi-norm $\|f\|_{\lv}$.
\begin{rem}\label{r-vlp}
Let $p(\cdot)\in\cp(\rn)$.
\begin{enumerate}
\item[(i)] It is easy to see that, for all $s\in (0,\fz)$ and $f\in\lv$,
$$\lf\||f|^s\r\|_{\lv}=\|f\|_{L^{sp(\cdot)}(\rn)}^s.$$
Moreover, for all $\lz\in{\mathbb C}$ and $f,\ g\in\lv$,
$\|\lz f\|_{\lv}=|\lz|\|f\|_{\lv}$ and
$$\|f+g\|_{\lv}^{\underline{p}}\le \|f\|_{\lv}^{\underline{p}}
+\|g\|_{\lv}^{\underline{p}},$$
here and hereafter,
\begin{align}\label{2.1y}
\underline{p}:=\min\{p_-,1\}
\end{align}
with $p_-$ as in \eqref{2.1x}.
Particularly, when $p_-\in[1,\fz)$,
$\lv$ is a Banach space (see \cite[Theorem 3.2.7]{dhr11}).
\item[(ii)] For any non-trivial
function $f\in \lv$, by \cite[Proposition 2.21]{cfbook},
we know that $\varrho_{p(\cdot)}(f/\|f\|_{\lv})=1$
and, if $\|f\|_{\lv}\le1$, then $\varrho_{p(\cdot)}(f)\le\|f\|_{\lv}$
(see \cite[Corollary 2.22]{cfbook}).
\item[(iii)] If there exist $\delta,\,c\in(0,\fz)$ such that $\int_\rn[|f(x)|/\delta]^{p(x)}\,dx\le c$, then it is easy
to see that $\|f\|_{\lv}\le C\delta$, where $C$ is a positive constant
independent of $\delta$, but depending on $p_-$ (or $p_+$) and $c$.
\item[(iv)] If $p_+\in(0,1)$, then it is easy to see that,
for all non-negative functions $f,\ g\in\lv$, the following reverse
Minkowski inequality holds true:
$$\|f\|_{\lv}+\|g\|_{\lv}\le\|f+g\|_{\lv}.$$
\end{enumerate}
\end{rem}
A function $p(\cdot)\in\cp(\rn)$ is said to satisfy the
\emph{globally log-H\"older continuous condition}, denoted by $p(\cdot)\in C^{\log}(\rn)$,
if there exist positive constants $C_{\log}(p)$ and $C_\fz$, and
$p_\fz\in\rr$ such that, for all $x,\ y\in\rn$,
\begin{equation}\label{elog}
|p(x)-p(y)|\le \frac{C_{\log}(p)}{\log(e+1/|x-y|)}
\end{equation}
and
\begin{equation}\label{edecay}
|p(x)-p_\fz|\le \frac{C_\fz}{\log(e+|x|)}.
\end{equation}
For any measurable set $E\subset\rn$ and $r\in(0,\fz)$, let $L^r(E)$
be the set of all measurable functions $f$ such that
$$\|f\|_{L^r(E)}:=\lf[\int_E|f(x)|^r\,dx\r]^{1/r}<\fz.$$
For $r\in(0,\fz)$, denote by $L_{\rm loc}^r(\rn)$ the set of all
$r$-locally integrable functions
on $\rn$.
Recall that the \emph{Hardy-Littlewood maximal operator}
$\cm$ is defined by setting,
for all $f\in L_{\rm loc}^1(\rn)$ and $x\in\rn$,
\begin{align}\label{2.2x}
\cm(f)(x):=\sup_{B\ni x}\frac1{|B|}\int_B |f(y)|\,dy,
\end{align}
where the supremum is taken over all balls $B$ of $\rn$ containing $x$.
\begin{rem}\label{r-hlb}
Let $p(\cdot)\in C^{\log}(\rn)$ and $1<p_-\le p_+<\fz$.
For any $r\in[1,\fz)$, it is easy to see that $rp(\cdot)\in C^{\log}(\rn)$
and hence, for all $f\in L^{rp(\cdot)}(\rn)$,
$$\|\cm(f)\|_{L^{rp(\cdot)}(\rn)}\le C\|f\|_{L^{rp(\cdot)}(\rn)},$$
where $C$ is a positive constant independent of $f$
(see, for example, \cite[Theorem 3.16]{cfbook}).
\end{rem}
The following result is just \cite[Lemma 2.6]{zyl}
(For the case when $p_-\in(1,\fz)$, see also \cite[Corollary 3.4]{Iz10}).
\begin{lem}\label{zhuolemma}
Let $p(\cdot)\in C^{\log}(\rn)$.
Then there exists a positive constant $C$ such that, for all balls $B_1$, $B_2$
of $\rn$ with $B_1\subset B_2$,
$$C^{-1}\lf(\frac{|B_1|}{|B_2|}\r)^{\frac 1{p_-}}
\le\frac{\lf\|\chi_{B_1}\r\|_{L^{p(\cdot)}(\rn)}}
{\lf\|\chi_{B_2}\r\|_{L^{p(\cdot)}(\rn)}}
\le C\lf(\frac{|B_1|}{|B_2|}\r)^{\frac 1{p_+}}.$$
\end{lem}
The following Fefferman-Stein vector-valued inequality of
the maximal operator $\cm$ on the variable Lebesgue space $\lv$
was obtained in \cite[Corollary 2.1]{cf06}.
\begin{lem}\label{mlm1}
Let $r\in(1,\fz)$.
Assume that $p(\cdot)\in C^{\log}(\rn)$ satisfies $1<p_-\le p_+<\fz$.
Then there exists a positive
constant $C$ such that, for all
sequences $\{f_j\}_{j=1}^\fz$ of measurable functions,
$$\lf\|\lf\{\sum_{j=1}^\fz
\lf[\cm(f_j)\r]^r\r\}^{1/r}\r\|_{\lv}
\le C\lf\|\lf(\sum_{j=1}
^\fz|f_j|^r\r)^{1/r}\r\|_{\lv},$$
where $\cm$ denotes the Hardy-Littlewood maximal operator as in \eqref{2.2x}.
\end{lem}
\begin{rem}\label{2.5.y}
Let $p(\cdot)\in C^{\log}(\rn)$ and $\beta\in[1,\fz)$. Then, by Lemma \ref{mlm1}
and the fact that, for all balls $B\subset\rn$ and $r\in(0,\underline{p})$,
$\chi_{\beta B}\le\beta^{\frac{n}{r}}[\cm(\chi_B)]^{\frac{1}{r}}$,
we conclude that there exists a positive
constant $C$ such that, for any sequence $\{B_j\}_{j\in\nn}$ of balls of $\rn$,
$$\lf\|\sum_{j\in\nn}\chi_{\beta B_j}\r\|_{\lv}\le
C\beta^{\frac{n}{r}}\lf\|\sum_{j\in\nn}\chi_{B_j}\r\|_{\lv}.$$
\end{rem}
\subsection{Variable weak Hardy spaces $W\!H^{p(\cdot)}(\rn)$}
\hskip\parindent
In this subsection, we introduce the variable weak Hardy space
via the radial grand maximal function. To this end,
we first recall the definition of the variable weak Lebesgue space $\wlv$, which
is a special case of the variable Lorentz space
$L_{p(\cdot),q(\cdot)}(\rn)$ studied by Kempka and
Vyb\'iral in \cite{kv14}.
\begin{defn}
Let $p(\cdot)\in\cp(\rn)$.
The \emph{variable weak Lebesgue space} $W\!L^{p(\cdot)}(\rn)$
is defined to be the set of all measurable functions $f$ such that
\begin{align}\label{wvlp}
\|f\|_{WL^{p(\cdot)}(\rn)}:=\sup_{\az\in(0,\fz)}\az\lf
\|\chi_{\{x\in\rn:\ |f(x)|>\az\}}\r\|_{L^{p(\cdot)}(\rn)}<\fz.
\end{align}
\end{defn}
\begin{rem}
\begin{enumerate}
\item[(i)] We point out that the variable weak Lebesgue space
$W\!L^{p(\cdot)}(\rn)$ is a suitable substitute
of the variable Lebesgue space $\lv$ when studying the boundedness of
the Hardy-Littlewood maximal operator $\cm$ on $\lv$ when $p_-=1$.
Indeed, if $p_-=1$, then $\cm$ is not bounded on $\lv$
(see \cite[Theorem 3.19]{cfbook}).
However, if $p(\cdot)\in C^{\log}(\rn)$ with $p_-=1$, then $\cm$
is bounded from $\lv$ to $\wlv$ (see \cite[Theorem 3.16]{cfbook}).
\item[(ii)] As a special case of the variable Lorentz space
$L_{p(\cdot),q(\cdot)}(\rn)$ in \cite{kv14},
the space $\wlv$ also naturally appears when considering the
{real interpolation} between $\lv$ and $L^\fz(\rn)$.
More precisely, if $p(\cdot)\in\cp(\rn)$ and
$\theta\in(0,1)$, then it was proved by Kempka and Vyb\'iral in
\cite[Theorem 4.1]{kv14} that
$$(\lv,L^{\fz}(\rn))_{\theta,\fz}=WL^{\wz p(\cdot)}(\rn),$$
where $\frac1{\wz p(\cdot)}:=\frac{1-\theta}{p(\cdot)}$
and $(\lv,L^{\fz}(\rn))_{\theta,\fz}$ denotes the
real interpolation between $\lv$ and $L^{\fz}(\rn)$.
\end{enumerate}
\end{rem}
\begin{rem}\label{10.13.x}
In \cite{lyj}, Liang et al. introduced the \emph{weak Musielak-Orlicz space}
$W\!L^{\vz}(\rn)$, with $\vz:\ \rn\times[0,\fz)\to[0,\fz)$ being a Musielak-Orlicz
growth function (see \cite[Definition 2.2]{lyj}), which is defined as
the set of all measurable functions $f$ such that
$$\|f\|_{WL^\vz(\rn)}:=\inf\lf\{\lz\in(0,\fz):\ \sup_{\az\in(0,\fz)}
\int_{\{x\in\rn:\ |f(x)|>\az\}}\vz\lf(x,\frac{\az}{\lz}\r)\,dx\le1\r\}<\fz.$$
Here we claim that, for any $p(\cdot)\in\cp(\rn)$,
when $\vz(x,t):=t^{p(x)}$ for all $x\in\rn$ and $t\in(0,\fz)$,
$W\!L^\vz(\rn)=W\!L^{p(\cdot)}(\rn)$.
To see this, it suffices to show that, for any measurable function $f$,
\begin{align}\label{1.23x}
\|f\|_{\wlv}
&=\inf\lf\{\lz\in(0,\fz):\ \sup_{\az\in(0,\fz)}
\int_{\{x\in\rn:\ |f(x)|>\az\}}\lf(\frac{\az}{\lz}\r)^{p(x)}\,dx\le1\r\}\noz\\
&=:\|f\|_{\wlv}^\ast.
\end{align}
Indeed, for any $\az\in(0,\fz)$, let $E_\az:=\{x\in\rn:\ |f(x)|>\az\}$.
Then, by Remark \ref{r-vlp}(ii), we know that, for any $\az\in(0,\fz)$,
\begin{align*}
\int_{E_\az}\lf[\frac{\az}{\|f\|_{\wlv}}\r]^{p(x)}\,dx
&\le\int_{E_\az}\lf[\frac{\az}{\az\lf
\|\chi_{E_\az}\r\|_{\lv}}\r]^{p(x)}\,dx\\
&=\int_{E_\az}\lf[\frac{1}{\lf
\|\chi_{E_\az}\r\|_{\lv}}\r]^{p(x)}\,dx=1,
\end{align*}
which shows that $\|f\|_{\wlv}^\ast\le \|f\|_{\wlv}$.
On the other hand, for any $\az\in(0,\fz)$, we easily find that
\begin{align*}
\alpha\|\chi_{E_\alpha}\|_{\lv}
&=\inf\lf\{\lz\in(0,\fz):\ \int_{E_\alpha}\lf(\frac{\alpha}{\lz}\r)
^{p(x)}\,dx\le1\r\}\\
&\le\inf\lf\{\lz\in(0,\fz):\ \sup_{\alpha\in(0,\fz)}\int_{E_\alpha}\lf(\frac{\alpha}{\lz}\r)
^{p(x)}\,dx\le1\r\}=\|f\|_{\wlv}^\ast,
\end{align*}
which implies that $\|f\|_{\wlv}\le\|f\|_{\wlv}^\ast$.
Therefore, \eqref{1.23x} holds true. This finishes
the proof of the above claim.
\end{rem}
Next we present some properties of the variable weak Lebesgue space $\wlv$.
\begin{lem}\label{mlm2}
Let $p(\cdot)\in \cp(\rn)$. Then $\|\cdot\|_{\wlv}$
defines a quasi-norm on $\wlv$, namely,
\begin{enumerate}
\item[{\rm (i)}] $\|f\|_{\wlv}=0$ if and only if $f=0$ almost everywhere;
\item[{\rm (ii)}] for all $\lz\in\mathbb C$ and
$f\in\wlv$, $\|\lz f\|_{\wlv}=|\lz|\|f\|_{\wlv}$;
\item[{\rm (iii)}] for any $f,\ g\in\wlv$,
$$\|f+g\|_{\wlv}^{\underline{p}}\le 2^{\underline{p}}
\lf[\|f\|_{\wlv}^{\underline{p}}+\|g\|_{\wlv}^{\underline{p}}\r],$$
where $\underline{p}$ is as in \eqref{2.1y}.
\end{enumerate}
\end{lem}
\begin{proof}
We only give the proofs of (ii) and (iii), since
(i) is obviously true, the details being omitted.
To prove (ii), without loss of generality, we may assume that
$\lz\neq0$. By the definition of $\|\cdot\|_{\wlv}$ in \eqref{wvlp}, we have
\begin{align*}
\|\lz f\|_{\wlv}&=\sup_{\az\in(0,\fz)}\az\lf
\|\chi_{\{x\in\rn:\ |\lz f(x)|>\az\}}\r\|_{\lv}\\
&=|\lz|\sup_{\az\in (0,\fz)}\frac{\az}{|\lz|}\lf
\|\chi_{\{x\in\rn:\ |f(x)|>\frac{\az}{|\lz|}\}}\r\|_{\lv}
=|\lz|\|f\|_{\wlv}.
\end{align*}
Thus, (ii) holds true.
To show (iii), for any $f,\ g\in\wlv$, by \eqref{wvlp} and Remark \ref{r-vlp}(i),
we find that
\begin{align*}
&\|f+g\|_{\wlv}^{\underline{p}}\\
&\hs=\sup_{\az\in(0,\fz)}\az^{\underline{p}}\lf
\|\chi_{\{x\in\rn:\ |f(x)+g(x)|>\az\}}\r\|_{\lv}^{\underline{p}}\\
&\hs\le\sup_{\az\in(0,\fz)}\az^{\underline{p}}\lf[\lf
\|\chi_{\{x\in\rn:\ |f(x)|>\frac{\az}{2}\}}\r\|_{\lv}^{\underline{p}}
+\lf\|\chi_{\{x\in\rn:\ |g(x)|>\frac{\az}{2}\}}\r\|_{\lv}^{\underline{p}}\r]\\
&\hs\le\sup_{\az\in(0,\fz)}\az^{\underline{p}}\lf
\|\chi_{\{x\in\rn:\ |f(x)|>\frac{\az}{2}\}}\r\|_{\lv}^{\underline{p}}
+\sup_{\az\in(0,\fz)}\az^{\underline{p}}\lf\|
\chi_{\{x\in\rn:\ |g(x)|>\frac{\az}{2}\}}\r\|_{\lv}^{\underline{p}}\\
&\hs\le 2^{\underline{p}}\lf[\|f\|_{\wlv}^{\underline{p}}
+\|g\|_{\wlv}^{\underline{p}}\r],
\end{align*}
namely, (iii) holds true. This finishes the proof of Lemma \ref{mlm2}.
\end{proof}
\begin{rem}\label{r-ar}
Let $p(\cdot)\in\cp(\rn)$.
Then, by the Aoki-Rolewicz theorem (see \cite{ta42, sr57}
and also \cite[Exercise 1.4.6]{g09-1}),
we find that there exists a positive constant $v\in(0,1)$
such that, for all $R\in\nn$ and $\{f_j\}_{j=1}^R$,
$$\lf\|\sum_{j=1}^R |f_j|\r\|_{\wlv}^{v}
\le 4\lf[\sum_{j=1}^R \|f_j\|_{\wlv}^{v}\r].$$
\end{rem}
\begin{lem}\label{mlmim}
Let $p(\cdot)\in \cp(\rn)$.
Then, for all $f\in\wlv$ and $s\in(0,\fz)$, it holds true that
$$\lf\||f|^s\r\|_{\wlv}=\|f\|_{WL^{sp(\cdot)}(\rn)}^s.$$
\end{lem}
\begin{proof}
By \eqref{wvlp} and Remark \ref{r-vlp}(i), we find that
\begin{align*}
\lf\||f|^s\r\|_{\wlv}&=\sup_{\az\in(0,\fz)}\az\lf\|
\chi_{\{x\in\rn:\ |f(x)|^s>\az\}}\r\|_{\lv}\\
&=\sup_{\bz\in(0,\fz)}\bz^s\lf\|
\chi_{\{x\in\rn:\ |f(x)|>\bz\}}\r\|_{\lv}\\
&=\sup_{\bz\in(0,\fz)}\bz^s\lf\|
\chi_{\{x\in\rn:\ |f(x)|>\bz\}}\r\|_{L^{sp(\cdot)}(\rn)}^s
=\|f\|_{WL^{sp(\cdot)}(\rn)}^s.
\end{align*}
This finishes the proof of Lemma \ref{mlmim}.
\end{proof}
From the Fatou lemma of $L^{p(\cdot)}(\rn)$ (see \cite[Theorem 2.61]{cfbook}),
we easily deduce the following Fatou lemma of $\wlv$, the details being omitted.
\begin{lem}\label{10.24.x1}
Let $p(\cdot)\in\cp(\rn)$ and $\{f_k\}_{k\in\nn}\subset\wlv$.
If $f_k\to f$ as $k\to\fz$ pointwise almost everywhere in $\rn$ and
$\liminf_{k\to\fz}\|f_k\|_{\wlv}$ is finite,
then $f\in\wlv$ and
$$\|f\|_{\wlv}\le\liminf_{k\to\fz}\|f_k\|_{\wlv}.$$
\end{lem}
In what follows, denote by $\cs(\rn)$ the \emph{space of all
Schwartz functions} and $\cs'(\rn)$ its \emph{topological dual space}
equipped with the weak-$*$ topology.
For any $N\in\nn$, let
\begin{align}\label{2.4x}
\cf_N(\rn):=\lf\{\psi\in\cs(\rn):\ \sum_{\bz\in\zz_+^n,\,|\bz|\le N}
\sup_{x\in\rn}\lf[(1+|x|)^N|D^\bz\psi(x)|\r]\le1\r\},
\end{align}
where, for any $\bz:=(\bz_1,\dots,\bz_n)\in\zz_+^n$,
$|\bz|:=\bz_1+\cdots+\bz_n$ and
$D^\bz:=(\frac\partial{\partial x_1})^{\bz_1}
\cdots(\frac\partial{\partial x_n})^{\bz_n}$.
Then, for all $f\in\cs'(\rn)$, the \emph{radial grand maximal function}
$f^\ast_{N,+}$
of $f$ is defined by setting, for all $x\in\rn$,
\begin{equation}\label{2.8x}
f_{N,+}^\ast(x):=\sup\lf\{|f\ast\psi_t(x)|:\
t\in(0,\fz)\ {\rm and}\ \psi\in\cf_N(\rn)\r\},
\end{equation}
where, for all $t\in(0,\fz)$ and $\xi\in\rn$,
$\psi_t(\xi):=t^{-n}\psi(\xi/t)$.
Now we introduce the variable weak Hardy space.
\begin{defn}
Let $p(\cdot)\in C^{\log}(\rn)$ and
$N\in(\frac{n}{\underline{p}}+n+1,\fz)$ be a positive integer,
where $\underline{p}$ is as in \eqref{2.1y}.
The \emph{variable weak Hardy space} $\whv$ is defined to be the set of all
$f\in\cs'(\rn)$ such that
$f_{N,+}^\ast\in\wlv$, equipped with the {quasi-norm}
$$\|f\|_{\whv}:=\|f_{N,+}^\ast\|_{\wlv}.$$
\end{defn}
\begin{rem}\label{1.29.x2}
\begin{enumerate}
\item[(i)] If $p(\cdot)\equiv p\in(0,\fz)$,
then the space $\whv$ is just the classical weak Hardy space
$W\!H^p(\rn)$ studied in \cite{frs74,fs86,liu91}. By Theorem \ref{mthm1}
below, we find that the space $\whv$ is independent of the choice of
$N\in(\frac{n}{\underline{p}}+n+1,\fz)$.
\item[(ii)] Very recently, Almeida et al. \cite{abr16} introduced the
variable anisotropic Hardy-Lorentz spaces on
$\rn$ associated with an expansive matrix $A$,
$H^{p(\cdot),q(\cdot)}(\rn, A)$, via the variable Lorentz spaces
$\cl^{p(\cdot),q(\cdot)}(\rn)$ in \cite{eks08}, where
$$p(\cdot),\ q(\cdot):\ (0,\fz)\to(0,\fz)$$
are bounded measurable functions. As was pointed out in
\cite[Remark 2.6]{kv14}, the space $\cl^{p(\cdot),q(\cdot)}(\rn)$
in \cite{eks08} never goes back to the space $\lv$,
since the variable exponent $p(\cdot)$ in $\cl^{p(\cdot),q(\cdot)}(\rn)$
is only defined on $(0,\fz)$ but not on $\rn$.
On the other hand, the space $\whv$ introduced in this article
is defined via the variable Lorentz space
$L_{p(\cdot),q(\cdot)}(\rn)$ from \cite{kv14} but with $q(\cdot)\equiv\fz$,
which is not covered by the space $H^{p(\cdot),q(\cdot)}(\rn, A)$
in \cite{abr16}.
Moreover, as was mentioned in \cite[p.\,5]{abr16},
the main procedure of \cite{abr16} requires the fact that
the set $L_\loc^1(\rn)\cap H^{p(\cdot),q(\cdot)}(\rn, A)$
is dense in $H^{p(\cdot),q(\cdot)}(\rn, A)$.
Thus, the method used in \cite{abr16} does not work for
$\whv$ in the present article,
due to the lack of a dense function subspace of $\whv$ even when
$p(\cdot)\equiv {\rm constant}\in(0,\fz)$.
\item[(iii)] Recall that Liang et al. \cite{lyj} introduced the weak
Musielak-Orlicz Hardy space $W\!H^\varphi(\rn)$ with a
Musielak-Orlicz function
$\varphi:\ \rn\times[0,\fz)\rightarrow[0,\fz)$. Observe that, when
\begin{equation}\label{liang1}
\vz(x,t):=t^{p(x)}\quad\mathrm{for\ all}\quad x\in\rn\quad\mathrm{and}\quad
t\in (0,\fz),
\end{equation}
then $W\!H^\varphi(\rn)=\whv$ (see also Remark \ref{10.13.x}).
However, a general Musielak-Orlicz
function $\varphi$ satisfying all the assumptions
in \cite{lyj} may not have the form as in \eqref{liang1}.
On the other hand, it was proved in \cite{yyz13} that
there exists a variable exponent function $p(\cdot)$
satisfying \eqref{elog} and \eqref{edecay} which were required in this article,
but $t^{p(\cdot)}$ is not a uniformly Muckenhoupt
weight which was required in \cite{lyj}. Thus, the weak Musielak-Orlicz
Hardy space $W\!H^\varphi(\rn)$ in \cite{lyj} and the variable weak Hardy space $\whv$ in this article can not cover each other.
\end{enumerate}
\end{rem}
\section{Maximal function characterizations of $\whv$\label{s-max}}
\hskip\parindent
In this section, we aim to characterize $W\!H^{p(\cdot)}(\rn)$
via radial or non-tangential maximal functions.
To this end, we first establish an interpolation theorem on $\wlv$ in
Subsection \ref{s3.1}. Via applying such an interpolation theorem,
the maximal function characterizations of $\whv$ are
obtained in Subsection \ref{s3.2}.
\subsection{An interpolation theorem on $\wlv$\label{s3.1}}
\hskip\parindent
In what follows, let $\mathscr{M}_0(\rn)$ be the linear space of
all almost everywhere finite measurable functions on $\rn$.
Let $T$ be an operator defined on $\mathscr{M}_0(\rn)$. Then $T$ is
called a \emph{sublinear operator} if,
for all $f,\ g\in \mathscr{M}_0(\rn)$ and all $\lz\in\cc$,
$$|T(f+g)|\le |T(f)|+|T(g)|\quad
{\rm and}\quad |T(\lz f)|=|\lz||T(f)|.$$
For $q_1(\cdot),\ q_2(\cdot)\in\cp(\rn)$, let
$$L^{q_1(\cdot)}(\rn)+L^{q_2(\cdot)}(\rn)
:=\lf\{f\in \mathscr{M}_0(\rn):
\ f=f_1+f_2,\ f_k\in L^{q_k(\cdot)}(\rn),\ k\in\{1,2\}\r\}.$$
The main result of this section is stated as follows.
\begin{thm}\label{mp1}
Let $p(\cdot)\in \cp(\rn)$ with $1<p_-\le p_+<\fz$,
$p_1\in(\frac{1}{p_-},1)$ and $p_2\in(1,\fz)$,
where $p_-$ and $p_+$ are as in \eqref{2.1x}.
Assume that $T$ is a sublinear operator defined on
$L^{p_1p(\cdot)}(\rn)+L^{p_2p(\cdot)}(\rn)$ satisfying that
there exist positive constants $C_1$ and $C_2$ such that,
for all $i\in\{1,2\}$, $f\in L^{p_ip(\cdot)}(\rn)$ and $\beta\in(0,\fz)$,
\begin{equation}\label{interp-1}
\beta\lf\|\chi_{\{x\in\rn:\ |T(f)(x)|>\beta\}}\r\|_{L^{p_ip(\cdot)}(\rn)}
\le C_i\|f\|_{L^{p_ip(\cdot)}(\rn)}.
\end{equation}
Then $T$ is bounded on $\wlv$ and there exists a positive constant
$C$ such that, for all $f\in \wlv$,
\begin{equation*}
\|T(f)\|_{\wlv}\le C\|f\|_{\wlv}.
\end{equation*}
\end{thm}
To prove Theorem \ref{mp1}, we need the following lemma, whose proof is quite easy,
the details being omitted.
\begin{lem}\label{r-2.1x}
Let $p(\cdot)\in \cp(\rn)$.
Then, for any $t\in(0,\fz)$ and $x\in\rn$, it holds true that
$$t^{p(x)}\sim\int_0^tr^{p(x)}\frac{dr}{r},$$
where the implicit equivalent positive
constants are independent of $t$ and $x$.
\end{lem}
\begin{proof}[Proof of Theorem \ref{mp1}]
Let $f\in \wlv$ and
$$\lz:=\|f\|_{\wlv}=\sup_{\bz\in(0,\fz)}\bz
\lf\|\chi_{\{x\in\rn:\ |f(x)|>\bz\}}\r\|_{\lv}.$$
Then, by Remark \ref{r-vlp}(ii), we easily know that, for all $\bz\in(0,\fz)$,
\begin{align}\label{3.1x}
\int_{\{x\in\rn:\ |f(x)|>\bz\}}\lf(\frac{\bz}{\lz}\r)^{p(x)}\,dx\le1.
\end{align}
Next we show that, for all $\az\in(0,\fz)$,
\begin{equation*}
\az\lf\|\chi_{\{x\in\rn:\ \lf|T(f)(x)\r|>\az\}}\r\|_{\lv}\ls\lz
\end{equation*}
with the implicit positive constant independent of $\az$ and $f$.
To this end, for any $\az\in(0,\fz)$, let
$$f_{\az,1}:=f\chi_{\{x\in\rn:\ |f(x)|>\az\}}\ \ \ \mbox{and}\ \ \ f_{\az,2}
:=f\chi_{\{x\in\rn:\ |f(x)|\le\az\}}.$$
We claim that, for $i\in\{1,2\}$,
\begin{align}\label{emb}
\int_{\rn}\lf[\frac{\lf|
f_{\az,i}(x)\r|/{\az}}{(\lz/{\az})
^{1/{p_i}}}\r]^{p_ip(x)}\,dx\ls 1.
\end{align}
Assuming that this claim holds true for the time being, then,
by Remark \ref{r-vlp}(iii), we find that,
for $i\in\{1,2\}$,
$$\lf\|f_{\az,i}/{\az}\r\|_{L^{p_ip(\cdot)}(\rn)}
\ls (\lz/{\az})^{1/{p_i}},$$
which shows that $f_{\az,i}\in L^{p_ip(\cdot)}(\rn)$
and ${\az}^{1-p_i}\lf\|f_{\az,i}\r\|_
{L^{p_ip(\cdot)}(\rn)}^{p_i}\ls\lz<\fz$.
From this and the fact that $T$ is sublinear, Remark \ref{r-vlp}(i)
and \eqref{interp-1}, we deduce that, for any $\az\in(0,\fz)$,
\begin{align*}
&\az\lf\|\chi_{\{x\in\rn:\ |T(f)(x)|>\az\}}\r\|_{\lv}\\
&\hs\ls\az\lf\|\chi_{\{x\in\rn:\
|T(f_{\az,1})(x)|>\az/2\}}\r\|_{\lv}+
\az\lf\|\chi_{\{x\in\rn:\
|T(f_{\az,2})(x)|>\az/2\}}\r\|_{\lv}\\
&\hs\sim\sum_{i=1}^2\az\lf\|\chi_{\{x\in\rn:\
|T(f_{\az,i})(x)|>\az/2\}}\r\|_{L^{p_ip(\cdot)}(\rn)}^{p_i}
\ls\sum_{i=1}^2{\az}^{1-p_i}\lf\|f_{\az,i}\r\|
_{L^{p_ip(\cdot)}(\rn)}^{p_i}\ls\lz.
\end{align*}
This further implies that $\|T(f)\|_{\wlv}\ls\|f\|_{\wlv}$,
which is the desired conclusion.
Therefore, to complete the proof of Theorem \ref{mp1},
it remains to prove the above claim.
To this end, when $i=1$,
by Lemma \ref{r-2.1x} and $p(x)\ge p_->1$ for almost every $x\in\rn$, we have
\begin{align}\label{emd}
&\int_{\rn}\lf[\frac{\lf|f_{\az,1}(x)\r|/{\az}}
{(\lz/{\az})^{1/{p_1}}}\r]^{p_1p(x)}\,dx\noz\\
&\hs=\int_{\rn}\frac{1}{p(x)}\int_0^\frac{[|f_{\az,1}(x)|/
{\az}]^{p_1}}{\lz/{\az}}t^{p(x)}\frac{dt}{t}\,dx
\le \frac{1}{p_-}\int_0^{\fz}\int
_{\{x\in\rn:\ [|f_{\az,1}(x)|/{\az}]
^{p_1}>\frac{t\lz}{\az}\}}
t^{p(x)}\frac{dx\,dt}{t}\noz\\
&\hs\le\int_0^{\az/\lz}
\int_{\{x\in\rn:\ [|f_{\az,1}(x)|
/{\az}]^{p_1}>\frac{t\lz}
{\az}\}}t^{p(x)}\frac{dx\,dt}{t}
+\int_{\az/\lz}^{\fz}\int_{\{x\in\rn:\
[|f_{\az,1}(x)|/{\az}]
^{p_1}>\frac{t\lz}{\az}\}}\cdots\noz\\
&\hs=:{\rm I}_1+{\rm I}_2.
\end{align}
By the definition of $f_{\az,1}$ and \eqref{3.1x}, we conclude that
\begin{align}\label{emd1}
{\rm I}_1&\le\int_0^{\az/\lz}
\int_{\{x\in\rn:\ |f(x)|>\az\}}\lf(\frac{\az}
{\lz}\r)^{p(x)}\lf(\frac{t\lz}{\az}\r)
^{p_-}\frac{dx\,dt}{t}\noz\\
&=\int_0^{\az/\lz}\lf(\frac{t\lz}{\az}\r)
^{p_-}\frac{dt}{t}
\int_{\{x\in\rn:\ |f(x)|>\az\}}\lf(\frac{\az}
{\lz}\r)^{p(x)}\,dx\noz\\
&\le\int_0^{\az/\lz}\lf(\frac{t\lz}{\az}\r)
^{p_-}\,\frac{dt}{t}\sim 1.
\end{align}
On the other hand, from the fact that $p_1\in(\frac{1}{p_-},1)$,
the definition of $f_{\az,1}$ and \eqref{3.1x} again, we deduce that
\begin{align}\label{emd2}
\hs{\rm I}_2&\le\int_{\az/\lz}^
{\fz}\int_{\{x\in\rn:\ |f_{\az,1}(x)|
>\az\lf(\frac{t\lz}{\az}\r)^{1/{p_1}}\}}
\lf[\frac{\az}{\lz}\lf
(\frac{t\lz}{\az}\r)^{1/{p_1}}\r]^{p(x)}
\lf(\frac{t\lz}{\az}\r)^
{(1-1/p_1)p_-}\,dx\frac{dt}{t}\noz\\
&\le\int_{\az/\lz}^{\fz}
\lf(\int_{\{x\in\rn:\ |f(x)|>\az\lf(\frac{t\lz}{\az}\r)^{1/{p_1}}\}}
\lf[\frac{\az}{\lz}\lf
(\frac{t\lz}{\az}\r)^{1/{p_1}}\r]^{p(x)}\,dx\r)
\lf(\frac{t\lz}{\az}\r)^
{(1-1/p_1)p_-}\frac{dt}{t}\noz\\
&\le\int_{\az/\lz}^
{\fz}\lf(\frac{t\lz}{\az}\r)^
{(1-1/p_1)p_-}\,\frac{dt}{t}
\sim 1.
\end{align}
Thus, by \eqref{emd}, \eqref{emd1}
and \eqref{emd2}, we obtain \eqref{emb} when $i=1$.
When $i=2$, by Lemma \ref{r-2.1x} and $p(x)\ge p_->1$ for almost every $x\in\rn$, we know that
\begin{align}\label{eme}
&\int_{\rn}\lf[\frac{|f_{\az,2}(x)|/{\az}}
{(\lz/{\az})^{1/{p_2}}}\r]^{p_2p(x)}\,dx\noz\\
&\hs=\int_{\rn}\frac1{p(x)}\int_0^
\frac{[|f_{\az,2}(x)|
/{\az}]^{p_2}}
{\lz/{\az}}t^{p(x)}\,\frac{dt}{t}\,dx
\le \int_0^{\fz}\int_
{\{x\in\rn:\ [|f_{\az,2}(x)|
/{\az}]^{p_2}>\frac{t\lz}{\az}\}}
t^{p(x)}\,dx\,\frac{dt}{t}\noz\\
&\hs=\int_0^{\az/\lz}\int_{\{x\in\rn:\
[|f_{\az,2}(x)|/{\az}]
^{p_2}>\frac{t\lz}{\az}\}}t^{p(x)}\,dx\,\frac{dt}{t}
+\int_{\az/\lz}^{\fz}\int_{\{x\in\rn:\
[|f_{\az,2}(x)|/{\az}]
^{p_2}>\frac{t\lz}{\az}\}}\cdots\noz\\
&\hs=:{\rm II}_1+{\rm II}_2.
\end{align}
By the definition of $f_{\az,2}$, \eqref{3.1x} and the fact that
$p_2\in(1,\fz)$, we find that
\begin{align}\label{eme1}
\hs\hs{\rm II}_1&\le\int_0^{\az/\lz}
\int_{\{x\in\rn:\ |f_{\az,2}(x)|
>\az\lf(\frac{t\lz}{\az}\r)^{1/{p_2}}\}}
\lf[\frac{\az}{\lz}\lf(\frac{t\lz}{\az}\r)^{1/{p_2}}\r]
^{p(x)}\lf(\frac{t\lz}{\az}\r)^
{(1-1/p_2)p_-}\,dx\,\frac{dt}{t}\noz\\
&\le\int_0^{\az/\lz}
\lf(\int_{\{x\in\rn:\ |f(x)|>\az\lf(\frac{t\lz}{\az}\r)^{1/{p_2}}\}}
\lf[\frac{\az}{\lz}\lf(\frac{t\lz}{\az}\r)^{1/{p_2}}\r]
^{p(x)}\,dx\r)\lf(\frac{t\lz}{\az}\r)^
{(1-1/p_2)p_-}\,\frac{dt}{t}\noz\\
&\ls\int_0^{\az/\lz}\lf(\frac{t\lz}{\az}\r)
^{(1-1/p_2)p_-}\,\frac{dt}{t}
\sim 1.
\end{align}
Observe that, when $t\in(\frac{\az}{\lz}, \fz)$,
$(|f_{\az,2}|/{\az})^{p_2}<1<\frac{t\lz}{\az}$
and hence ${\rm II}_2=0$.
From this, \eqref{eme} and \eqref{eme1},
we deduce that \eqref{emb} holds true when $i=2$,
which implies that the above claim \eqref{emb}
holds true.
This finishes the proof of Theorem \ref{mp1}.
\end{proof}
As a simple consequence of Theorem \ref{mp1} and Remark \ref{r-hlb},
we immediately obtain the following boundedness of $\cm$ on $\wlv$,
which is of independent interest, the details being omitted.
\begin{cor}\label{mc1}
Let $p(\cdot)\in C^{\log}(\rn)$ satisfy $1<p_-\le p_+<\fz$, where
$p_-$ and $p_+$ are as in \eqref{2.1x}.
Then the Hardy-Littlewood maximal operator $\cm$ is bounded on $\wlv$.
\end{cor}
Moreover, using Theorem \ref{mp1} and Lemma \ref{mlm1}, we now establish the
following vector-valued inequality of the Hardy-Littlewood maximal
operator $\cm$ on variable weak Lebesgue spaces.
\begin{prop}\label{mlmveq}
Let $p(\cdot)\in C^{\log}(\rn)$ satisfy $1<p_-\le p_+<\fz$,
with $p_-$ and $p_+$ as in \eqref{2.1x}, and $r\in(1,\fz)$.
Then there exists a positive
constant $C$ such that, for all sequences $\{f_j\}_{j\in\nn}$
of measurable functions,
$$\lf\|\lf\{\sum_{j\in\nn}\lf[\cm (f_j)\r]^r\r\}^{1/r}\r\|_{\wlv}
\le C\lf\|\lf(\sum_{j\in\nn}|f_j|^r\r)^{1/r}\r\|_{\wlv},$$
where $\cm$ denotes the Hardy-Littlewood maximal operator as in \eqref{2.2x}.
\end{prop}
\begin{proof}
To prove this proposition, let $\{f_j\}_{j\in\nn}$ be a given arbitrary sequence of
measurable functions and, for any measurable function
$g$ and $x\in\rn$, define
$$A(g)(x):=\lf\{\sum_{j\in\nn}
\lf[\cm(g\eta_j)(x)\r]^r\r\}^{\frac{1}{r}},$$
where $r\in(1,\fz)$ and, for any $y\in\rn$,
$$\eta_j(y):=\frac{f_j(y)}
{[\sum_{j\in\nn}|f_j(y)|^r]^{1/r}}\quad
{\rm if}\quad \lf[\sum_{j\in\nn}\lf|f_j(y)\r|^r\r]^{1/r}\neq0,$$
and $\eta_j(y):=0$ otherwise.
Then, by the Minkowski inequality, we find that, for any measurable functions
$g_1$, $g_2$ and $x\in\rn$,
\begin{align*}
&A(g_1+g_2)(x)\\
&\hs=\lf\{\sum_{j\in\nn}\lf[
\cm\lf([g_1+g_2]\eta_j\r)(x)\r]^r\r\}^{\frac{1}{r}}
\le\lf\{\sum_{j\in\nn}\lf[\cm(g_1\eta_j)(x)+
\cm(g_2\eta_j)(x)\r]^r\r\}^{\frac{1}{r}}\\
&\hs\le\lf\{\sum_{j\in\nn}
\lf[\cm(g_1\eta_j)(x)\r]^r\r\}^{\frac{1}{r}}
+\lf\{\sum_{j\in\nn}\lf[\cm(g_2\eta_j)(x)\r]^r\r\}
^{\frac{1}{r}}
=A(g_1)(x)+A(g_2)(x).
\end{align*}
Thus, $A$ is sublinear. Moreover, by Lemma \ref{mlm1},
we know that, for any $p_1\in(\frac{1}{p_-},1)$, $p_2\in(1,\fz)$ and
measurable function $h$,
\begin{align*}
\lf\|A(h)\r\|_{L^{p_ip(\cdot)}(\rn)}
&=\lf\|\lf\{\sum_{j\in\nn}\lf[\cm(h\eta_j)\r]^r\r\}^{\frac{1}{r}}\r\|
_{L^{p_ip(\cdot)}(\rn)}\\
&\ls\lf\|\lf(\sum_{j\in\nn}|h\eta_j|^r\r)^{\frac{1}{r}}\r\|
_{L^{p_ip(\cdot)}(\rn)}
\sim\|h\|_{L^{p_ip(\cdot)}(\rn)},
\end{align*}
which implies that the operator $A$ is bounded on
$L^{p_ip(\cdot)}(\rn)$, where $i\in\{1,2\}$.
Now, letting $g:=(\sum_{j=1}^\fz|f_j|^r)^{1/r}$,
then, by Theorem \ref{mp1}, we conclude that
\begin{align*}
\lf\|\lf\{\sum_{j\in\nn}\lf[\cm(f_j)\r]^r\r\}^{\frac{1}{r}}\r\|_{\wlv}
&=\lf\|A(g)\r\|_{\wlv}\\
&\ls\|g\|_{\wlv}
\sim\lf\|\lf(\sum_{j\in\nn}\lf|f_j\r|^r\r)^{1/r}\r\|_{\wlv},
\end{align*}
which completes the proof of Proposition \ref{mlmveq}.
\end{proof}
\subsection{Maximal function characterizations of $\whv$\label{s3.2}}
\hskip\parindent
We begin with the following definitions of the radial maximal
function and the non-tangential maximal function.
\begin{defn}
Let $\psi\in\cs(\rn)$ and $\int_\rn \psi(x)\,dx\neq0$.
Let $f\in\cs'(\rn)$.
The \emph{radial maximal function} of $f$
associated to $\psi$ is defined by setting, for all $x\in\rn$,
\begin{align}\label{3.5x}
\psi_{+}^\ast(f)(x):=\sup_{t\in(0,\fz)}\lf|f*\psi_t(x)\r|
\end{align}
and, for any $a\in(0,\fz)$, the \emph{non-tangential maximal function} of $f$
associated to $\psi$ is defined by setting, for all $x\in\rn$,
$$\psi_{\triangledown,a}^\ast(f)(x)
:=\sup_{t\in(0,\fz),\,|y-x|<at}\lf|f*\psi_t(y)\r|.$$
When $a=1$, we simply use $\psi_{\triangledown}^\ast(f)$ to denote
$\psi_{\triangledown,a}^\ast(f)$.
\end{defn}
In what follows, for any $N\in\nn$ and $a\in(0,\fz)$,
the \emph{non-tangential grand maximal function} of $f\in\cs'(\rn)$
is defined by setting, for all $x\in\rn$,
$$f_{N,\triangledown,a}^{\ast}(x):=\sup_{\psi\in\cf_N(\rn)}
\sup_{t\in(0,\fz),\,|y-x|<at}\lf|f*\psi_t(y)\r|,$$
where $\cf_N(\rn)$ is as in \eqref{2.4x}.
When $a=1$, we simply denote $f_{N,\triangledown,a}^{\ast}$ by
$f_{N,\triangledown}^\ast$.
\begin{rem}\label{r-m-fun}
Let $f\in\cs'(\rn)$.
\begin{enumerate}
\item[(i)] From the definitions of $f_{N,+}^\ast$ and $f_{N,\triangledown}^\ast$,
and \cite[Proposition 2.1]{zyl} (see also \cite[Lemma 7.9]{cw14}),
we deduce that there exists a positive constant
$C$ such that, for all $x\in\rn$,
$$C^{-1}f_{N,\triangledown}^\ast(x)\le f_{N,+}^\ast(x)
\le Cf_{N,\triangledown}^\ast(x).$$
\item[(ii)] For any $a\in(0,\fz)$ and $\psi\in\cs(\rn)$, it is easy to see that
$\psi_{\triangledown,a}^\ast(f)\le C f_{N,\triangledown}^\ast$ pointwise,
where $C$ is a positive constant independent of $f$.
\end{enumerate}
\end{rem}
A distribution $f\in\cs'(\rn)$ is called a \emph{bounded distribution} if, for all
$\phi\in\cs(\rn)$, $f\ast \phi\in L^\fz(\rn)$. For a bounded distribution $f$,
its \emph{non-tangential maximal function},
with respect to {Poisson kernels} $\{P_t\}_{t>0}$, is defined by setting,
for all $x\in\rn$,
$$\cn (f)(x):=\sup_{t\in(0,\fz),\,|y-x|<t}\lf|f*P_t(y)\r|,$$
where, for all $x\in\rn$ and $t\in(0,\fz)$,
\begin{align}\label{1.23y}
P_t(x):=\frac{\Gamma([n+1]/2)}{\pi^{(n+1)/2}}
\frac{t}{(t^2+|x|^2)^{(n+1)/2}}
\end{align}
and $\Gamma$ denotes the Gamma function.
The following conclusion is the main result of this subsection, which
gives out the maximal function characterizations of the space $\whv$.
\begin{thm}\label{mthm1}
Let $p(\cdot)\in C^{\log}(\rn)$.
Suppose that $N\in(\frac{n}{\underline{p}}+n+1,\fz)$
is a positive integer, where $\underline{p}$ is as in \eqref{2.1y}.
Then the following items are equivalent:
\begin{enumerate}
\item[{\rm (i)}] $f\in\whv$, namely, $f\in\cs'(\rn)$ and $f_{N,+}^\ast\in\wlv$;
\item[{\rm (ii)}] $f$ is a bounded distribution and $\cn (f)\in\wlv$;
\item[{\rm (iii)}] $f\in\cs'(\rn)$ and there exists a $\psi\in\cs(\rn)$ with
$\int_{\rn}\psi(x)\,dx=1$ such that $\psi_{+}^\ast(f)\in\wlv$.
\end{enumerate}
Moreover, for any $f\in \whv$, it holds true that
$$\lf\|f_{N,+}^\ast\r\|_{\wlv}\sim\lf\|\cn(f)\r\|_{\wlv}
\sim\lf\|\psi_{+}^\ast(f)\r\|_{\wlv},$$
where the implicit equivalent positive constants are independent of $f$.
\end{thm}
\begin{proof}
{\bf STEP 1}: In this step, we show
${\rm (i)}\Rightarrow{\rm (ii)}\Rightarrow{\rm (iii)}.$
Suppose that ${\rm (i)}$ holds true, namely, $f\in\cs'(\rn)$ and $f_{N,+}^\ast\in\wlv$
with $N$ as in Theorem \ref{mthm1}.
To prove (ii), we first show that $f$ is a bounded distribution.
Indeed, by Remark \ref{r-m-fun}(i), we easily know that
there exists a positive constant $C_{(N)}$ such that,
for any $\phi\in\cs(\rn)$, $x\in\rn$ and $y\in B(x, 1)$,
$|f*{\phi}(x)|\le\frac1{C_{(N)}}f_{N,+}^\ast(y)$,
since
$$|f*{\phi}(x)|\ls f_{N,\triangledown}^\ast(y)\ls f_{N,+}^\ast(y).$$
Thus, for any $x\in\rn$, we have
$$B(x, 1)\subset\{y\in\rn:\ f_{N,+}^\ast(y)
\ge C_{(N)}|f*\phi(x)|\}=:\Omega_{f,x}.$$
By this and Remark \ref{r-vlp}(ii), we conclude that
\begin{align}\label{max-f3}
&\min\{\lf|f*\phi(x)\r|^{p_-}, \lf|f*\phi(x)\r|^{p_+}\}\noz\\
&\hs\le\min\{\lf|f*\phi(x)\r|^{p_-}, \lf|f*\phi(x)\r|^{p_+}\}
\frac{1}{|B(x,1)|}\int_{\rn}\chi_{\Omega_{f,x}}(y)\,dy\noz\\
&\hs\ls\min\{\lf|f*\phi(x)\r|^{p_-}, \lf|f*\phi(x)\r|^{p_+}\}
\int_{\Omega_{f,x}}\lf[\frac{1}{\|\chi_{\Omega_{f,x}}\|_{\lv}}\r]^{p(y)}
\|\chi_{\Omega_{f,x}}\|_{\lv}^{p(y)}\,dy\noz\\
&\hs\ls\min\{\lf|f*\phi(x)\r|^{p_-},\lf|f*\phi(x)\r|^{p_+}\}
\max\lf\{\lf\|\chi_{\Omega_{f,x}}\r\|_{\lv}^{p_-},
\lf\|\chi_{\Omega_{f,x}}\r\|_{\lv}^{p_+}\r\}\noz\\
&\hs\ls\max\lf\{\|f_{N,+}^\ast\|_{\wlv}^{p_-},\|f_{N,+}^\ast\|
_{\wlv}^{p_+}\r\}<\fz,
\end{align}
which implies that $f\ast \phi\in L^\fz(\rn)$.
Therefore, $f$ is a bounded distribution.
Next, we show that $\cn(f)\in \wlv$. By \cite[p.\,98]{stein93},
we know that, for all $x\in\rn$,
$$P_1(x)=\sum_{k=0}^{\fz}2^{-k}\psi_{2^k}^{(k)}(x),$$
where $\{\psi^{(k)}\}_{k\in\nn}\st\cs(\rn)$ have uniformly bounded seminorms
in $\cs(\rn)$ and $P_1$ is the Poisson kernel as in \eqref{1.23y} with $t=1$.
From this decomposition, it follows that, for any $t\in(0,\fz)$, $x\in\rn$ and
$y\in B(x,t)$,
\begin{align*}
\lf|f*P_t(y)\r|\le\sum_{k=0}^{\fz}2^{-k}\lf|f*{\psi}_{2^kt}^{(k)}(y)\r|
\le\sum_{k=0}^{\fz}2^{-k}(\psi^{(k)})_{\triangledown}^\ast (f)(x),
\end{align*}
which implies that, for all $x\in\rn$,
\begin{equation}\label{emeqp}
\cn(f)(x)\le \sum_{k=0}^{\fz}2^{-k}(\psi^{(k)})_{\triangledown}^\ast (f)(x).
\end{equation}
Since $\{\psi^{(k)}\}_{k\in\nn}$ have uniformly bounded seminorms in $\cs(\rn)$,
it follows, from \eqref{emeqp}, Remarks \ref{10.24.x1},
\ref{r-ar} and \ref{r-m-fun}, that
\begin{align*}
\lf\|\cn (f)\r\|_{\wlv}^v
&\le\lf\|\sum_{k=0}^{\fz}2^{-k}
(\psi^{(k)})_{\triangledown}^\ast (f)\r\|_{\wlv}^v\\
&\ls\sum_{k=0}^{\fz}2^{-kv}
\lf\|(\psi^{(k)})_{\triangledown}^\ast (f)\r\|_{\wlv}^v
\ls\lf\|f_{N,+}^\ast\r\|_{\wlv}^v,
\end{align*}
where $v$ is as in Remark \ref{r-ar}.
This shows that $\cn(f)\in\wlv$ and hence (ii) holds true.
Finally, assume that (ii) holds true, namely,
$f$ is a bounded distribution and $\cn (f)\in\wlv$.
Then, by \cite[p.\,99]{stein93}, we know that there exists $\psi\in\cs(\rn)$
with $\int_\rn\psi(x)\,dx=1$ such that, for all $x\in\rn$,
$\psi_{+}^\ast(f)(x)\ls \cn (f)(x)$.
Therefore, (iii) holds true, which completes the proof of STEP 1.
{\bf STEP 2}: In this step, we prove ${\rm (iii)}\Rightarrow{\rm (i)}$.
Assume that (iii) holds true, namely, $f\in\cs'(\rn)$ and there exists
$\psi\in\cs(\rn)$ with $\int_{\rn}\psi(x)\,dx=1$ such that
$\psi_{+}^\ast(f)\in\wlv$.
Since $N\in(\frac{n}{\underline{p}}+n+1,\fz)$, it follows that there exists a positive constant
$T>\frac{n}{\underline{p}}$ such that $N>T+n+1$.
From this and \cite[(3.1)]{cw14},
it follows that, for all $x\in\rn$,
\begin{equation}\label{emeq1}
f_{N,+}^\ast(x)\ls M_{\psi,T}(f)(x),
\end{equation}
where
$$M_{\psi,T}(f)(x):=\sup_{t\in(0,\fz),\,y\in\rn}|f*\psi_t(x-y)
|\lf(1+\frac{|y|}{t}\r)^{-T},\quad\forall\, x\in\rn.$$
On the other hand, by the proof of \cite[Theorem 2.1.4(c)]{Gra14},
we find that, for $q:=\frac{n}{T}$ and all $x\in\rn$,
$[M_{\psi,T}(f)(x)]^q\le\cm([\psi_{\triangledown}^\ast(f)]^q)(x).$
Thus, by the fact that $T>\frac n{\underline p}$,
Lemma \ref{mlmim} and Corollary \ref{mc1},
we conclude that
\begin{align}\label{max-f2}
\lf\|M_{\psi,T}(f)\r\|_{\wlv}
&=\lf\|\lf[M_{\psi,T}(f)\r]^q\r\|
_{WL^{p(\cdot)/q}(\rn)}^{1/q}
\le\lf\|\cm\lf(\lf[\psi_{\triangledown}^\ast(f)\r]^q\r)\r\|
_{WL^{p(\cdot)/q}(\rn)}^{1/q}\noz\\
&\ls\lf\|\lf[\psi_{\triangledown}^\ast(f)\r]^q\r\|
_{WL^{p(\cdot)/q}(\rn)}^{1/q}
\sim\lf\|\psi_{\triangledown}^\ast(f)\r\|_{\wlv}.
\end{align}
Now we claim that
\begin{align}\label{emeq3}
\lf\|\psi_{\triangledown}^\ast(f)\r\|_{\wlv}
\ls\lf\|\psi_{+}^\ast(f)\r\|_{\wlv}.
\end{align}
Assuming that this claim holds true for the time being, then,
due to \eqref{emeq1} and \eqref{max-f2}, we have
$$\lf\|f_{N,+}^\ast\r\|_{\wlv}\ls
\lf\|\psi_{+}^\ast(f)\r\|_{\wlv}<\fz,$$
which implies that (i) holds true.
To show the claim \eqref{emeq3}, we first assume that
$\psi_{\triangledown}^\ast(f)\in\wlv$, which will be proved later.
For any $\eta\in(0,\fz)$, let
$E:=\{x\in\rn:\ f_{N,+}^\ast(x)<\eta \psi_{\triangledown}^\ast(f)(x)\}.$
Then, by \eqref{emeq1} and \eqref{max-f2}, we know that there exists
a positive constant $C_0$ such that
\begin{align*}
\lf\|\psi_{\triangledown}^\ast(f)\chi_{E^{\com}}\r\|_{\wlv}
&\le\frac{1}{\eta}\lf\|f_{N,+}^\ast\chi_{E^{\com}}\r\|_{\wlv}\\
&\le\frac{1}{\eta}\lf\|f_{N,+}^\ast\r\|_{\wlv}
\le\frac{C_0}{\eta}\lf\|\psi_{\triangledown}^\ast(f)\r\|_{\wlv}.
\end{align*}
Thus, by Lemma \ref{mlm2}(iii), we find that
\begin{align*}
\lf\|\psi_{\triangledown}^\ast(f)\r\|_{\wlv}
&\le \widetilde{C}\lf[\lf\|\psi_{\triangledown}^\ast(f)\chi_E\r\|_{\wlv}
+\lf\|\psi_{\triangledown}^\ast(f)\chi_{E^{\com}}\r\|_{\wlv}\r]\\
&\le \widetilde{C}\lf\|\psi_{\triangledown}^\ast(f)\chi_E\r\|_{\wlv}
+\frac{\widetilde{C}C_0}{\eta}\lf\|\psi_{\triangledown}^\ast(f)\r\|_{\wlv},
\end{align*}
where $\widetilde{C}$ is a positive constant independent of $f$ and $\eta$.
By this and via choosing $\eta:=2\widetilde{C}C_0$, we conclude that
\begin{align}\label{emeq5}
\lf\|\psi_{\triangledown}^\ast(f)\r\|_{\wlv}
\le2\widetilde{C}\lf\|\psi_{\triangledown}^\ast(f)\chi_E\r\|_{\wlv}.
\end{align}
On the other hand, by \cite[p.\,96]{stein93},
we know that, for any $q\in(0,\underline{p})$ and all $x\in E$,
$$\psi_{\triangledown}^\ast(f)(x)
\ls\lf[\cm\lf(\lf[\psi_{+}^\ast(f)\r]^{q}\r)(x)\r]^{1/{q}},$$
which, combined with Lemma \ref{mlmim} and Corollary \ref{mc1}, implies that
\begin{align*}
\lf\|\psi_{\triangledown}^\ast(f)\chi_E\r\|_{\wlv}
&\ls\lf\|\lf[\cm\lf(\lf[\psi_{+}^\ast(f)\r]^{q}\r)\r]
^{1/{q}}\r\|_{\wlv}
\sim\lf\|\cm\lf(\lf[\psi_{+}^\ast(f)\r]^{q}\r)\r\|
_{WL^{p(\cdot)/{q}}(\rn)}^{1/{q}}\\
&\ls\lf\|\lf[\psi_{+}^\ast(f)\r]^{q}\r\|_
{WL^{p(\cdot)/{q}}(\rn)}^{1/{q}}\sim\lf\|\psi_{+}^\ast(f)\r\|_{\wlv}.
\end{align*}
From this and \eqref{emeq5}, we deduce that \eqref{emeq3} holds true.
Next we show that $\psi_{\triangledown}^\ast(f)\in\wlv$.
To this end, for any $\epsilon\in(0,\frac{1}{3}),\ L\in(0,\fz)$,
$T\in(\frac{n}{\underline p},\fz)$,
$\wz N\in\mathbb N$ and $x\in\rn$, let
$$M_{\psi}^{\epz, L}(f)(x):=\sup_{t\in(0,1/\epsilon)}\lf|f*\psi_t(x)\r|
\frac{t^L}{(t+\epsilon+\epsilon|x|)^L},$$
$$M_{\wz N}^{\epz, L}(f)(x):=\sup_{\psi\in\cf_{\wz N}(\rn)}
M_{\psi}^{\epz, L}(f)(x),$$
$$M_{\psi, 1}^{\epz, L}(f)(x):=
\sup_{t\in(0,1/\epz),\,|y-x|<t}\lf|f*{\psi}_t(y)\r|
\frac{t^L}{(t+\epz+\epz|y|)^L}$$
and
$$\bar {M}_{\psi, T}^{\epz, L}(f)(x):=
\sup_{t\in(0,1/\epz),\,y\in\rn}\lf|f*{\psi}_t(x-y)\r|
\lf(1+\frac{|y|}{t}\r)^{-T}\frac{t^L}{(t+\epz+\epz|x-y|)^L}.$$
By \cite[p.\,45]{Gra14}, we know that there exist positive constants $m$ and $l$
such that, for all $\epsilon\in(0,\frac{1}{3})$ and $x\in\rn$,
$$M_{\psi, 1}^{\epz, L}(f)(x)
\sim\sup_{t\in(0,1/\epz),\,|y-x|<t}\lf|f*{\psi}_t(y)\r|
\lf(\frac{t}{t+\epz}\r)^L\frac{1}{(1+\epz|y|)^L}
\ls\frac{C_{(f,\psi,\epsilon,n,l,m,L)}}{(1+\epsilon|x|)^{L-m}}.$$
Observe that, for all $x\in\rn$,
$$(1+\epsilon|x|)^{m-L}
\le\epsilon^{m-L}(1+|x|)^{m-L}\ls\epsilon^{m-L}
[\cm(\chi_{B(0,1)})(x)]^{\frac{L-m}{n}}.$$
By this via taking $L\in(m+\frac{n}{p_-},\fz)$, Lemma \ref{mlmim}
and Corollary \ref{mc1}, we conclude that, for all $\epsilon\in(0,\frac{1}{3})$,
$$\lf\|M_{\psi, 1}^{\epz, L}(f)\r\|_{\wlv}
\ls\lf\|\cm(\chi_{B(0,1)})\r\|
_{WL^{\frac{(L-m)p(\cdot)}{n}}(\rn)}^{\frac{L-m}{n}}
\ls\|\chi_{B(0,1)}\|_{\wlv}<\fz,$$
which implies that, for all $\epsilon\in(0,\frac{1}{3})$,
$M_{\psi, 1}^{\epz, L}(f)\in\wlv$. Moreover, by \cite[p.\,460]{cw14}, we know that
there exists $\wz N\in[T+L+n+1,\fz)$ such that, for all
$\epsilon\in(0,\frac1{3})$ and $x\in\rn$,
\begin{align}\label{emeqex}
M_{\wz N}^{\epz, L}(f)(x)\ls \bar {M}_{\psi, T}^{\epz, L}(f)(x).
\end{align}
For any $\lz\in(0,\fz)$, let
$F:=\{x\in\rn:\ M_{\wz N}^{\epz, L}(f)(x)<\lz M_{\psi, 1}^{\epz, L}(f)(x)\}$.
Then, by an argument similar to that used in the proof of
\cite[(3.7)]{cw14}, we know that, for all $x\in F$,
\begin{align}\label{10.1.x}
M_{\psi, 1}^{\epz, L}(f)(x)
\ls \lf\{\cm\lf(\lf[\psi_{+}^\ast(f)\r]
^{\underline{p}}\r)(x)\r\}^{1/{\underline{p}}}.
\end{align}
On the other hand, observe that, for all $\epsilon\in(0,1/3)$, $t\in(0,1/\epz)$
and $x,\ y\in\rn$, it holds true that, for all $z\in B(x-y,t)$,
$$\lf|\psi_t*f(x-y)\r|\frac{t^L}{(t+\epsilon+\epsilon|x-y|)^L}
\le M_{\psi, 1}^{\epz, L}(f)(z).$$
By this and the fact that $B(x-y,t)\subset B(x,|y|+t)$, we find that,
for $q=n/T$, $\epsilon\in(0,1/3)$, $t\in(0,1/\epz)$
and $x,\ y\in\rn$,
\begin{align*}
&\lf[\lf|\psi_t*f(x-y)\r|\frac{t^L}{(t+\epsilon+\epsilon|x-y|)^L}\r]^q\\
&\hs\le\frac{|B(x,|y|+t)|}{|B(x-y,t)|}\frac{1}{|B(x,|y|+t)|}
\int_{B(x,|y|+t)}\lf[M_{\psi, 1}^{\epz, L}(f)\r]^q(z)\,dz\\
&\hs\le\lf(1+\frac{|y|}{t}\r)^n\cm\lf(\lf[M_{\psi, 1}^{\epz, L}(f)\r]^q\r)(x),
\end{align*}
namely,
$$\lf[\lf|\psi_t*f(x-y)\r|\frac{t^L}{(t+\epsilon+\epsilon|x-y|)^L}
\lf(1+\frac{|y|}{t}\r)^{-T}\r]^q
\le\cm\lf(\lf[M_{\psi, 1}^{\epz, L}(f)\r]^q\r)(x),$$
which further implies that
$$\lf[\bar {M}_{\psi, T}^{\epz, L}(f)(x)\r]^q
\le\cm\lf(\lf[M_{\psi, 1}^{\epz, L}(f)\r]^q\r)(x).$$
From this, Lemma \ref{mlmim}, the fact that $q<\underline{p}$
and Corollary \ref{mc1}, we deduce that
\begin{align*}
\lf\|\bar {M}_{\psi, T}^{\epz, L}(f)\r\|_{\wlv}
&=\lf\|\lf[\bar {M}_{\psi, T}^{\epz, L}(f)\r]^q\r\|
_{WL^{p(\cdot)/q}(\rn)}^{1/q}
\le\lf\|\cm\lf(\lf[M_{\psi, 1}^{\epz, L}(f)\r]^q\r)\r\|
_{WL^{p(\cdot)/q}(\rn)}^{1/q}\\
&\ls\lf\|\lf[M_{\psi, 1}^{\epz, L}(f)\r]^q\r\|
_{WL^{p(\cdot)/q}(\rn)}^{1/q}
\sim\lf\|M_{\psi, 1}^{\epz, L}(f)\r\|_{\wlv},
\end{align*}
which, combined with \eqref{emeqex}, implies that
\begin{align}\label{emeqex1}
\lf\|M_{\wz N}^{\epz, L}(f)\r\|_{\wlv}
\ls\lf\|M_{\psi, 1}^{\epz, L}(f)\r\|_{\wlv}.
\end{align}
By \eqref{10.1.x}, \eqref{emeqex1}, the fact that
$M_{\psi, 1}^{\epz, L}(f)\in\wlv$ and an argument
similar to that used in the proof of \eqref{emeq3},
we conclude that, for all $\epz\in(0,1/3)$,
$$\lf\|M_{\psi, 1}^{\epz, L}(f)\r\|_{\wlv}
\ls \lf\|\psi_{+}^\ast(f)\r\|_{\wlv}<\fz$$
with the implicit positive constant independent of $\epz$.
By this, the fact that $M_{\psi, 1}^{\epz, L}(f)$ increases pointwise to
$\psi_{\triangledown}^\ast(f)$ as $\epz\rightarrow0$
for any $L\in(0,\fz)$ and Remark \ref{10.24.x1}, we find that
\begin{align*}
\lf\|\psi_{\triangledown}^\ast(f)\r\|_{\wlv}
&\le\liminf_{\epsilon\to0}\lf\|M_{\psi, 1}^{\epz, L}(f)\r\|_{\wlv}
\ls\lf\|\psi_{+}^\ast(f)\r\|_{\wlv}<\fz,\noz
\end{align*}
which implies that $\psi_{\triangledown}^\ast(f)\in\wlv$.
This finishes the proof of \eqref{emeq3} and hence STEP 2.
Thus, we complete the proof of Theorem \ref{mthm1}.
\end{proof}
By Remark \ref{r-m-fun} and Theorem \ref{mthm1}, we obtain the following
conclusion.
\begin{cor}\label{1.29.x1}
Let $p(\cdot)$ and $N$ be as in Theorem \ref{mthm1} and $a\in(0,\fz)$.
Then $f\in\whv$ if and only if one of the following items holds true:
\begin{enumerate}
\item[{\rm(i)}] $f\in\cs'(\rn)$ and there exists a $\psi\in\cs(\rn)$ with $\int_{\rn}\psi(x)\,dx=1$
such that $\psi_{\triangledown,a}^*(f)\in\wlv$;
\item[{\rm(ii)}] $f\in\cs'(\rn)$ and $f^*_{N,\triangledown}\in\wlv$.
\end{enumerate}
Moreover, for any $f\in\whv$, it holds true that
$$\|f\|_{\whv}\sim\|f^*_{N,\triangledown}\|_{\wlv}
\sim\|\psi_{\triangledown,a}^*(f)\|_{\wlv}$$
with the implicit equivalent positive constants independent of $f$.
\end{cor}
\section{Atomic characterizations of $\whv$\label{s-atom}}
\hskip\parindent
In this section, we establish the atomic characterization of $\whv$.
We begin with recalling the notion of $(p(\cdot),q,s)$-atoms introduced
by Nakai and Sawano in \cite[Definition 1.4]{ns12}.
\begin{defn}\label{atd1}
Let $p(\cdot)\in\cp(\rn)$,
$q\in(1,\fz]$ and
\begin{equation}\label{4.1.x}
s\in\lf(\frac{n}{p_-}-n-1,\fz\r)\cap{\zz}_+.
\end{equation}
A measurable function $a$ on $\rn$
is called a \emph{$(p(\cdot),q,s)$-atom} if there exists a ball $B$ such that
\begin{enumerate}
\item[{\rm (i)}] $\supp a \st B$;
\item[{\rm (ii)}] $\|a\|_{L^q(\rn)}\le \frac{|B|^{1/q}}{\|\chi_B\|_{\lv}}$;
\item[{\rm (iii)}] $\int_{\mathbb R^n}a(x)x^\az\,dx=0$ for all $\az\in{\zz}_+^n$
with $|\az|\le s$.
\end{enumerate}
\end{defn}
We now introduce the notion of the variable weak atomic Hardy space.
\begin{defn}\label{atd2}
Let $p(\cdot)\in C^{\log}(\rn)$, $q\in(1,\fz]$ and $s$ be
as in \eqref{4.1.x}.
The \emph{variable weak atomic Hardy space} $\wha$ is
defined as the space of all $f\in\cs'(\rn)$ which can be decomposed as
\begin{equation}\label{4.1.y}
f=\sum_{i\in\zz}\sum_{j\in\nn}\lij\aij\quad \mbox{in}\quad \cs'(\rn),
\end{equation}
where $\{\aij\}_{i\in\zz,j\in\nn}$ is a sequence of
$(p(\cdot),q,s)$-atoms, associated with balls $\{\Bij\}_{i\in\zz,\ j\in\nn}$,
satisfying that there exists a positive constant $c\in(0,1]$ such that,
for all $x\in\rn$ and $i\in\zz$,
$\sum_{j\in\nn}\chi_{c\Bij}(x)\le A$ with $A$ being a positive constant
independent of $x$ and $i$ and, for all $i\in\zz$ and $j\in\nn$,
$\lij:=\wz A 2^i\|\chi_\Bij\|_{\lv}$ with $\wz A$ being a positive constant
independent of $i$ and $j$.
Moreover, for any $f\in\wha$, define
$$\|f\|_{\wha}:=\inf\lf[{\sup_{i\in\zz}{\lf\|\lf\{\sum_{j\in\nn}
\lf[\frac{\lij\chi_\Bij}
{\|\chi_\Bij\|_{\lv}}\r]^{\underline{p}}\r\}^{1/{\underline{p}}}\r\|_{\lv}}}\r],$$
where the infimum is taken over all decompositions of $f$ as above.
\end{defn}
From the definition of $\wha$ and Remark \ref{2.5.y},
we easily deduce the following conclusion,
the details being omitted.
\begin{rem}\label{atomlm3}
Let $f\in\wha$. Then there exists a sequence $\{\aij\}_{i\in\zz,j\in\nn}$ of
$(p(\cdot),q,s)$-atoms, associated with balls $\{\Bij\}_{i\in\zz,j\in\nn}$,
satisfying that, for all $i\in\zz$ and $x\in\rn$,
$\sum_{j\in\nn}\chi_{c\Bij}(x)\le A$ with
$c\in(0,1]$ and $A$ being positive constants independent of $i$ and $x$
such that $f$ admits a decomposition as in \eqref{4.1.y}
with $\lij:=\wz A2^i\|\chi_\Bij\|_{\lv}$ for all
$i\in\zz$ and $j\in\nn$, where $\wz A$ is a positive
constant independent of $i$ and $j$, and
\begin{align}\label{atom-x1}
\|f\|_{\wha}
\sim\sup_{i\in\zz}{\lf\|\lf\{\sum_{j\in\nn}
\lf[\frac{\lij\chi_\Bij}
{\|\chi_\Bij\|_{\lv}}\r]^{\underline{p}}\r\}^{1/{\underline{p}}}
\r\|_{\lv}}
\end{align}
with the implicit equivalent positive constants independent of $f$.
Moreover, by the fact that $\sum_{j\in\nn}\chi_{c\Bij}\le A$
for all $i\in\zz$, the definition of $\{\lz_{i,j}\}_{i\in\zz,j\in\nn}$
and Remark \ref{2.5.y}, we further conclude that
\begin{align}
\|f\|_{\wha}&\sim\sup_{i\in\zz}2^i\lf\|\lf(\sum_{j\in\nn}
\chi_\Bij\r)^{1/{\underline{p}}}\r\|_{\lv}
\sim\sup_{i\in\zz}2^i\lf\|\lf(\sum_{j\in\nn}
\chi_{c\Bij}\r)^{1/{\underline{p}}}\r\|_{\lv}\noz\\
&\sim\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}
\chi_{c\Bij}\r\|_{\lv}
\sim\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}
\chi_{\Bij}\r\|_{\lv},\noz
\end{align}
where the implicit equivalent positive constants are independent of $f$.
\end{rem}
The main result of this section is stated as follows.
\begin{thm}\label{atthm1}
Let $p(\cdot)\in C^{\log}(\rn)$, $q\in(\max\{p_+,1\},\fz]$ with $p_+$
as in \eqref{2.1x} and $s$ be as in
\eqref{4.1.x}.
Then $\whv=\wha$ with equivalent quasi-norms.
\end{thm}
To prove Theorem \ref{atthm1}, we need the following useful technical lemma,
which is a variant of \cite[Lemma 4.1]{s10} and can be proved via
combining \cite[Lemma 4.1]{s10} and Lemma \ref{mlm1},
the details being omitted.
\begin{lem}\label{atlm2}
Let $p(\cdot)\in C^{\log}(\rn)$, $r\in(0,\underline{p}]$ and
$q\in[1,\fz]\cap(p_+,\fz]$. Then there exists
a positive constant $C$ such that, for all sequences
$\{B_j\}_{j\in\nn}$ of balls, numbers $\{\lz_j\}_{j\in\nn}\subset\mathbb C$
and measurable functions $\{a_j\}_{j\in\nn}$ satisfying that,
for each $j\in\nn$, $\supp a_j\st B_j$ and $\|a_j\|_{L^q(\rn)}\le |B_j|^{1/q}$,
it holds true that
$$\lf\|\lf(\sum_{j\in\nn}|\lz_j a_j|^{r}\r)^{\frac{1}{r}}
\r\|_{\lv}\le C\lf\|\lf(\sum_{j\in\nn}|\lz_j\chi_{B_j}|^{r}\r)
^{\frac{1}{r}}\r\|_{\lv}.$$
\end{lem}
In what follows, we use $\vec0_n$ to denote the origin of $\rn$
and, for any $\varphi\in\cs(\rn)$,
we use $\widehat{\varphi}$ to denote its
\emph{Fourier transform},
which is defined by setting, for all $\xi\in\rn$,
$$\widehat{\varphi}(\xi):=\int_{\rn}e^{-2\pi ix\xi}\varphi(x)\,dx.$$
We also need the following Calder\'on reproducing formula,
which was obtained by Calder\'on \cite[p.\,219]{c77}
(see also \cite[Lemma 4.1]{ct75}).
\begin{lem}\label{l-1.24x}
Let $\psi\in\cs(\rn)$ be such that $\supp \psi\st B(\vec0_n,1)$
and $\int_\rn\psi(x)\,dx=0$.
Then there exists a
function $\phi\in\cs(\rn)$ such that $\widehat{\phi}$
has compact support away from the origin and,
for all $x\in\rn\setminus\{\vec0_n\}$,
$$\int_0^\fz \widehat{\psi}(tx)\widehat{\phi}(tx)\,\frac{dt}t=1.$$
\end{lem}
Recall that, for any $d\in\zz_+$, $p(\cdot)\in\cp(\rn)$,
a locally integrable function $f$ on $\rn$ is
said to belong to the \emph{Campanato space}
$\cl_{1,p(\cdot),d}(\rn)$ if
$$\|f\|_{\cl_{1,p(\cdot),d}(\rn)}
:=\sup_{Q\subset\rn}\frac{1}{\|\chi_Q\|_{\lv}}
\int_{Q}|f(x)-P_Q^df(x)|\,dx<\fz,$$
where the supremum is taken over all cubes $Q$ of $\rn$ and
$P_Q^d$ denotes the \emph{unique polynomial} $P$ having degree at most $d$
and satisfies that, for any polynomial $R$ on $\rn$ with order at most
$d$, $\int_Q[f(x)-P(x)]R(x)\,dx=0$ (see \cite[Definition 6.1]{ns12}).
Now we turn to the proof of Theorem \ref{atthm1}.
For $N\in(\frac{n}{\underline{p}}+n+1,\fz)$
and $h\in\cs'(\rn)$, we denote $h_{N,+}^\ast$ simply by $h^\ast$.
\begin{proof}[Proof of Theorem \ref{atthm1}]
{\bf STEP 1:} In this step, we show that $\wha\st\whv$.
Let $f\in\wha$. Then, by Remark \ref{atomlm3}, we know that there exist
a sequence $\{\aij\}_{i\in\zz,j\in\nn}$ of $(p(\cdot),q,s)$-atoms,
associated with balls $\{\Bij\}_{i\in\zz,j\in\nn}$,
and $\{\lz_{i,j}\}_{i\in\zz,j\in\nn}\st\cc$ such that
\eqref{4.1.y} holds true in $\cs'(\rn)$ and
\begin{align}\label{atom-c1}
\|f\|_{\wha}\sim\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}\chi_\Bij\r\|_{\lv}.
\end{align}
To prove $f\in\whv$, by the definition of $\whv$, it suffices to show that
$$\sup_{\az\in(0,\fz)}\az\lf\|\chi_{\{x\in\rn:\ |f^*(x)|>\az\}}
\r\|_{\lv}\ls\|f\|_{\wha}.$$
For any given $\az\in(0,\fz)$, we choose $i_0\in\zz$ such that
$2^{i_0}\le\az<2^{i_0+1}$ and write
$$f=\sum_{i=-\fz}^{i_0-1}\sum_{j\in\nn}\lij\aij
+\sum_{i=i_0}^{\fz}\sum_{j\in\nn}\lij\aij=:f_1+f_2.$$
Thus, by Remark \ref{r-vlp}(i), we find that
\begin{align}\label{eqatomsum}
&\lf\|\chi_{\{x\in\rn:\ f^*(x)>\az\}}\r\|_{\lv}\noz\\
&\hs\ls\lf\|\chi_{\{x\in\rn:\ f_1^*(x)
>\frac{\az}{2}\}}\r\|_{\lv}
+\lf\|\chi_{\{x\in A_{i_0}:\ f_2^*(x)
>\frac{\az}{2}\}}\r\|_{\lv}\noz\\
&\hs\hs\hs\hs+\lf\|\chi_{\{x\in(A_{i_0})^{\com}:\ f_2^*(x)
>\frac{\az}{2}\}}\r\|_{\lv}\noz\\
&\hs=:{\rm I}_1+{\rm I}_2+{\rm I}_3,
\end{align}
where $A_{i_0}:=\bigcup_{i={i_0}}^\fz\bigcup_{j\in\nn}(2\Bij)$.
For ${\rm I}_1$, it is easy to see that
\begin{align}\label{eqatomi1}
{\rm I}_1&\ls\lf\|\chi_{\{x\in\rn:\ \sum_{i=-\fz}
^{i_0-1}\sum_{j\in\nn}\lij(\aij)^{\ast}(x)\chi_{2\Bij}(x)
>\frac{\az}{4}\}}\r\|_{\lv}\noz\\
&\hs\hs\hs+\lf\|\chi_{\{x\in\rn:\ \sum_{i=-\fz}^{i_0-1}
\sum_{j\in\nn}\lij(\aij)^{\ast}(x)\chi_{(2\Bij)^\com}(x)
>\frac{\az}{4}\}}\r\|_{\lv}\noz\\
&=:{\rm I}_{1,1}+{\rm I}_{1,2}.
\end{align}
To estimate ${\rm I}_{1,1}$, for any $b\in(0,\underline{p})$,
let $\q1\in(1,\min\{\frac{q}{\max\{p_+,1\}},\frac{1}{b}\})$ and
$a\in(0,1-\frac1{\q1})$.
Then, by the H\"older inequality, we find that, for all $x\in\rn$,
\begin{align*}
&\sum_{i=-\fz}^{i_0-1}\sum_{j\in\nn}\lij(\aij)
^{\ast}(x)\chi_{2\Bij}(x)\\
&\hs\le \lf(\sum_{i=-\fz}^{i_0-1}2^{ia{\q1}'}\r)
^{1/{\q1}'}\lf\{\sum_{i=-\fz}^{i_0-1}2^{-ia{\q1}}
\lf[\sum_{j\in\nn}\lij(\aij)^{\ast}(x)\chi_{2\Bij}(x)\r]
^{\q1}\r\}^{1/{\q1}}\\
&\hs=\frac{2^{i_0a}}{(2^{a{\q1}'}-1)^{1/{\q1}'}}
\lf\{\sum_{i=-\fz}^{i_0-1}2^{-ia{\q1}}
\lf[\sum_{j\in\nn}\lij(\aij)^{\ast}(x)\chi_{2\Bij}(x)\r]
^{\q1}\r\}^{1/{\q1}},
\end{align*}
where ${\q1}'$ denotes the conjugate exponent of $\q1$, namely,
$\frac{1}{\q1}+\frac{1}{{\q1}'}=1$.
From this, the facts that $\q1b<1$ and $f^*(x)\ls\cm f(x)$
for all $x\in\rn$, Remark \ref{r-vlp}(i) and \cite[Theorem 2.61]{cfbook},
we deduce that
\begin{align*}
{\rm I}_{1,1}
&\le\lf\|\chi_{\{x\in\rn:\ \frac{2^{i_0a}}
{(2^{a{\q1}'}-1)^{1/{\q1}'}}
[\sum_{i=-\fz}^{i_0-1}2^{-ia{\q1}}
\{\sum_{j\in\nn}\lij(\aij)^{\ast}(x)
\chi_{2\Bij}(x)\}^{\q1}]^{1/{\q1}}>
2^{i_0-2}\}}\r\|_{\lv}\\
&\ls 2^{-i_0\q1(1-a)}\lf
\|\sum_{i=-\fz}^{i_0-1}2^{-ia{\q1}}\lf
[\sum_{j\in\nn}\lij(\aij)^{\ast}\chi_{2\Bij}\r]
^{\q1}\r\|_{\lv}\\
&\ls 2^{-i_0\q1(1-a)}\lf\|\sum_{i=-\fz}^{i_0-1}
2^{(1-a)i{\q1}b}\sum_{j\in\nn}\lf
[\|\chi_{\Bij}\|_{\lv}\cm(\aij)\chi_{2\Bij}\r]
^{\q1b}\r\|
^{\frac{1}{b}}_{L^{\frac{p(\cdot)}
b}(\rn)}\\
&\ls 2^{-i_0\q1(1-a)}\Bigg[\sum_{i=-\fz}
^{i_0-1}2^{(1-a)i{\q1}b}\\
&\quad\quad\quad\times\lf.\lf\|\lf\{\sum_{j\in\nn}
\lf[\|\chi_{\Bij}\|_{\lv}\cm(\aij)\chi_{2\Bij}\r]^
{\q1b}\r\}^{\frac{1}{b}}\r\|^{b}_{\lv}\r]^{\frac{1}{b}}.
\end{align*}
Now let $r:=\frac{q}{\q1}$. Then $r\in(1,\fz)$ and, by
the boundedness of $\cm$ on $L^r(\rn)$, we find that,
for all $i\in\zz$ and $j\in\nn$,
\begin{align*}
\lf\|\lf[\|\chi_{\Bij}\|_{\lv}\cm(\aij)\chi_{2\Bij}\r]^{\q1}\r\|_{L^r(\rn)}
&\ls\|\chi_{\Bij}\|_{\lv}^{\q1}\lf\|\cm(\aij)\chi_{2\Bij}\r\|^{\q1}_{L^q(\rn)}\noz\\
&\ls |\Bij|^{\frac{1}{r}}.
\end{align*}
Therefore, by Lemma \ref{atlm2}, Remark \ref{2.5.y} and \eqref{atom-c1},
we conclude that
\begin{align*}
{\rm I}_{1,1}&\ls 2^{-i_0\q1(1-a)}\lf
[\sum_{i=-\fz}^{i_0-1}2^{(1-a)i{\q1}b}
\lf\|\lf(\sum_{j\in\nn}\chi_{2\Bij}\r)
^{\frac{1}{b}}\r\|
^{b}_{\lv}\r]^
{\frac{1}{b}}\\
&\ls 2^{-i_0\q1(1-a)}\lf
[\sum_{i=-\fz}^{i_0-1}2^{(1-a)i{\q1}b}
\lf\|\lf(\sum_{j\in\nn}\chi_{c\Bij}\r)
^{\frac{1}{b}}\r\|
^{b}_{\lv}\r]^
{\frac{1}{b}}\\
&\ls 2^{-i_0\q1(1-a)}
\lf\{\sum_{i=-\fz}^{i_0-1}2^{[(1-a){\q1}-1]ib}
\r\}^{\frac{1}{b}}\sup_{i\in\zz}2^i\lf\|
\sum_{j\in\nn}\chi_{\Bij}\r\|_{\lv}
\ls\az^{-1}\|f\|_{\wha},
\end{align*}
which implies that
\begin{align}\label{eqatomi11}
\az{\rm I}_{1,1}\ls\|f\|_{\wha}.
\end{align}
To deal with ${\rm I}_{1,2}$, we need some estimates on $(\aij)^*$.
Let $\phi\in\cf_N(\rn)$ and, for any $i\in\zz$ and $j\in\nn$,
$B_{i,j}:=B(x_{i,j},r_{i,j})$ with some $x_{i,j}\in\rn$
and $r_{i,j}\in(0,\fz)$.
Then, from the vanishing moment condition of $\aij$, the Taylor remainder theorem and
the H\"older inequality,
we deduce that, for any $i\in\zz\cap[i_0,\fz)$, $j\in\nn$, $t\in(0,\fz)$ and
$x\in (2\Bij)^\com$,
\begin{align}\label{11.14.x2}
\hs\hs\lf|\aij * \phi_t(x)\r|
&=\lf|\int_\Bij \aij(y)
\lf[\phi\lf(\frac{x-y}{t}\r)
-\sum_{|\beta|\le s}\frac{D^\beta\phi
(\frac{x-\xij}{t})}{\beta!}
\lf(\frac{\xij-y}{t}\r)^\beta\r]\,\frac{dy}{t^n}\r|\noz\\
&\ls\int_\Bij|\aij(y)|\frac{|y-\xij|^{s+1}}
{|x-\xij|^{n+s+1}}\,dy\noz\\
&\ls\frac{(r_{i,j})^{s+1}}{|x-\xij|^{n+s+1}}
\lf[\int_\Bij|\aij(y)|^q\,dy\r]^{1/q}
\lf(\int_\Bij\,1\,dy\r)^{1/q'}\noz\\
&\ls\|\chi_{\Bij}\|_{\lv}^{-1}
\lf(\frac{\rij}{|x-\xij|}\r)^{n+s+1},
\end{align}
which implies that, for any $x\in (2\Bij)^\com$,
\begin{align}\label{eatomst}
(\aij)^*(x)&\ls\|\chi_{\Bij}\|_{\lv}^{-1}
\lf(\frac{\rij}{|x-\xij|}\r)^{n+s+1}\noz\\
&\ls\|\chi_{\Bij}\|_{\lv}^{-1}
\lf[\cm\lf(\chi_{\Bij}\r)(x)\r]^{\frac{n+s+1}{n}}.
\end{align}
By this, the H\"older inequality, Remark \ref{r-vlp}(i), Lemma \ref{mlm1} and
\eqref{atom-c1}, we find that, for any $b\in(0,\frac{n}{n+s+1})$,
$\Q1\in(\frac{n}{[n+s+1]b}, \frac{1}{b})$ and $a\in(0,1-\frac1{\Q1})$,
\begin{align*}
{\rm I}_{1,2}
&\le \lf\|
\chi_{\{x\in\rn:\ \frac{2^{i_0a}}{(2^{a{\Q1}'}-1)^{1/{\Q1}'}}
\{\sum_{i=-\fz}^{i_0-1}2^{-ia{\Q1}}
[\sum_{j\in\nn}\lij(\aij)^{\ast}(x)
\chi_{({2\Bij})^\com}(x)]^{\Q1}\}^
{1/{\Q1}}>2^{i_0-2}\}}\r\|_{\lv}\\
&\ls 2^{-i_0\Q1(1-a)}\lf
\|\sum_{i=-\fz}^{i_0-1}2^{-ia{\Q1}}\lf
[\sum_{j\in\nn}\lij(\aij)^{\ast}
\chi_{({2\Bij})^\com}\r]^{\Q1}\r\|_{\lv}\\
&\ls 2^{-i_0\Q1(1-a)}\lf\{
\sum_{i=-\fz}^{i_0-1}2^{(1-a)i{\Q1}b}
\lf\|\sum_{j\in\nn}\lf[\cm\lf(\chi_{\Bij}\r)\r]^
{\frac{(n+s+1)\Q1 b}{n}}\r\|_{L^{\frac{p(\cdot)}
{b}}(\rn)}\r\}^{\frac{1}{b}}\\
&\ls 2^{-i_0\Q1(1-a)}\lf
[\sum_{i=-\fz}^{i_0-1}2^{(1-a)i{\Q1}b}
\lf\|\sum_{j\in\nn}\chi_{\Bij}\r\|_
{L^{\frac{p(\cdot)}{b}}(\rn)}\r]^{\frac{1}{b}}\\
&\ls 2^{-i_0\Q1(1-a)}\lf\{\sum_{i=-\fz}
^{i_0-1}2^{[(1-a){\Q1}-1]ib}
2^{i{b}}\lf\|\lf(\sum_{j\in\nn}
\chi_{c\Bij}\r)^{\frac{1}{b}}\r\|_
{\lv}^{b}\r\}^{\frac{1}{b}}\\
&\ls\az^{-1}\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}
\chi_{\Bij}\r\|_{\lv}\sim\az^{-1}\|f\|_{\wha},
\end{align*}
that is,
\begin{align}\label{eqatomi12}
\az{\rm I}_{1,2}\ls\|f\|_{\wha}.
\end{align}
By this, combined with \eqref{eqatomi1} and \eqref{eqatomi11}, we conclude that
\begin{align}\label{eqatomi}
\az{\rm I}_1\ls\|f\|_{\wha}.
\end{align}
For ${\rm I}_2$, we choose $r_1\in[\frac{1}{\underline{p}},\fz)$.
Then, by Remark \ref{2.5.y} and \eqref{atom-c1}, we conclude that
\begin{align*}
{\rm I}_2&\le\|\chi_{A_{i_0}}\|_{L^{p(\cdot)}(\rn)}
\le\lf\|\sum_{i={i_0}}^\fz\sum_{j\in\nn}
\chi_{2\Bij}\r\|_{L^{p(\cdot)}(\rn)}
\ls\lf\|\sum_{i={i_0}}^\fz\sum_{j\in\nn}
\chi_{\Bij}\r\|_{L^{p(\cdot)}(\rn)}\\
&\ls\lf[\sum_{i={i_0}}^\fz\lf\|\sum_{j\in\nn}\chi_{\Bij}
\r\|_{L^{p(\cdot)}(\rn)}^{\frac{1}{r_1}}\r]^{r_1}
\sim\lf\{\sum_{i={i_0}}^\fz2^{-\frac{i}{r_1}}
\lf[2^i\lf\|\sum_{j\in\nn}\chi_{\Bij}
\r\|_{L^{p(\cdot)}(\rn)}\r]^{\frac{1}{r_1}}\r\}^{r_1}\\
&\ls\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}\chi_\Bij
\r\|_{\lv}\lf(\sum_{i={i_0}}^\fz
2^{-\frac{i}{r_1}}\r)^{r_1}
\ls\az^{-1}\|f\|_{\wha},
\end{align*}
which implies that
\begin{align}\label{eqatomii}
\az {\rm I}_2\ls\|f\|_{\wha}.
\end{align}
For ${\rm I}_3$, since $\underline{p}\in(\frac{n}{n+s+1},1]$, it follows that
there exists $r_2\in(0,\fz)$ such that
$r_2\in(\frac{n}{\underline{p}(n+s+1)},1)$.
By this, the value of $\lz_{i,j}$, Lemma \ref{mlm1}, \eqref{eatomst} and \eqref{atom-c1}, we find that
\begin{align*}
{\rm I}_3&\le\lf\|\chi_{\{x\in (A_{i_0})
^\complement:\ \sum_{i=i_0}^\fz\sum_{j\in\nn}
\lij(\aij)^*(x)>\az/2\}}\r\|_{\lv}\\
&\ls\az^{-r_2}\lf\|\sum_{i=i_0}
^\fz\sum_{j\in\nn}[\lij(\aij)^*]^{r_2}\chi_{(A_{i_0})^\com}
\r\|_{\lv}\noz\\
&\ls\az^{-r_2}\lf[\sum_{i=i_0}^
\fz\lf\|\lf\{\sum_{j\in\nn}[\lij(\aij)^*]^{r_2}\chi_{(A_{i_0})^\com}\r\}^
{\frac{n}{r_2(n+s+1)}}\r\|_{L^{\frac{r_2(n+s+1)}{n}p(\cdot)}(\rn)}\r]
^{\frac{r_2(n+s+1)}{n}}\\
&\ls\az^{-r_2}\lf\{\sum_{i=i_0}^\fz
2^{\frac{in}{n+s+1}}\lf\|\sum_{j\in\nn}\lf[\cm\lf(\chi_{\Bij}\r)\r]
^{\frac{r_2(n+s+1)}{n}}
\r\|_{L^{p(\cdot)}(\rn)}^{{\frac{n}{r_2(n+s+1)}}}\r\}
^{\frac{r_2(n+s+1)}{n}}\\
&\ls\az^{-r_2}\lf[\sum_{i=i_0}^\fz2^{\frac{in}{n+s+1}}\lf
\|\sum_{j\in\nn}\chi_{\Bij}\r\|
_{\lv}^\frac{n}{r_2(n+s+1)}\r]^{\frac{r_2(n+s+1)}{n}}\noz\\
&\ls\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}\chi_\Bij
\r\|_{\lv}\az^{-r_2}\lf[\sum_{i=i_0}^\fz2^{\frac{in}{n+s+1}}
2^{-\frac{in}{(n+s+1)r_2}}\r]
^{\frac{r_2(n+s+1)}{n}}\noz\\
&\ls\az^{-1}\|f\|_{\wha},
\end{align*}
namely,
\begin{align}\label{eqatomiii}
\az {\rm I}_3\ls\|f\|_{\wha}.
\end{align}
Combining with \eqref{eqatomsum}, \eqref{eqatomi}, \eqref{eqatomii} and
\eqref{eqatomiii},
we obtain
\begin{align*}
\|f\|_{\whv}
&=\sup_{\az\in(0,\fz)}
\az\lf\|\chi_{\{x\in \rn:\ f^*(x)>\az\}}\r\|_{\lv}\\
&\ls\sup_{\az\in(0,\fz)}\az({\rm I}_1
+{\rm I}_2+{\rm I}_3)
\ls\|f\|_{\wha},
\end{align*}
which implies that $f\in\whv$. This finishes the proof of STEP 1.
{\bf STEP 2:} In this step, we prove that $\whv\st\wha$.
To complete the proof of STEP 2, it suffices to show $\whv\st\whz$,
since, due to the obvious fact that each $(p(\cdot),\fz,s)$-atom
is also a $(p(\cdot),q,s)$-atom for any $q\in(1,\fz)$, $\whz\st\wha$.
Let $\psi\in\cs(\rn)$ be such that $\supp \psi\st B(\vec0_n,1)$,
$\int_\rn\psi(x)x^\gz\,dx=0$ for all $\gz\in\zz_+^n$ with
$|\gz|\le s$.
Then, by Lemma \ref{l-1.24x}, we know that
there exists $\phi\in\cs(\rn)$ such that $\supp\wh \phi$ has compact support
away from the origin and,
for all $x\in\rn\setminus\{\vec0_n\}$,
$$\int_0^\fz\wh\psi(tx)\wh\phi(tx)\dt=1.$$
Define a function $\eta$ on $\rn$ by setting, for all $x\in\rn\setminus\{\vec0_n\}$,
$$\wh\eta(x):=\int_1^\fz\wh\psi(tx)\wh\phi(tx)\dt$$
and $\wh\eta(\vec0_n):=1$. Then, by \cite[p.\,219]{c77}, we find that $\eta$
is infinitely differentiable, has compact support and equals 1 near the origin.
Let $x_0:=(\overbrace{2,\,...\,,2}^{n\,{\rm times}})\in\rn$ and $f\in\whv$.
Following \cite{c77}, for all
$x\in\rn$ and $t\in(0,\fz)$, let $\wz \phi(x):=\phi(x-x_0)$,
$\wz \psi(x):=\psi(x+x_0)$,
$$
F(x,t):=f*\wz\phi_t(x)\quad \mbox{and}\quad G(x,t):=f*\eta_t(x).
$$
Then, by \cite[p.\,220]{c77}, we have
$$f(\cdot)=\int_0^{\fz}\int_{\rn}F(y,t)\wz \psi(\cdot-y)\frac{dy\,dt}{t}\quad \mbox{in}\quad \cs'(\rn).$$
For all $x\in\rn$, let
$$M_{\triangledown} (f)(x):=\sup_{t\in(0,\fz),\,|y-x|\le
3(|x_0|+1)t}[|F(y,t)|+|G(y,t)|].$$
Then $M_{\triangledown} (f)$ is lower semi-continuous and,
due to Corollary \ref{1.29.x1}, $M_{\triangledown} (f)\in\wlv$;
moreover,
\begin{align}\label{maxeq}
\|M_{\triangledown} (f)\|_{\wlv}\sim\|f\|_{\whv}.
\end{align}
For all $i\in\zz$, let $\boz_i:=\{x\in\rn:\ M_{\triangledown} f(x)>2^i\}$.
Then $\boz_i$ is open. By \eqref{wvlp} and \eqref{maxeq}, we further find that
\begin{align}\label{9.16.x}
\sup_{i\in\zz}2^i\|\chi_{\boz_i}\|_{\lv}
\le\|M_{\triangledown} (f)\|_{\wlv}\ls\|f\|_{\whv}.
\end{align}
Since $\boz_i$ is a proper open subset of $\rn$,
by the Whitney decomposition (see, for example, \cite[p.\,463]{g09-1}),
we know that there exists a sequence $\{\qij\}_{j\in\nn}$ of cubes
such that, for all $i\in\zz$,
\begin{enumerate}
\item[{\rm (i)}] $\bigcup_{j\in\nn} \qij = \boz_i$ and
$\{\qij\}_{j\in\nn}$ have disjoint interiors;
\item[{\rm (ii)}] for all $j\in\nn$, $\sqrt{n}\,l_{\qij}
\le \dist(\qij,\, \boz_i^{\complement}) \le 4 \sqrt{n}\,l_{\qij }$,
where $l_{\qij }$ denotes the length of the cube $\qij$ and
$\dist(\qij,\, \boz_i^{\complement}):=\inf\{|x-y|:\ x\in\qij,\ y\in \boz_i^{\complement}\}$;
\item[{\rm (iii)}] for any $j,\, k\in\nn$, if the boundaries of
two cubes $\qij$ and $Q_{i,k}$ touch, then
$\frac14\le\frac{l_{\qij}}{ l_{Q_{i,k}}}\le 4;$
\item[{\rm (iv)}] for a given $j\in\nn$, there exist at most
$12 n$ different cubes $Q_{i,k}$ that touch $\qij$.
\end{enumerate}
Now, for any $\epsilon\in(0,\fz)$, $i\in\zz$, $j\in\nn$ and $x\in\rn$, let
$$\dist\lf(x,\boz_i^\com\r):=\inf\lf\{|x-y|:\ y\in \boz_i^\com\r\},$$
$$\wz\boz_i:=\lf\{(x,t)\in\rr^{n+1}_+:\ 0<2t(|x_0|+1)<\dist(x, \boz_i^\com)\r\},$$
$$\wqij:=\lf\{(x,t)\in\rr_+^{n+1}
:\ x\in\qij,\ (x,t)\in\wz\boz_i\bh\wz\boz_{i+1}\r\}$$
and
$$\bije(x):=\int_{\epsilon}^{\fz}\int_\rn\chi_{\wqij}(y,t) F(y,t)
\wz\psi_t(x-y)\frac{dy\,dt}{t}.$$
Then, by an argument similar to that used in \cite[pp.\,221-222]{c77}
(see also \cite[p.\,650]{lyj}),
we conclude that there exist positive constants $C_1$ and $C_2$
such that, for all $\ez\in(0,\fz)$, $i\in\zz$ and $j\in\nn$,
$\supp \bije\st C_1\qij$, $\|\bije\|_{L^\fz(\rn)}\le C_22^i$,
$\int_\rn\bije(x)x^\gz\,dx=0$ for all $\gz\in\zz_+^n$ satisfying
$|\gz|\le s$ and
$$f=\lim_{\ez\to0}\sum_{i\in\zz}\sum_{j\in\nn}\bije\ \mbox{ in }\ \cs'(\rn).$$
Moreover, by the argument used in \cite[p.\,650]{lyj}, we find that
there exist $\{\bij\}_{i\in\zz,j\in\nn}\st L^{\fz}(\rn)$ and
a sequence $\{\ez_k\}_{k\in\nn}\st(0,\fz)$ such that
$\ez_k\to 0$ as $k\to \fz$ and,
for any $i\in\zz$, $j\in\nn$ and $g\in L^1(\rn)$,
\begin{equation}\label{10.7.x4}
\lim_{k\to\fz}\la b_{i,j}^{\ez_k},g\ra = \la \bij,g\ra,
\end{equation}
$\supp \bij\st C_1\qij$, $\|\bij\|_{L^{\fz}(\rn)}\le C_22^i$ and,
for all $\gz\in\zz_+^n$ with $|\gz|\le s$,
$$\int_\rn\bij(x)x^\gz\,dx
=\la b_{i,j}, x^\gz\chi_{C_1\qij}\ra
=\lim_{k\to\fz}\int_\rn b_{i,j}^{\ez_k}(x)x^\gz\,dx=0.$$
Next we show that
\begin{align}\label{10.7.x1}
\lim_{k\to\fz}\sum_{i\in\zz}\sum_{j\in\nn} b_{i,j}^{\ez_k}
=\sum_{i\in\zz}\sum_{j\in\nn} b_{i,j} \ \mbox{ in }\ \cs'(\rn).
\end{align}
Since $\|b_{i,j}^{\ez_k}\|_{L^\fz(\rn)}\ls2^i$,
$\|\bij\|_{L^\fz(\rn)}\ls2^i$ and, for all $\gz\in\zz_+^n$ with $|\gz|\le s$,
$$\int_\rn b_{i,j}^{\ez_k}(x)x^\gz\,dx=0=\int_\rn\bij(x)x^\gz\,dx,$$
it follows, from Remarks \ref{r-vlp}(iv) and \ref{r-hlb} and \eqref{9.16.x},
that, for all $\zeta\in\cs(\rn)$ and $k,\ N\in\nn$,
\begin{align*}
&\sum_{|i|>N}\sum_{j\in\nn}
[|\la b_{i,j}^{\ez_k},\zeta\ra|+|\la\bij,\zeta\ra|]\\
&\hs=\sum_{i=-\fz}^{-N-1}\sum_{j\in\nn}
[|\la b_{i,j}^{\ez_k},\zeta\ra|+|\la\bij,\zeta\ra|]
+\sum_{i=N+1}^{\fz}\sum_{j\in\nn}\lf\{
\lf|\int_{C_1\qij}b_{i,j}^{\ez_k}(x)
[\zeta(x)-P_{C_1\qij}^s\zeta(x)]\,dx\r|\r.\\
&\quad\quad+\lf.\lf|\int_{C_1\qij}\bij(x)
[\zeta(x)-P_{C_1\qij}^s\zeta(x)]\,dx\r|\r\}\\
&\hs\ls\sum_{i=-\fz}^{-N-1}2^i\int_{\rn}|\zeta(x)|\,dx
+\sum_{i=N+1}^{\fz}\sum_{j\in\nn}2^i
\int_{C_1\qij}|\zeta(x)-P_{C_1\qij}^s\zeta(x)|\,dx\\
&\hs\ls2^{-N}\|\zeta\|_{L^1(\rn)}+\sum_{i=N+1}^{\fz}\sum_{j\in\nn}
2^i\|\chi_{\qij}\|_{L^{\frac{p(\cdot)}{r}}(\rn)}
\|\zeta\|_{\cl_{1,\frac{p(\cdot)}{r},s}(\rn)}\\
&\hs\ls2^{-N}\|\zeta\|_{L^1(\rn)}
+\|\zeta\|_{\cl_{1,\frac{p(\cdot)}{r},s}(\rn)}
\sum_{i=N+1}^{\fz}2^i\|\chi_{\boz_i}\|_{L^{\frac{p(\cdot)}{r}}(\rn)}\\
&\hs\ls2^{-N}\|\zeta\|_{L^1(\rn)}
+\|\zeta\|_{\cl_{1,\frac{p(\cdot)}{r},s}(\rn)}
\lf[\sup_{i\in\zz}2^i\|\chi_{\boz_i}\|_{\lv}\r]
^{r}\sum_{i=N+1}^{\fz}2^{-i(r-1)}\\
&\hs\ls2^{-N}\|\zeta\|_{L^1(\rn)}
+2^{-N(r-1)}\|\zeta\|_{\cl_{1,\frac{p(\cdot)}{r},s}(\rn)}\|f\|_{\whv}^r,
\end{align*}
which tends to $0$ as $N\to\fz$,
where $r$ is chosen such that $r\in(\max\{p_+,1\},\fz)$ and, in the last inequality,
we used the fact that, for any $\zeta\in\cs(\rn)$,
$\|\zeta\|_{\cl_{1,\frac{p(\cdot)}{r},s}(\rn)}$ is finite
(see \cite[Lemma 2.8]{zyl}). Similarly, we have
$$\sum_{|i|\le N}\sum_{j\in\nn}[|\la b_{i,j}
^{\ez_k},\zeta\ra|+|\la\bij,\zeta\ra|]<\fz.$$
Therefore, by the argument same as that used in \cite[p.\,651]{lyj},
we conclude that \eqref{10.7.x1} holds true.
For all $i\in\zz$ and $j\in\nn$,
let $B_{i,\,j}$ be the ball having the same center
as $Q_{i,\,j}$ with the radius $5\sqrt n C_1 l_{Q_{i,\,j}}$,
$$\aij:=\frac{\bij}{C_2 2^i \|\chi_{\Bij}\|_{\lv}}\quad\mbox{and}
\quad\lij:=C_2 2^i \|\chi_{\Bij}\|_{\lv}.$$
Then, from the properties of $\bij$, it follows that
$\aij$ is a $(p(\cdot),\fz,s)$-atom associated with the ball
$\Bij$ and, due to \eqref{10.7.x1},
$f=\sum_{i\in\zz}\sum_{j\in\nn}\lij\aij$ in $\cs'(\rn)$.
Moreover, by Definition \ref{atd2}, Remark \ref{2.5.y} and \eqref{9.16.x},
we find that
\begin{align*}
\|f\|_{\whz}&\ls\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}\chi_\Bij\r\|_{\lv}
\ls\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}
\chi_{\qij}\r\|_{\lv}\\
&\ls\sup_{i\in\zz}2^i
\|\chi_{\boz_i}\|_{\lv}\ls \|f\|_{\whv}.
\end{align*}
Thus, $f\in\whz$ and $\|f\|_\whz\ls \|f\|_\whv$. This shows
$$\whv\st\whz,$$
which completes the proof of Theorem \ref{atthm1}.
\end{proof}
\section{Molecular characterizations of $\whv$\label{s-mole}}
\hskip\parindent
In this section, we establish the molecular characterization of $\whv$
and begin with the following definition of molecules.
\begin{defn}\label{mod1}
Let $p(\cdot)\in C^{\log}(\rn)$, $q\in (1,\fz]$, $s\in{\zz}_+$ and
$\epsilon\in (0, \fz)$. A measurable function $m$ is called a
$(p(\cdot), q, s,\epsilon)$-\emph{molecule} associated
with some ball $B\subset\rn$ if
\begin{enumerate}
\item[{\rm (i)}] for each $j\in\nn$, $\|m\|_{L^q(U_j(B))}\le 2^{-j\epsilon}
|U_j(B)|^{\frac{1}{q}}\|\chi_B\|_{\lv}^{-1}$,
where $U_0(B):=B$ and, for all $j\in\nn$,
$U_j(B):=(2^j B) \backslash (2^{j-1} B)$;
\item[{\rm (ii)}] $\int_{\rn}m(x)x^\beta dx=0$ for all
$\beta\in\zz_+^n$ with $|\beta|\leq s$.
\end{enumerate}
\end{defn}
\begin{defn}\label{mod2}
Let $p(\cdot)\in C^{\log}(\rn)$, $q\in(1,\fz]$,
$s\in(\frac{n}{p_-}-n-1,\fz)\cap{\zz}_+$ with $p_-$ as in \eqref{2.1x},
and $\epsilon\in (0, \fz)$.
The \emph{variable weak molecular Hardy space}
$\whm$ is defined as the space of all $f\in\cs'(\rn)$
which can be decomposed as $f=\sum_{i\in\zz}\sum_{j\in\nn}\lij\mij$
in $\cs'(\rn)$, where $\{\mij\}_{i\in\zz,j\in\nn}$ is a sequence of
$(p(\cdot),q,s,\epsilon)$-molecules associated with balls
$\{\Bij\}_{i\in\zz,j\in\nn}$,
$\{\lij\}_{i\in\zz,j\in\nn}:=\{\wz A2^i\|\chi_{\Bij}\|_{\lv}\}_{i\in\zz,j\in\nn}$ with
$\wz A$ being a positive constant independent of $i,\ j$, and there exist
positive constants $A$ and $C$ such that, for all $i\in\zz$ and
$x\in\rn, \sum_{j\in\nn}\chi_{C\Bij}(x)\le A$.
Moreover, for any $f\in\whm$, define
$$\|f\|_{\whm}:=\inf\lf[\sup_{i\in\zz}{\lf\|\lf\{\sum_{j\in\nn}
\lf[\frac{\lij\chi_\Bij}
{\|\chi_\Bij\|_{\lv}}\r]^{\underline{p}}\r\}^{1/{\underline{p}}}\r\|
_{\lv}}\r],\noz\\$$
where the infimum is taken over all decompositions of $f$ as above.
\end{defn}
\begin{thm}\label{mothm1}
Let $p(\cdot)\in C^{\log}(\rn)$, $q\in(\max\{p_+,1\},\fz]$,
$s\in(\frac{n}{p_-}-n-1,\fz)\cap{\zz}_+$ and
$\epz\in (n+s+1, \fz)$, where $p_+$ and $p_-$ are as in \eqref{2.1x}.
Then $\whv=\whm$ with equivalent quasi-norms.
\end{thm}
\begin{proof}
Notice that a $(p(\cdot), \fz,s)$-atom
is also a $(p(\cdot), q, s,\epsilon)$-molecule.
Thus, we have
$$\whv\subset\whz\subset\whm.$$
Therefore, to prove this theorem, it suffices to show $\whm\subset\whv$.
Let $m$ be any fixed $(p(\cdot),q,s,\epsilon)$-molecule
associated with some ball $B:=B(x_B,r_B),$
where $x_B\in\rn$ and $r_B\in(0,\fz)$.
Then we claim that $m$ is an infinite linear combination of
$(p(\cdot),q,s)$-atoms.
To prove this, for all $k\in\zz_+$, let $m_k:=m\chi_{U_k(B)}$ with
$U_k(B)$ as in Definition \ref{mod1}(i),
and $\cp_k$ be the {linear vector space} generated by the set
$\{x^\gamma\chi_{U_k(B)}\}_{|\gamma|\le s}$ of ``polynomials". It is well known
(see, for example, \cite{tg80}) that, for any given $k\in\zz_+$,
there exists a unique {polynomial}
$P_k\in\cp_k$ such that, for all multi-indices $\beta$ with $|\beta|\le s$,
\begin{align}\label{11.9.x1}
\int_{\rn}x^\beta[m_k(x)-P_k(x)]\,dx=0,
\end{align}
where $P_k$ is given by
\begin{align}\label{11.9.x5}
P_k:=\sum_{\beta\in\zz_+^n,|\beta|\le s}
\lf[\frac{1}{|U_k(B)|}\int_{\rn}x^\beta m_k(x)\,dx\r]Q_{\beta,k}
\end{align}
and $Q_{\beta,k}$ is the unique polynomial in $\cp_k$ satisfying
that, for all multi-indices $\beta$ with $|\beta|\le s$ and the
\emph{Kronecker delta} $\delta_{\gamma,\beta}$,
\begin{align}\label{11.9.x3}
\int_{\rn}x^\gamma Q_{\beta,k}(x)\,dx=|U_k(B)|\delta_{\gamma,\beta},
\end{align}
where, when $\gamma=\beta$, $\delta_{\gamma,\beta}:=1$ and, when $\gamma\neq \beta$,
$\delta_{\gamma,\beta}:=0$.
Recall that it was proved in \cite[p.\,83]{tg80} that, for all $k\in\zz_+$,
\begin{align*}
\sup_{x\in U_k(B)}|P_k(x)|\ls \frac{1}{|U_k(B)|}\|m_k\|_{L^1(\rn)}.
\end{align*}
From this and the H\"older inequality, we deduce that, for all $k\in\zz_+$,
\begin{align}\label{11.9.x2}
\|m_k-P_k\|_{L^q(U_k(B))}
&\le\|m_k\|_{L^q(U_k(B))}+\|P_k\|_{L^q(U_k(B))}\le \wz C\|m_k\|_{L^q(U_k(B))}\noz\\
&\le \wz C2^{-k\epsilon}|2^kB|^{1/q}\|\chi_{B}\|_{\lv}^{-1},
\end{align}
where $\wz C$ is a positive constant independent of $m$, $B$ and $k$.
Obviously, for all $k\in\zz_+$, $\supp (m_k-P_k)\st U_k(B)$.
By this and \eqref{11.9.x1}, for any given $k\in\zz_+$, if we let
$$a_k:=\frac{2^{k\epsilon}\|\chi_{B}\|_{\lv}(m_k-P_k)}{\wz C\|\chi_{2^kB}\|_{\lv}},$$
we then conclude that $a_k$ is a $(p(\cdot),q,s)$-atom. Therefore,
\begin{align}\label{11.9.y2}
\sum_{k=0}^{\fz}(m_k-P_k)=\sum_{k=0}^{\fz}\mu_ka_k
\end{align}
is an infinite linear combination of $(p(\cdot),\fz,s)$-atoms,
where, for any $k\in\zz_+$,
$$\mu_k:=\wz C2^{-k\epsilon}\|\chi_{2^kB}\|_{\lv}/\|\chi_{B}\|_{\lv}.$$
Next we prove that $\sum_{k=0}^\fz P_k$ can be divided
into an infinite linear combination of $(p(\cdot),\fz,s)$-atoms.
For any $j\in\zz_+$ and $\ell\in\zz_+^n$, let
$$N_\ell^j:=\sum_{k=j}^\fz\int_{U_k(B)}m_k(x)x^\ell\,dx.$$
Then, for any $\ell\in\zz_+^n$ with $|\ell|\le s$, it holds true that
\begin{align}\label{11.9.x6}
N_\ell^0=\sum_{k=0}^\fz\int_{U_k(B)}m_k(x)x^\ell\,dx
=\int_{\rn}m(x)x^\ell\,dx=0.
\end{align}
From this and \eqref{11.9.x5}, we further deduce that
\begin{align}\label{11.9.x7}
\sum_{k=0}^{\fz}P_k&=\sum_{\ell\in\zz_+^n,|\ell|\le s}
\sum_{k=0}^{\fz}|U_k(B)|^{-1}Q_{\ell,k}\int_{\rn}m_k(x)x^\ell\,dx\noz\\
&=\sum_{\ell\in\zz_+^n,|\ell|\le s}\sum_{k=0}^{\fz}N_\ell^{k+1}
\lf[|U_{k+1}(B)|^{-1}Q_{\ell,k+1}\chi_{U_{k+1}(B)}(x)-|U_k(B)|^{-1}Q_{\ell,k}\chi_{U_k(B)}(x)\r]\noz\\
&=:\sum_{\ell\in\zz_+^n,|\ell|\le s}\sum_{k=0}^{\fz}b_{\ell}^k.
\end{align}
By an argument similar to that used in the proof of \cite[(4.35)]{hyy}, we conclude that,
for any $k\in\zz_+$ and $\ell\in\zz_+^n$ with $|\ell|\le s$,
\begin{equation}\label{5.7x}
\|b_\ell^k\|_{L^\fz(\rn)}\ls 2^{-k\epsilon}\|\chi_B\|_{\lv}^{-1}\quad
{\rm and}\quad \supp b_\ell^k\subset 2^{k+1}B;
\end{equation}
moreover, for all $\gamma\in\zz_+^n$ with $|\gamma|\le s$,
$\int_\rn b_\ell^k(x)x^\gamma\,dx=0$.
For all $k\in\zz_+$ and $\ell\in\zz_+^n$ with $|\ell|\le s$, let
$$\mu_\ell^k:=2^{-k\epsilon}\frac{\|\chi_{2^{k+1}B}\|_{\lv}}{\|\chi_{B}\|_{\lv}}
\quad{\rm and}\quad
a_\ell^k:=2^{k\epsilon}b_{\ell}^k\frac{\|\chi_{B}\|_{\lv}}{\|\chi_{2^{k+1}B}\|_{\lv}}.$$
Then, for any $k\in\zz_+$ and $\ell\in\zz_+^n$ with $|\ell|\le s$,
by \eqref{5.7x} and the definition of $a_\ell^k$, we find that $a_\ell^k$
is a $(p(\cdot),\fz,s)$-atom supported on $2^{k+1}B$ up to a positive constant multiple.
Thus,
\begin{equation}\label{1.29x}
\sum_{k=0}^{\fz}P_k=\sum_{\ell\in\zz_+^n,|\ell|\le s}\sum_{k=0}^{\fz}
\mu_\ell^ka_\ell^k
\end{equation}
forms an infinite linear combination of $(p(\cdot),\fz,s)$-atoms.
Combining \eqref{11.9.y2} and \eqref{1.29x}, we conclude that
\begin{align}\label{11.14x1}
m=\sum_{k=0}^{\fz}m_k=\sum_{k=0}^{\fz}(m_k-P_k)+\sum_{k=0}^{\fz}P_k=\sum_{k=0}^{\fz}\mu_ka_k+
\sum_{\ell\in\zz_+^n,|\ell|\le s}\sum_{k=0}^{\fz}\mu_\ell^ka_\ell^k.
\end{align}
This shows that a $(p(\cdot),q,s,\epsilon)$-molecule can be divided into an infinite
linear combination of $(p(\cdot),q,s)$-atoms. Therefore, the above claim holds true.
Now we prove $\whm\subset\whv$. Let $f\in\whm$. Then, by an argument
similar to that used in Remark \ref{atomlm3}, we know that there exist $\{\lij\}_{i\in\zz,j\in\nn}$ and a sequence $\{\mij\}_{i\in\zz,j\in\nn}$ of $(p(\cdot), q, s,\epsilon)$-molecules
associated with balls $\{\Bij\}_{i\in\zz,j\in\nn}$
such that, for all $i\in\zz$, $\sum_{j\in\nn}\chi_{c\Bij}\ls 1$ with
$c\in(0,1]$ being a positive constant independent of $i$, and
$f=\sum_{i\in\zz}\sum_{j\in\nn}\lij\mij$ in $\cs'(\rn)$, moreover,
\begin{align}\label{11.17.x1}
\|f\|_{\whm}\sim\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}\chi_\Bij\r\|_{\lv}.
\end{align}
Next we prove that, for any $\az\in (0, \fz)$,
\begin{align}\label{10.11.z}
\az\lf\|\chi_{\{x\in\rn: f^*(x)>\az\}}\r\|
_{\lv}\ls\|f\|_{\whm},
\end{align}
where the implicit positive constant is independent of $\az$
and $f^*:=f^*_{N,+}$ with $N$ as in Theorem \ref{mthm1}.
For any $\az\in(0,\fz)$, let $i_0\in\zz$
such that $2^{i_0}\le\az<2^{i_0+1}$. Then we have
$$f=\sum_{i=-\fz}^{i_0-1}\sum_{j\in\nn}\lij\mij
+\sum_{i=i_0}^{\fz}\sum_{j\in\nn}\lij\mij=:f_1+f_2$$
and
\begin{align}\label{emodep1}
\lf\|\chi_{\{x\in\rn:\ f^*(x)>\az\}}\r\|_{\lv}
&\ls\lf\|\chi_{\{x\in\rn:\ f_1^*(x)>\frac{\az}{2}\}}\r\|_{\lv}
+\lf\|\chi_{\{x\in\rn:\ f_2^*(x)>\frac{\az}{2}\}}\r\|_{\lv}\noz\\
&=:{\rm I}_1+{\rm I}_2.
\end{align}
We first estimate ${\rm I}_1$.
To this end, we need another estimate for $(\mij)^*$. From \eqref{11.14x1}, we deduce that,
for all $i\in\zz$ and $j\in\nn$, there exists a sequence of multiples
of $(p(\cdot),q,s)$-atoms, $\{a_{i,j}^l\}_{l\in\zz_+}$, associated with balls
$\{2^{l+1}\Bij\}_{l\in\zz_+}$ such that
$$\|a_{i,j}^l\|_{L^q(\rn)}\ls
\frac{2^{-l\epsilon}|2^{l+1}\Bij|^{1/q}}{\|\chi_{\Bij}\|_{\lv}}$$
and $\mij=\sum_{l\in\zz_+}a_{i,j}^l$ almost everywhere in $\rn$.
Then, for all $i\in\zz\cap(-\fz,i_0-1]$ and $j\in\nn$, we have
\begin{align}\label{10.11.x}
(\mij)^*\le\sum_{l\in\zz_+}(a_{i,j}^l)^*
=\sum_{l\in\zz_+}\sum_{k\in\zz_+}(a_{i,j}^l)^*
\chi_{U_k(2^l\Bij)}=:\sum_{l\in\zz_+}\sum_{k=0}^2{\rm J}_{l,k}
+\sum_{l\in\zz_+}\sum_{k=3}^{\fz}{\rm J}_{l,k},
\end{align}
where $U_k(2^l\Bij)$ is defined as in Definition \ref{mod1}(i) with $B$
replaced by $2^l\Bij$.
Thus, it follows that
\begin{align}\label{11.16.x3}
{\rm I}_1&=\lf\|\chi_{\{x\in\rn:\ f_1^*(x)>\frac{\az}{2}\}}\r\|_{\lv}
\le\lf\|\chi_{\{x\in\rn:\ \sum_{i=-\fz}^{i_0-1}\sum_{j\in\nn}\lij(\mij)^*(x)
>\frac{\az}{2}\}}\r\|_{\lv}\noz\\
&\ls\lf\|\chi_{\{x\in\rn:\ \sum_{i=-\fz}^{i_0-1}
\sum_{j\in\nn}\sum_{l\in\zz_+}
\sum_{k=0}^2\lij{\rm J}_{l,k}>\frac{\az}{4}\}}\r\|_{\lv}\noz\\
&\quad+\lf\|\chi_{\{x\in\rn:\ \sum_{i=-\fz}^{i_0-1}\sum_{j\in\nn}
\sum_{l\in\zz_+}\sum_{k=3}^{\fz}\lij{\rm J}_{l,k}
>\frac{\az}{4}\}}\r\|_{\lv}\noz\\
&=:{\rm I}_{1,1}+{\rm I}_{1,2}.
\end{align}
For ${\rm I}_{1,1}$, by an argument similar to that used in the proof of \eqref{eqatomi11},
we conclude that
\begin{align}\label{11.15.x1}
\az{\rm I}_{1,1}\ls\|f\|_{\whm}.
\end{align}
On the other hand, by an argument similar to that used in the proof of \eqref{11.14.x2},
we conclude that, for any $i\in\zz$, $j\in\nn$, $l\in\zz_+$, $k\in[3,\fz)\cap\zz_+$,
$x\in U_k(2^l\Bij)$ and $y\in2^{l+1}\Bij$,
\begin{align}\label{mostar}
{\rm J}_{l,k}
&\ls\frac{|y-\xij|^{s+1}}{|x-\xij|^{n+s+1}}
\int_{2^{l+1}\Bij} |a_{i,j}^l(y)|\,dy\chi_{U_k(2^l\Bij)}(x)\noz\\
&\ls\frac{|y-\xij|^{s+1}}{|x-\xij|^{n+s+1}}\|a_{i,j}^l\|
_{L^q(\rn)}|2^{l+1}\Bij|^{1/{q'}}\chi_{U_k(2^l\Bij)}(x)\noz\\
&\ls\frac{2^{-l(n+\epsilon)-k(n+s+1)}}{\rij^n\|\chi_{\Bij}\|_{\lv}}
|2^{l+1}\Bij|\chi_{U_k(2^l\Bij)}(x)\noz\\
&\ls\frac{2^{-l\epsilon-k(n+s+1)}}
{\|\chi_{\Bij}\|_{\lv}}\chi_{U_k(2^l\Bij)}(x),
\end{align}
which, combined with \eqref{10.11.x}, \eqref{11.17.x1}, Remark \ref{2.5.y},
the fact that $\epsilon\in(n+s+1,\fz)$ and via choosing
$r\in(\frac{n}{n+s+1},\underline p)$, implies that
\begin{align*}
\az{\rm I}_{1,2}
&\ls\az^{1-1/r}\lf\|\sum_{i=-\fz}^{i_0-1}\sum_{j\in\nn}\sum_{l\in\zz_+}
\sum_{k=3}^{\fz}2^i2^{-l\epsilon}2^{-k(n+s+1)}
\chi_{U_k(2^l\Bij)}\r\|_{L^{\frac{p(\cdot)}{r}}(\rn)}^{1/r}\\
&\ls\az^{1-1/r}\lf[\sum_{l\in\zz_+}\sum_{k=3}^{\fz}2^{-l\epsilon}2^{-k(n+s+1)}
\sum_{i=-\fz}^{i_0-1}2^i\lf\|\sum_{j\in\nn}
\chi_{U_k(2^l\Bij)}\r\|_{L^{\frac{p(\cdot)}{r}}(\rn)}\r]^{1/r}\\
&\ls\az^{1-1/r}\lf[\sum_{l\in\zz_+}
\sum_{k=3}^{\fz}2^{-l\epsilon}2^{-k(n+s+1)}2^{\frac{n(k+l)}{r}}
\sum_{i=-\fz}^{i_0-1}2^i\lf\|\sum_{j\in\nn}
\chi_{\Bij}\r\|_{L^{\frac{p(\cdot)}{r}}(\rn)}\r]^{1/r}\\
&\ls\az^{1-1/r}\lf[\sum_{i=-\fz}^{i_0-1}2^i\lf\|\sum_{j\in\nn}
\chi_{\Bij}\r\|_{L^{\frac{p(\cdot)}{r}}(\rn)}\r]^{1/r}\\
&\ls\az^{1-1/r}\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}\chi_{\Bij}\r\|_{\lv}
\lf[\sum_{i=-\fz}^{i_0-1}2^{i(1-r)}\r]^{1/r}
\sim\|f\|_{\whm}.
\end{align*}
From this, \eqref{11.16.x3} and \eqref{11.15.x1}, we deduce that
\begin{align}\label{11.16.x4}
\az{\rm I}_1\ls\|f\|_{\whm}.
\end{align}
We next estimate ${\rm I_2}$. By \eqref{10.11.x}, we know that
\begin{align}\label{11.16.x1}
{\rm I}_2&\ls\lf\|\chi_{\{x\in\rn:\ \sum_{i=i_0}^{\fz}
\sum_{j\in\nn}\sum_{l\in\zz_+}
\sum_{k=0}^2\lij{\rm J}_{l,k}>\frac{\az}{4}\}}\r\|_{\lv}\noz\\
&\quad\hs+\lf\|\chi_{\{x\in\rn:\ \sum_{i=i_0}^{\fz}\sum_{j\in\nn}
\sum_{l\in\zz_+}\sum_{k=3}^{\fz}\lij{\rm J}_{l,k}
>\frac{\az}{4}\}}\r\|_{\lv}\noz\\
&=:{\rm I}_{2,1}+{\rm I}_{2,2}.
\end{align}
Let $\q1\in(1,\min\{\frac{q}{\max\{p_+,1\}},\frac{1}{b}\})$ and
$a\in(1-\frac1{\q1},\fz)$ for any $b\in(0,\underline{p})$,
where $p_+$ and $\underline{p}$ are as in \eqref{2.1x}, respectively, \eqref{2.1y}.
Then, with an argument similar to that used in the proof of \eqref{11.15.x1}, we obtain
\begin{align*}
{\rm I}_{2,1}&\ls 2^{-i_0\q1(1-a)}
\lf\{\sum_{i=i_0}^{\fz}2^{[(1-a){\q1}-1]ib}
\r\}^{\frac{1}{b}}\sup_{i\in\zz}2^i\lf\|
\lf(\sum_{j\in\nn}\chi_{\Bij}\r)
^{\frac{1}{b}}\r\|_{\lv}\\
&\ls\az^{-1}\|f\|_{\whm},
\end{align*}
which shows that
\begin{align}\label{11.16.x5}
\az{\rm I}_{2,1}\ls\|f\|_{\whm}.
\end{align}
On the other hand, let $a\in(\frac{1}{\underline p},\fz)$ and
$b\in(1-\frac{1}{a},\fz)$. By the H\"older inequality,
we find that, for all $x\in\rn$,
\begin{align}\label{10.11.y}
\sum_{i=i_0}^{\fz}\sum_{j\in\nn}
\sum_{l\in\zz_+}\sum_{k=3}^{\fz}\lij{\rm J}_{l,k}
&\le\lf(\sum_{i=i_0}^{\fz}2^{iba'}\r)^{1/a'}\lf
[\sum_{i=i_0}^{\fz}2^{-iba}\lf(\sum_{j\in\nn}
\sum_{l\in\zz_+}\sum_{k=3}^{\fz}\lij{\rm J}_{l,k}\r)^a\r]^{1/a}\noz\\
&=\frac{2^{i_0b}}{(2^{ba'}-1)^{1/a'}}\lf[\sum_{i=i_0}^{\fz}
2^{-iba}\lf(\sum_{j\in\nn}\sum_{l\in\zz_+}
\sum_{k=3}^{\fz}\lij{\rm J}_{l,k}\r)^a\r]^{1/a}.
\end{align}
Choose $d\in(\frac{n}{n+s+1},1)$. Then, by \eqref{10.11.y},
\eqref{mostar} and Remark \ref{2.5.y},
we find that
\begin{align*}
\hs\hs{\rm I}_{2,2}
&\le\lf\|\chi_{\{x\in\rn:\ \frac{2^{i_0b}}{(2^{ba'}-1)^{1/a'}}
\lf[\sum_{i=i_0}^{\fz}2^{-iba}\lf(\sum_{j\in\nn}\sum_{l\in\zz_+}
\sum_{k=3}^{\fz}\lij{\rm J}_{l,k}\r)^a\r]^{1/a}>2^{i_0-2}\}}\r\|_{\lv}\\
&\ls 2^{-i_0a(1-b)}\lf\|\sum_{i=i_0}^{\fz}2^{-iba}\lf(\sum_{j\in\nn}
\sum_{l\in\zz_+}\sum_{k=3}^{\fz}\lij{\rm J}_{l,k}\r)^a\r\|_{\lv}\\
&\ls2^{-i_0a(1-b)}\lf\|\sum_{i=i_0}^{\fz}2^{i(1-b)}
\sum_{l\in\zz_+}\sum_{k=3}^{\fz}2^{-l\epsilon}2^{-k(n+s+1)}
\sum_{j\in\nn}\chi_{U_k(2^l\Bij)}\r\|_{L^{ap(\cdot)(\rn)}}^a\\
&\ls2^{-i_0a(1-b)}\lf[\sum_{i=i_0}^{\fz}2^{i(1-b)}
\sum_{l\in\zz_+}\sum_{k=3}^{\fz}2^{-l\epsilon}2^{-k(n+s+1)}
\lf\|\sum_{j\in\nn}\chi_{U_k(2^l\Bij)}\r\|_{L^{ap(\cdot)(\rn)}}\r]^a\\
&\ls2^{-i_0a(1-b)}\lf[\sum_{i=i_0}^{\fz}2^{i(1-b)}
\lf\{\sum_{l\in\zz_+}\sum_{k=3}^{\fz}2^{-l\epsilon}2^{-k(n+s+1)}2^{\frac{n(k+l)}{d}}\r\}
\lf\|\sum_{j\in\nn}\chi_{\Bij}\r\|_{L^{ap(\cdot)(\rn)}}\r]^a\\
&\ls2^{-i_0a(1-b)}\lf[\sum_{i=i_0}^{\fz}2^{i(1-b)}
\lf\|\sum_{j\in\nn}\chi_{\Bij}\r\|_{L^{ap(\cdot)(\rn)}}\r]^a\\
&\ls2^{-i_0a(1-b)}\lf[\sum_{i=i_0}^{\fz}2^{i(1-b-\frac{1}{a})}\r]^a
\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}\chi_{\Bij}\r\|_{\lv}\\
&\ls\az^{-1}\sup_{i\in\zz}2^i\lf\|\sum_{j\in\nn}\chi_{\Bij}\r\|_{\lv}
\sim\az^{-1}\|f\|_{\whm},
\end{align*}
which, combined with \eqref{11.16.x5} and \eqref{11.16.x1},
implies that $\az{\rm I}_2\ls\|f\|_{\whm}$.
This, together with \eqref{emodep1} and \eqref{11.16.x4},
shows that \eqref{10.11.z} holds true and hence
finishes the proof of Theorem \ref{mothm1}.
\end{proof}
\section{Littlewood-Paley function characterizations of $\whv$\label{s-palay}}
\hskip\parindent
In this section, we establish Littlewood-Paley
function characterizations of $\whv$ in Theorems
\ref{lpthm1}, \ref{lpthm2} and \ref{10.10.x}
as an application of the atomic characterization of $\whv$ in Theorem \ref{atthm1}.
Let $\phi\in\mathcal{S}(\mathbb R^n)$ be a radial function satisfying
\begin{align}\label{12.7.x1}
\supp\phi\subset\{x\in\mathbb R^n:\ |x|\le 1\},
\end{align}
\begin{align}\label{12.7.x2}
\int_{\mathbb R^n}\phi(x)x^\gz \,dx=0\ \ \mbox{for all}\ \
\gz\in\zz_+^n\ \ \mbox{with}\ \ |\gz|
\le\max\lf\{\lf\lfloor\frac{n}{p_-}-n-1\r\rfloor,0\r\}
\end{align}
and
\begin{align}\label{12.7.x3}
\int_0^\infty|\widehat{\phi}(\xi t)|^2\,\frac{dt}{t}=1\ \ \mbox{for all}
\ \ \xi\in\rn\setminus\{\vec0_n\}.
\end{align}
Here and hereafter, the \emph{symbol $\lfloor s \rfloor$} for any $s\in\rr$ denotes the maximal integer not larger than
$s$. Recall that, for all $f\in\mathcal{S}'(\mathbb R^n)$,
the \emph{Littlewood-Paley $g$-function},
the \emph{Lusin area function} and the \emph{Littlewood-Paley $g_\lz^*$-function}
of $f$ with $\lz\in(0,\fz)$ are defined,
respectively, by setting, for all $x\in\rn$,
\begin{align*}
g(f)(x):=\lf[\int_0^\infty\lf|f\ast\phi_t(x)\r|^2\dt\r]^{1/2},
\end{align*}
\begin{align*}
S(f)(x):=\lf[\int_{\Gamma(x)}
|f\ast\phi_t(y)|^2\,\frac{\,dy\,dt}{t^{n+1}}\r]^{1/2}
\end{align*}
and
\begin{align*}
g_\lz^*(f)(x):=\lf[\int_0^\infty\int_{\mathbb R^n}
\lf(\frac{t}{t+|x-y|}\r)^{\lz
n}\lf|f\ast\phi_t(y)\r|^2\,\frac{\,dy\,dt}{t^{n+1}}\r]^{1/2},
\end{align*}
where, for any $x\in\rn$, $\Gamma(x):=\{(y,t)\in\rn\times(0,\fz):\ |y-x|<t\}$
and, for any $t\in(0,\fz)$, $\phi_t(\cdot):=\frac{1}{t^n}\phi(\frac{\cdot}{t})$.
Recall that $f\in\mathcal{S}'(\mathbb R^n)$ is said to
\emph{vanish weakly at infinity} if, for every $\phi\in\mathcal{S}(\mathbb R^n)$,
$f\ast\phi_t\to 0$ in $\mathcal{S}'(\mathbb R^n)$ as $t\to \infty$
(see, for example, \cite[p.\,50]{fs82}).
The main results of this section are stated as follows.
\begin{thm}\label{lpthm1}
Let $p(\cdot)\in C^{\log}(\rn)$.
Then $f\in\whv$ if and only if
$f\in\cs'(\rn)$, $f$ vanishes weakly at infinity and $S(f)\in\wlv$.
Moreover, for all $f\in\whv$,
$$C^{-1}\|S(f)\|_{\wlv}\le\|f\|_{\whv}\le C\|S(f)\|_{\wlv},$$
where $C$ is a positive constant independent of $f$.
\end{thm}
\begin{thm}\label{lpthm2}
Let $p(\cdot)\in C^{\log}(\rn)$.
Then $f\in\whv$ if and only if
$f\in\cs'(\rn)$, $f$ vanishes weakly at infinity
and $g(f)\in\wlv$. Moreover, for all $f\in\whv$,
$$C^{-1}\|g(f)\|_{\wlv}\le\|f\|_{\whv}\le C\|g(f)\|_{\wlv},$$
where $C$ is a positive constant independent of $f$.
\end{thm}
\begin{thm}\label{10.10.x}
Let $p(\cdot)\in C^{\log}(\rn)$ and $\lambda\in(1+\frac{2}{{\min\{p_-,2\}}}, \fz)$.
Then $f\in\whv$ if and only if $f\in\cs'(\rn)$, $f$ vanishes weakly at infinity and
$g_\lambda^{\ast}(f)\in\wlv$. Moreover, for all $f\in\whv$,
$$C^{-1}\|g_\lz^\ast(f)\|_{\wlv}\le\|f\|_{\whv}\le C\|g_\lz^\ast(f)\|_{\wlv},$$
where $C$ is a positive constant independent of $f$.
\end{thm}
\begin{rem}
In \cite[Theorem 4.13]{lyj}, Liang et al. established the $g_\lz^\ast$-function
characterization of the weak Hardy space $W\!H^p(\rn)$ with constant exponent
$p\in(0,1]$, as a special
case of the weak Musielak-Orlicz Hardy space $W\!H^\vz(\rn)$, with the best known range
for $\lz\in(2/p,\fz)$. However, it is still unclear whether or not the
$g_\lz^\ast$-function, when
$\lz\in(\frac{2}{\min\{p_-,2\}},1+\frac{2}{\min\{p_-,2\}}]$,
can characterize $\whv$, since the method used in the proof of
Theorem \ref{10.10.x} does not work in this case, while the method used in
\cite[Theorem 4.13]{lyj} strongly depends on the properties of uniformly
Muckenhoupt weights, which are not satisfied by $t^{p(\cdot)}$
with $p(\cdot)\in C^{\log}(\rn)$ (see Remark \ref{1.29.x2}(iii)).
\end{rem}
To prove Theorems \ref{lpthm1}, \ref{lpthm2} and \ref{10.10.x},
we need some technical lemmas.
\begin{lem}\label{lppro1}
Let $p(\cdot)\in C^{\log}(\rn)$. If $f\in\whv$,
then $f$ vanishes weakly at infinity.
\end{lem}
\begin{proof}
Let $f\in\whv$. Then, by Remark \ref{r-m-fun}(i), we know that,
for any $\phi\in\cs(\rn)$, $t\in(0, \fz)$, $x\in\rn$ and $y\in B(x, t)$,
$|f*{\phi_t}(x)|\ls f_{N,\triangledown}^\ast(y)\ls f_{N,+}^\ast(y)$,
where $N\in(\frac{n}{\underline{p}}+n+1,\fz)$ with $\underline{p}$ as in \eqref{2.1y}.
Thus, there exists a positive constant
$C_0$, independent of $x$, $t$ and $f$, such that
$$B(x, t)\subset\{y\in\rn:\
f_{N,+}^\ast(y)\ge C_0|f*{\phi_t}(x)|\}.$$
By this, Remark \ref{r-vlp}(ii) and an argument similar to that used in the
proof of \eqref{max-f3}, we conclude that, for all $x\in\rn$,
\begin{align*}
&\min\{\lf|f*{\phi_t}(x)\r|^{p_-}, \lf|f*{\phi_t}(x)\r|^{p_+}\}\\
&\hs\ls\frac{1}{|B(x,t)|}\max\lf\{\|f\|_{\whv}^{p_-},\|f\|
_{\whv}^{p_+}\r\}\rightarrow0
\end{align*}
as $t\rightarrow\fz$, which implies that $f$ vanishes weakly at infinity,
where $p_+$ and $p_-$ are as in \eqref{2.1x}.
This finishes the proof of Lemma \ref{lppro1}.
\end{proof}
The following inequality of the Lusin area function on classical
Lebesgue spaces is well known,
whose proof can be found, for example, in \cite[Chapter 7]{fs82} (see also \cite{st89}).
\begin{lem}\label{lm-9.22}
Let $q\in(1,\fz)$. Then there exists a positive constant $C$ such that,
for all $f\in L^q(\rn)$,
$$C^{-1}\|f\|_{L^q(\rn)}\le\|S(f)\|_{L^q(\rn)}\le C\|f\|_{L^q(\rn)}.$$
\end{lem}
\begin{proof}[Proof of Theorem \ref{lpthm1}]
We first prove the sufficiency.
Let $f\in\cs'(\rn)$, $f$ vanish
weakly at infinity and $S(f)\in\wlv$. Then
we prove that $f\in \wha$ for some $q$ and $s$ as in
Theorem \ref{atthm1} and
\begin{equation}\label{area-4}
\|f\|_{\whv}\sim\|f\|_{\wha}\ls\|S(f)\|_{\wlv}.
\end{equation}
Denote by $\cq$ the set of all dyadic cubes in $\rn$.
For any $i\in\zz$, let $$\Omega_i:=\{x\in\rn:\ S(f)(x)>2^i\}$$
and
$$\cq_i:=\lf\{Q\in\cq:\ |Q\cap\Omega_i|\ge\frac{|Q|}{2}\ {\rm and}\
|Q\cap\Omega_{i+1}|<\frac{|Q|}{2}\r\}.$$
For any $i\in\zz$, we use $\{\qij\}_{j}$ to denote the maximal dyadic cubes in $\cq_i$, namely,
there does not exist $Q\in\cq_i$ such that $\qij\subsetneqq Q$. For any $Q\in\cq$, let
$\ell(Q)$ denote its side length and
$$Q^+:=\{(y,t)\in\mathbb R^{n+1}_+:\ y\in Q,\, \sqrt n{\ell(Q)}<t\le
2\sqrt n\ell(Q)\}$$
and, for all $i\in\zz$ and $j$, let
$$\Bij:=\bigcup_{Q\in\cq_i,\,Q\st Q_{i,j}}Q^+.$$
Here we point out that $Q^+$ for different $Q\in \cq_i$ and $Q\subset Q_{i,j}$
are mutually disjoint.
Then, by the proof of \cite[Theorem 4.5]{lyj}, we find that
\begin{equation}\label{area-3}
f=:\sum_{i\in\zz}\sum_{j}\lz_{i,j}\aij\quad \mbox{in}\quad \cs'(\rn),
\end{equation}
where, for any $i\in\zz$, $j$ and $x\in\rn$,
$\lz_{i,j}:=2^i\|\chi_{4\sqrt nQ_{i,j}}\|_{\lv}$,
\begin{align*}
\aij(x):=&\frac1{\lz_{i,j}}\int_{\Bij}f*\phi_t(y)\phi_t(x-y)\frac{dy\,dt}{t}\\
=&\frac1{\lz_{i,j}}\sum_{Q\in\cq_i,Q\st \qij}\int_{Q^+}f*\phi_t(y)
\phi_t(x-y)\frac{\,dy\,dt}{t}
=:\frac1{\lz_{i,j}}\sum_{Q\in\cq_i,Q\st \qij}e_Q(x),
\end{align*}
and $\phi$ is as in \eqref{12.7.x1}, \eqref{12.7.x2}
and \eqref{12.7.x3}.
Moreover, for any $i\in\zz$ and $j$,
$\supp\aij\subset\wz{Q}_{i,j}:=4{\sqrt n}\qij$ and
$$\int_{\rn}\aij(x)x^{\beta} dx=0\quad\mbox{for}\quad|\beta|\le s.$$
Next we estimate $\|\aij\|_{L^q(\rn)}$ for all $i\in\zz$ and $j$.
Let
$$\wz\boz_i:=\lf\{x\in\rn:\ \cm(\chi_{\boz_i})(x)\ge \frac{1}{2}\r\}.$$
Observe that $|Q\cap \Omega_i|\ge \frac{|Q|}2$ for any $Q\in \cq_i$, which implies
that $Q\st \wz \Omega_i$. From this and the fact that
$|Q\cap \Omega_{i+1}|<\frac{|Q|}2$, we deduce that, for all $i\in\zz$
and $x\in Q\in\cq_i$,
\begin{equation}\label{area-1}
\cm\lf(\chi_{Q\cap(\wz\boz_i\setminus\boz_{i+1})}\r)(x)\ge \frac{\chi_Q(x)}{2}.
\end{equation}
For any $Q\in\cq_i$, let
$$c_Q:=\lf[\int_{Q^+}|\phi_t*f(y)|^2\frac{dy\,dt}{t^{n+1}}\r]^{1/2}.$$
Notice that, for all $i\in\zz$, $j$ and $x\in\rn$,
\begin{equation*}
S(a_{i,j})(x)\ls \frac1{\lz_{i,j}}
\lf\{\sum_{Q\in\cq_i,Q\st\qij}\lf[\cm(c_Q\chi_Q)(x)\r]^2
\r\}^{\frac12}
\end{equation*}
(see \cite[(4.9)]{lyj}). From this, Lemma \ref{lm-9.22}, \eqref{area-1}
and the Fefferman-Stein
vector-valued inequality (see, for example, \cite[p.\,51, Theorem 1(c)]{stein93}),
we deduce that, for all $i\in\zz$ and $j$,
\begin{align}\label{area-2}
\|\aij\|_{L^q(\rn)}
&\ls\|S(\aij)\|_{L^q(\rn)}
\ls\frac1{\lz_{i,j}}\lf\|\lf\{\sum_{Q\in\cq_i,Q\st \qij}
\lf[\cm(c_Q\chi_Q)\r]^2\r\}^{1/2}
\r\|_{L^q(\rn)}\noz\\
&\ls\frac1{\lz_{i,j}}\lf\|\lf[\sum_{Q\in\cq_i,Q\st \qij}
(c_Q)^2\chi_Q\r]^{1/2}
\r\|_{L^q(\rn)}\noz\\
&\ls\frac1{\lz_{i,j}}\lf\|\lf\{\sum_{Q\in\cq_i,Q\st \qij}
\lf[c_Q^2\cm\lf(\chi_{Q\cap(\wz\boz_i\setminus\boz_{i+1})}\r)\r]^2\r\}^{1/2}
\r\|_{L^q(\rn)}\noz\\
&\ls\frac1{\lz_{i,j}}\lf\|\lf[\sum_{Q\in\cq_i,Q\st \qij}
(c_Q)^2\chi_{Q\cap(\wz\boz_i\setminus\boz_{i+1})}\r]^{1/2}
\r\|_{L^q(\rn)}.
\end{align}
Since, for all $x\in Q\st\qij$, if $(y,t)\in Q^+$, then
$|x-y|<\sqrt n \ell(Q)\le t$, it follows that
$Q^+\st \Gamma(x)$, which, combined with the fact that
$\{Q^+:\ Q\in \cq_i,\ Q\st \qij\}_{i\in\zz,j}$ are disjoint,
further implies that, for all $i\in\zz$, $j$ and $x\in\rn$,
\begin{align*}
\sum_{Q\in\cq_i,Q\st \qij}
(c_Q)^2\chi_{Q\cap(\wz\boz_i\setminus\boz_{i+1})}(x)
&=\sum_{Q\in\cq_i,Q\st \qij}\int_{Q^+}|\phi_t*f(y)|^2\frac{dy\,dt}{t^{n+1}}
\chi_{Q\cap(\wz\boz_i\setminus\boz_{i+1})}(x)\\
&\le [S(f)(x)]^2\chi_{\qij\cap(\wz\boz_i\setminus\boz_{i+1})}(x)
\ls 2^{2i}\chi_{\qij}(x).
\end{align*}
Thus, by this and \eqref{area-2}, we conclude that, for all $i\in\zz$ and $j$,
\begin{equation*}
\|a_{i,j}\|_{L^q(\rn)}\ls\frac1{\lz_{i,j}}\lf\|2^i\chi_{\qij}\r\|_{L^q(\rn)}
\ls\frac{|\wz{Q}_{i,j}|^{\frac{1}{q}}}{\|\chi_{\wz{Q}_{i,j}}\|_{\lv}}.
\end{equation*}
Therefore, for any $i\in\zz$ and $j$, $a_{i,j}$ is a $(p(\cdot),q,s)$-atom
up to a harmless constant multiple
and hence \eqref{area-3} forms an atomic decomposition of $f$.
On the other hand, by Remark \ref{2.5.y},
$\lf|\qij\cap\boz_i\r|\ge\frac{\lf|\qij\r|}{2}$,
Lemma \ref{zhuolemma} and the fact that $\{\qij\}_{j}$
have disjoint interiors, we find that, for any $i\in\zz$,
\begin{align*}
&\lf\|\lf\{\sum_{j}\lf[\frac{\lz_{i,j}\chi_{\wz{Q}_{i,j}}}
{\|\chi_{\wz{Q}_{i,j}}\|_{\lv}}\r]
^{\underline{p}}\r\}^{\frac1{\underline{p}}}\r\|_{\lv}\\
&\hs\sim2^i\lf\|\lf(\sum_{j}
\chi_{\wz{Q}_{ij}}\r)^{\frac1{\underline{p}}}\r\|_{\lv}
\ls2^i\lf\|\lf(\sum_{j}
\chi_{\qij}\r)^{\frac1{\underline{p}}}\r\|_{\lv}\\
&\hs\ls 2^i\lf\|\lf(\sum_{j}
\chi_{\qij\cap\boz_i}\r)^{\frac1{\underline{p}}}\r\|_{\lv}
\ls2^i\|\chi_{\boz_i}\|_{\lv}\ls\|S(f)\|_{\wlv},
\end{align*}
which, together with Theorem \ref{atthm1}, implies that $f\in\wha=\whv$
and \eqref{area-4} holds true.
This finishes the proof of the sufficiency of Theorem \ref{lpthm1}.
Next we prove the necessity of Theorem \ref{lpthm1}.
Let $f\in \whv$. Obviously, by Lemma \ref{lppro1}, we know that
$f$ vanishes weakly at infinity. Due to Theorem \ref{atthm1},
we can decompose $f$ as follows
$$f=\sum_{i=-\fz}^{i_0-1}\sum_{j\in\nn}\lij\aij
+\sum_{i=i_0}^{\fz}\sum_{j\in\nn}\lij\aij=:f_1+f_2,$$
where $\{\lz_{i,j}\}_{i\in\zz,j\in\nn}$ and $\{a_{i,j}\}_{i\in\zz,j\in\nn}$
are as in Theorem \ref{atthm1} satisfying \eqref{4.1.y}.
Thus, we obtain
\begin{align}\label{lydecom}
&\lf\|\chi_{\{x\in\rn:\ S(f)(x)>\az\}}\r\|_{\lv}\noz\\
&\hs\ls\lf\|\chi_{\{x\in\rn:\ S(f_1)(x)>\frac{\az}{2}\}}\r\|_{\lv}
+\lf\|\chi_{\{x\in A_{i_0}:\ S(f_2)(x)>\frac{\az}{2}\}}\r\|_{\lv}\noz\\
&\hs\hs\hs+\lf\|\chi_{\{x\in(A_{i_0})^{\com}:\ S(f_2)(x)
>\frac{\az}{2}\}}\r\|_{\lv}\noz\\
&\hs=:{\rm I}_1+{\rm I}_2+{\rm I}_3,
\end{align}
where $A_{i_0}:=\bigcup_{i={i_0}}^\fz\bigcup_{j\in\nn}(4\Bij)$
and $\{\Bij\}_{i\in\zz,j\in\nn}$ are the balls as in Theorem \ref{atthm1}.
It is easy to see that
\begin{align}\label{lydecom1}
{\rm I}_1&\ls\lf\|\chi_{\{x\in\rn:\ \sum_{i=-\fz}
^{i_0-1}\sum_{j\in\nn}\lij S(\aij)(x)\chi_{4\Bij}(x)
>\frac{\az}{4}\}}\r\|_{\lv}\noz\\
&\hs\hs+\lf\|\chi_{\{x\in\rn:\ \sum_{i=-\fz}^{i_0-1}
\sum_{j\in\nn}\lij S(\aij)(x)\chi_{(4\Bij)^\com}(x)
>\frac{\az}{4}\}}\r\|_{\lv}\noz\\
&=:{\rm I}_{1,1}+{\rm I}_{1,2}.
\end{align}
For ${\rm I}_{1,1}$, by Lemmas \ref{lm-9.22} and \ref{atlm2}, Remark \ref{2.5.y}
and an argument similar to that
used in the proof of \eqref{eqatomi11}, we conclude that
\begin{align}\label{area-5}
{\rm I}_{1,1}\ls\az^{-1}\|f\|_{\wha}.
\end{align}
For ${\rm I}_{1,2}$, we first write that, for any $i\in\zz$, $j\in\nn$ and $x\in\rn$,
\begin{align}\label{lysaij}
\hs\hs\hs\lf[S(\aij)(x)\r]^2
&=\int_0^{\frac{|x-\xij|}{4}}
\int_{|y-x|<t}|\aij\ast\phi_t(y)|^2\dytn
+\int_{\frac{|x-\xij|}{4}}^\infty\int_{|y-x|<t}\cdots\noz\\
&=:\mj_1+\mj_2,
\end{align}
where $\xij$ denotes the center of $\Bij$.
From the Taylor remainder theorem, we deduce that, for
all $i\in\zz$, $j\in\nn$, $N\in\zz_+$, $t\in(0, \fz)$,
$x\in (4\Bij)^{\com}$, $|y-x|<t$ and $z\in\Bij$,
\begin{align}\label{lyphi}
\lf|\phi\lf(\frac{y-z}{t}\r)-\sum_{|\alpha|\le s}
\frac{\partial^\alpha\phi(\frac{y-\xij}{t})}{\alpha!}
\lf(\frac{\xij-z}{t}\r)^\alpha\r|\ls
\lf(\frac{t}{|\xi|}\r)^{N}\frac{|z-\xij|^{s+1}}{t^{s+1}},
\end{align}
where $\xi=(y-\xij)+\theta(\xij-z)$ for some $\theta\in[0,1]$.
When $x\in (4\Bij)^{\com}$ and $|y-x|<t\le\frac{|x-\xij|}4$, we have
$|y-\xij|\sim|x-\xij|$ and, in this case,
$|\xi|\ge |y-\xij|-|\xij-z|\ge \frac12|x-\xij|$.
Thus, by this and the vanishing moment condition of $\aij$,
\eqref{lyphi} in the case that $N=n+s+2$ and the H\"older inequality,
we know that, for all $i\in\zz$, $j\in\nn$, $x\in (4\Bij)^{\com}$
and $|y-x|<t\le\frac{|x-\xij|}4$,
\begin{align*}
\lf|\aij\ast\phi_t(y)\r|
&\ls t\int_\Bij|\aij(z)|\frac{|z-\xij|^{s+1}}
{|x-\xij|^{n+s+2}}\,dz
\ls\frac{t(\rij)^{s+1}}{|x-\xij|^{n+s+2}}\|\aij\|
_{L^q(\rn)}|\Bij|^{1/{q'}}\\
&\ls\frac{t}{|x-\xij|}\lf(\frac{\rij}{|x-\xij|}\r)^{n+s+1}\|
\chi_{\Bij}\|_{\lv}^{-1}.
\end{align*}
From this, we further deduce that, for all $i\in\zz$, $j\in\nn$
and $x\in(4\Bij)^{\com}$,
\begin{align}\label{lyesti}
\mj_1&\ls
\|\chi_{\Bij}\|_{\lv}^{-2}\lf(\frac{\rij}{|x-\xij|}\r)^{2(n+s+1)}
\frac{1}{|x-\xij|^2}\int_0^{\frac{|x-\xij|}{4}}t\,dt\noz\\
&\sim\|\chi_{\Bij}\|_{\lv}^{-2}
\lf(\frac{\rij}{|x-\xij|}\r)^{2(n+s+1)}\noz\\
&\ls\|\chi_{\Bij}\|_{\lv}^{-2}
\lf[\cm\lf(\chi_{\Bij}\r)(x)\r]^{\frac{2(n+s+1)}{n}}.
\end{align}
When $t\geq \frac{|x-\xij|}{4}$, by \eqref{lyphi}
in the case that $N=0$ and the H\"older inequality,
together with the vanishing moment condition of $\aij$,
we also find that,
for all $i\in\zz$, $j\in\nn$, $x\in (4\Bij)^{\complement}$
and $|y-x|<t$,
\begin{align*}
\lf|\aij\ast\phi_t(y)\r|
\ls \lf(\frac{\rij}{t}\r)^{n+s+1}\|\chi_{\Bij}\|
_{\lv}^{-1},
\end{align*}
which implies that
\begin{align}\label{lyesti2}
\hs\hs\mj_2&\ls\|\chi_{\Bij}\|_{\lv}^{-2}(\rij)^{2(n+s+1)}
\int_{\frac{|x-\xij|}{4}}^\infty t^{-2(n+s+1)-1}\,dt \noz\\
&\sim\|\chi_{\Bij}\|_{\lv}^{-2}\lf(\frac{\rij}
{|x-\xij|}\r)^{2(n+s+1)}\noz\\
&\ls\|\chi_{\Bij}\|_{\lv}^{-2}
\lf[\cm\lf(\chi_{\Bij}\r)(x)\r]^{\frac{2(n+s+1)}{n}}.
\end{align}
Thus, by \eqref{lysaij}, \eqref{lyesti} and \eqref{lyesti2},
we conclude that, for all
$i\in\zz$, $j\in\nn$ and $x\in(4\Bij)^\com$,
\begin{align}\label{lysaij1}
|S(\aij)(x)|\ls\|\chi_{\Bij}\|_{\lv}^{-1}
\lf[\cm\lf(\chi_{\Bij}\r)(x)\r]^{\frac{n+s+1}{n}}.
\end{align}
From this, the H\"older inequality, Remark \ref{r-vlp}(i),
Lemma \ref{mlm1} and an argument similar to that used in
the proof of \eqref{eqatomi12}, we deduce that
${\rm I}_{1,2}\ls\az^{-1}\|f\|_{\wha}$.
Combining this, \eqref{lydecom1} and \eqref{area-5}, we further conclude that
\begin{align}\label{lye1}
\az{\rm I}_1\ls\|f\|_{\wha}.
\end{align}
By an argument similar to that used in
the proof of \eqref{eqatomii}, we also find that
\begin{align}\label{lyeerr}
{\rm I}_2
\ls\|\chi_{A_{i_0}}\|_{\lv}\ls\az^{-1}\|f\|_{\wha}.
\end{align}
Let $r_2\in(\frac{n}{\underline{p}(n+s+1)},1)$. Then, by \eqref{lysaij1},
Lemma \ref{mlm1} and an argument similar to that used
in the proof of \eqref{eqatomiii}, we know that
\begin{align}\label{lyeez}
{\rm I}_3&
=\lf\|\chi_{\{x\in{(A_{i_0})^{\com}}:\
S(f_2)(x)>\frac{\az}{2}\}}\r\|_{\lv}\noz\\
&\ls\lf\|\lf[\frac{\sum_{i=i_0}^
\fz\sum_{j\in\nn}\lij S(\aij)}
{\az}\r]^{r_2}\chi_{(A_{i_0})^\com}\r\|_{\lv}\noz\\
&\ls\az^{-r_2}\lf[\sum_{i=i_0}^\fz2^{ir_2}\r.\noz\\
&\hs\hs\times\lf.\lf
\|\lf\{\sum_{j\in\nn}\lf[\cm\lf(\chi_{\Bij}\r)\r]
^{\frac{r_2(n+s+1)}{n}}\r\}^{\frac{n}{r_2(n+s+1)}}
\r\|_{L^{\frac{r_2(n+s+1)}{n}p(\cdot)}(\rn)}\r]
^{\frac{r_2(n+s+1)}{n}}\noz\\
&\ls\az^{-1}\|f\|_{\wha}.
\end{align}
Finally, combining \eqref{lydecom}, \eqref{lye1},
\eqref{lyeerr} and \eqref{lyeez}, we conclude that
$$\|S(f)\|_{\wlv}\ls\|f\|_{\whv},$$
which completes the proof of the necessity and hence the proof of Theorem \ref{lpthm1}.
\end{proof}
By an argument similar to that used in the proof
of the necessity part of Theorem \ref{lpthm1}, we obtain the following
boundedness of the Littlewood-Paley $g$-function from
$\whv$ to $\wlv$, the details being omitted.
\begin{prop}\label{lppro2}
Let $p(\cdot)\in C^{\log}(\rn)$.
If $f\in\whv$, then $g(f)\in\wlv$ and
$$\|g(f)\|_{\wlv}\le C\|f\|_{\whv},$$
where $C$ is a positive constant independent of $f$.
\end{prop}
To prove Theorem \ref{lpthm2}, we borrow some ideas from Ullrich \cite{u12}
and begin with the following notation.
For any $\phi\in\cs(\rn)$ and $f\in\cs'(\rn)$,
we let, for all $t,\ a\in(0,\fz)$ and $x\in\rn$,
$$(\phi^*_tf)_a(x):=\sup_{y\in\rn}\frac{|\phi_t*f(x+y)|}{(1+|y|/t)^a}$$
and
\begin{equation}\label{1.21-x}
g_{a,*}(f)(x):=\lf\{\int_{0}^{\fz}[(\phi^*_tf)_a(x)]^2\frac{dt}{t}\r\}^{1/2},
\end{equation}
where $\phi_t(\cdot):=\frac{1}{t^n}\phi(\frac{\cdot}{t})$.
The following estimate is a special case of \cite[Lemma 3.5]{lsuyy},
which is further traced back to \cite[(2.66)]{u12} and the argument
used in the proof of \cite[Theorem 2.6]{u12}.
\begin{lem}\label{lm-12.7}
Let $\phi\in\cs(\rn)$ satisfy
\eqref{12.7.x1}, \eqref{12.7.x2} and \eqref{12.7.x3} and $N_0\in\nn$. Then, for all
$t\in[1,2]$, $a\in(0,N_0]$, $l\in\zz$, $f\in\cs'(\rn)$ and $x\in\rn$, it holds true that
\begin{equation}\label{1.22x}
\lf[(\phi^*_{2^{-l}t}f)_a(x)\r]^r\le C_{(N_0,r)}\sum_{k=0}^{\fz}2^{-kN_0r}
2^{(k+l)n}\int_{\rn}\frac{|(\phi_{2^{-(k+l)}})_t*f(y)|^r}{(1+2^l|x-y|)^{ar}}\,dy,
\end{equation}
where $r$ is an arbitrary fixed positive number and $C_{(N_0,r)}$ a positive
constant independent of $\phi$, $l$, $t$, $f$ and $x$, but may depend on $N_0$ and $r$.
\end{lem}
\begin{proof}
By \eqref{12.7.x2}, we know that, for all $\gz\in\zz_+^n$ with
$|\gz|\le\max\{\lfloor\frac{n}{p_-}-n-1\rfloor,0\}$,
$$D^{\gz}\widehat\phi(\vec0_n)=\int_{\rn}(-2\pi i\xi)^{\gz}\phi(\xi)\,d\xi
=(-2\pi i)^{|\gz|}\int_{\rn}\xi^{\gz}\phi(\xi)\,d\xi=0,$$
where $p_-$ is as in \eqref{2.1x}.
On the other hand, since $\phi$ satisfies \eqref{12.7.x3}, it follows
that there exists $\xi_0\in\rn\setminus\{\vec0_n\}$ such that
$|\widehat{\phi}(\xi_0)|>0$. From this and
the continuity of $\widehat\phi$, we deduce that
there exists $\delta\in(0,\fz)$ such that
$\vec0_n\notin B(\xi_0,\delta)$ and,
for all $\xi\in B(\xi_0,\delta)$, $|\widehat{\phi}(\xi)|>0$.
By this, combined with the fact that $\widehat{\phi}$
is radial due to $\phi$ being radial,
we further conclude that,
$$\mbox{for\ all}\quad \xi\in B(\vec0_n,|\xi_0|+\delta)\backslash B(\vec0_n,|\xi_0|-\delta),\quad
|\widehat{\phi}(\xi)|>0.$$
Thus, if a radial Schwartz function $\phi$ satisfies
\eqref{12.7.x1}, \eqref{12.7.x2} and \eqref{12.7.x3},
then $\phi$ also satisfies the assumptions in \cite[Lemma 3.5]{lsuyy}.
Therefore, by \cite[Lemma 3.5]{lsuyy}, we find that \eqref{1.22x} holds true,
which completes the proof of Lemma \ref{lm-12.7}.
\end{proof}
We now prove Theorem \ref{lpthm2}.
\begin{proof}[Proof of Theorem \ref{lpthm2}]
For any $f\in\whv$, by Lemma \ref{lppro1} and Proposition \ref{lppro2},
we know that $f\in\cs'(\rn)$, $f$ vanishes weakly at infinity and $g(f)\in\wlv$.
Thus, to prove Theorem \ref{lpthm2},
by Theorem \ref{lpthm1}, it suffices to show that, for any $f\in\cs'(\rn)$, which
vanishes weakly at infinity, it holds true that
\begin{align}\label{lyine}
\|S(f)\|_{\wlv}\ls\|g(f)\|_{\wlv}.
\end{align}
To this end, let $f\in\cs'(\rn)$ vanish weakly at infinity. It is easy to know that,
for any $a\in(0,\fz)$ and almost every $x\in\rn$,
$S(f)(x)\ls g_{a,*}(f)(x)$.
Thus, to prove \eqref{lyine}, it suffices to show that
\begin{align}\label{elypo}
\lf\|g_{a,*}(f)\r\|_{\wlv}\ls\lf\|g(f)\r\|_{\wlv}
\end{align}
holds true for some $a\in(\frac{n}{\min\{p_-,2\}},\fz)$. We now prove \eqref{elypo}.
Since $a\in(\frac{n}{\min\{p_-,2\}},\fz)$,
it follows that there exists $r\in\lf(0,{\min\{p_-,2\}}\r)$
such that $a\in(\frac{n}{r},\fz)$. Choosing $N_0$ sufficiently large,
then, by Lemma \ref{lm-12.7} and the
Minkowski integral inequality, we find that, for all $x\in\rn$,
\begin{align*}
g_{a,*}(f)(x)
&=\lf\{\sum_{j\in\zz}\int_1^2[(\phi_{2^{-j}t}^*f)_a(x)]^2\frac{dt}{t}\r\}^{1/2}\\
&\ls\lf\{\sum_{j\in\zz}\int_1^2\lf[\sum_{k=0}^{\fz}2^{-kN_0r}2^{(k+j)n}
\int_{\rn}\frac{|(\phi_{2^{-(k+j)}})_t*f(y)|^r}{(1+2^j|x-y|)^{ar}}\,dy\r]
^{\frac{2}{r}}\frac{dt}{t}\r\}^{1/2}\\
&\ls\lf[\sum_{j\in\zz}\lf\{\sum_{k=0}
^{\fz}2^{-kN_0r}2^{(k+j)n}\int_{\rn}
\frac{[\int_1^2|\lf(\phi_{2^{-(k+j)}}\r)_t*f(y)|^2\frac{dt}{t}]
^{\frac{r}{2}}}{(1+2^j|x-y|)^{ar}}\,dy\r\}^{\frac{2}{r}}\r]^{1/2},
\end{align*}
which, together with Lemma \ref{mlmim} and Remarks \ref{r-ar} and \ref{10.24.x1},
implies that
\begin{align*}
&\lf\|g_{a,*}(f)\r\|_{\wlv}^{rv}\\
&\hs\ls\lf\|\sum_{k=0}
^{\fz}2^{-k(N_0r-n)}\lf[\sum_{j\in\zz}2^{j\frac{2n}{r}}
\lf\{\int_{\rn}\frac{[\int_1^2|\lf(\phi_{2^{-(k+j)}}\r)_t*f(y)|
^2\frac{dt}{t}]^{\frac{r}{2}}}{(1+2^j|\cdot-y|)^{ar}}\,dy\r\}
^{\frac{2}{r}}\r]^{\frac{r}{2}}\r\|
_{WL^{\frac{p(\cdot)}{r}}(\rn)}^v\\
&\hs\ls\sum_{k=0}
^{\fz}2^{-kv(N_0r-n)}\lf\|\lf[\sum_{j\in\zz}2^{j\frac{2n}{r}}
\lf\{\int_{\rn}\frac{[\int_1^2|\lf(\phi_{2^{-(k+j)}}\r)_t*f(y)|
^2\frac{dt}{t}]^{\frac{r}{2}}}{(1+2^j|\cdot-y|)^{ar}}\,dy\r\}
^{\frac{2}{r}}\r]^{\frac{r}{2}}\r\|
_{WL^{\frac{p(\cdot)}{r}}(\rn)}^v\\
&\hs\ls\sum_{k=0}
^{\fz}2^{-kv(N_0r-n)}\\
&\hs\hs\times\!\lf\|\lf[\sum_{j\in\zz}2^{j\frac{2n}{r}}
\lf\{\sum_{i=0}^{\fz}2^{-iar}
\int_{|\cdot-y|\sim2^{i-j}}\lf[\int_1^2|\lf(\phi_{2^{-(k+j)}}\r)_t*f(y)|
^2\frac{dt}{t}\r]^{\frac{r}{2}}\,dy\r\}
^{\frac{2}{r}}\r]^{\frac{r}{2}}\r\|
_{WL^{\frac{p(\cdot)}{r}}(\rn)}^v,
\end{align*}
where $v$ is as in Remark \ref{r-ar} and
$|\cdot-y|\sim2^{i-j}$ means that $|x-y|<2^{-j}$ when $i=0$,
or $2^{i-j-1}\le|x-y|<2^{i-j}$ when $i\in\nn$. Applying the Minkowski
series inequality, Proposition \ref{mlmveq}, and Remarks \ref{r-ar}
and \ref{10.24.x1}, we conclude that
\begin{align*}
\lf\|g_{a,*}(f)\r\|_{\wlv}^{rv}
&\ls\sum_{k=0}^{\fz}2^{-kv(N_0r-n)}\lf\|\sum_{i=0}^{\fz}2^{-iar+in}\r.\\
&\hs\times\lf.\lf\{\sum_{j\in\zz}
\lf[\cm\lf(\lf[\int_1^2|\lf(\phi_{2^{-(k+j)}}\r)_t*f|
^2\frac{dt}{t}\r]^{\frac{r}{2}}\r)\r]
^{\frac{2}{r}}\r\}^{\frac{r}{2}}\r\|
_{WL^{\frac{p(\cdot)}{r}}(\rn)}^v\\
&\ls\sum_{k=0}^{\fz}2^{-kv(N_0r-n)}\\
&\hs\times\sum_{i=0}^{\fz}2^{(-iar+in)v}
\lf\|\lf\{\sum_{j\in\zz}
\lf[\int_1^2|\lf(\phi_{2^{-(k+j)}}\r)_t*f|
^2\frac{dt}{t}\r]\r\}^{\frac{r}{2}}\r\|
_{WL^{\frac{p(\cdot)}{r}}(\rn)}^v\\
&\ls\lf\|g(f)\r\|_{\wlv}^{rv},
\end{align*}
which implies that \eqref{elypo} holds true.
This finishes the proof of Theorem \ref{lpthm2}.
\end{proof}
Applying Theorems \ref{lpthm1} and \ref{lpthm2}, we now prove Theorem \ref{10.10.x}.
\begin{proof}[Proof of Theorem \ref{10.10.x}]
To prove this theorem, we only need to show the necessity,
since the sufficiency is easy because of Theorem \ref{lpthm1} and
the obvious fact that, for all $f\in\cs'(\rn)$ and
$x\in\rn$, $S(f)(x)\le g_\lz^{\ast}(f)(x)$.
To show the necessity, for any $f\in\whv$, by Lemma \ref{lppro1}, we know that
$f$ vanishes weakly at infinity. From the fact that
$\lambda\in(1+\frac{2}{{\min\{p_-,2\}}}, \fz)$,
we deduce that there exists
$a\in(\frac{n}{{\min\{p_-,2\}}},\fz)$ such that $\lz\in(1+\frac{2a}{n},\fz)$
and, for all $x\in\rn$,
\begin{align*}
g_\lambda^{\ast}(f)(x)&=\lf[\int_0^\infty\int_{\mathbb R^n}
\lf(\frac{t}{t+|x-y|}\r)^{\lz n}
\lf|f\ast\phi_t(y)\r|^2\,\frac{\,dy\,dt}{t^{n+1}}\r]^{1/2}\\
&\ls\lf\{\int_0^{\fz}[(\phi_t^*f)_a(x)]^2\int_{\rn}\lf(1+\frac{|x-y|}{t}\r)
^{2a-\lz n}\frac{dy\,dt}{t^{n+1}}\r\}^{1/2}\\
&\sim\lf\{\int_0^{\fz}[(\phi_t^*f)_a(x)]^2
\frac{dt}{t}\r\}^{1/2}\sim g_{a,*}(f)(x),
\end{align*}
which, together with \eqref{elypo} and Theorem \ref{lpthm2},
implies that
$$\|g_\lambda^{\ast}(f)\|_{\wlv}\ls\|f\|_{\whv}.$$
This finishes the proof of Theorem \ref{10.10.x}.
\end{proof}
\section{Boundedness of Calder\'on-Zygmund operators\label{s-bou}}
\hskip\parindent
In this section, as an application of the variable weak Hardy space $\whv$, we
establish
the boundedness of the Calder\'on-Zygmund operators from the variable Hardy
space $\hv$ to $\whv$. We begin with recalling the definition of
the variable Hardy space $\hv$ (see \cite[Definition 1.1]{ns12}).
\begin{defn}
Let $p(\cdot)\in C^{\log}(\rn)$ and $N\in(\frac{n}{p_-}+n+1,\fz)\cap\nn$
with $p_-$ as in \eqref{2.1x}.
The \emph{variable Hardy space} $\hv$ is defined to be the set of all
$f\in\cs'(\rn)$ such that the (quasi-)norm
$$\|f\|_{\hv}:=\|f_{N,+}^\ast\|_{\lv}<\fz,$$
where $f_{N,+}^\ast$ is as in \eqref{2.8x}.
\end{defn}
Recall that the \emph{variable atomic Hardy space}
$\ha$ is defined as
the space of all $f\in\cs'(\rn)$ such that
$f=\sum_{j\in\nn}\lz_j a_j$ in $\cs'(\rn)$, where $\{\lz_j\}_{j\in\nn}$
is a sequence of non-negative numbers, $\{a_j\}_{j\in\nn}$ is a sequence
of $(p(\cdot),q,s)$ atoms, associated with balls $\{B_j\}_{j\in\nn}$ of $\rn$,
satisfying that
\begin{enumerate}
\item[{\rm (i)}] $\supp a_j \st B_j$;
\item[{\rm (ii)}] $\|a_j\|_{L^q(\rn)}\le\frac{|B_j|^{1/q}}{\|\chi_{B_j}\|_{\lv}}$;
\item[{\rm (iii)}] $\int_{\mathbb R^n}a_j(x)x^\az\,dx=0$ for all $\az\in{\zz}_+^n$
with $|\az|\le s$.
\end{enumerate}
Moreover, for any $f\in\ha$, let
$$\|f\|_{\ha}=\inf\lf\{\lf\|\lf\{\sum_{j\in\nn}
\lf[\frac{\lz_j\chi_{B_j}}{\|\chi_{B_j}\|_{\lv}}\r]
^{\underline{p}}\r\}^{\frac{1}{\underline{p}}}\r\|_{\lv}\r\},$$
where $\underline{p}$ is as in \eqref{2.1y} and the infimum is taken over all
admissible decompositions of $f$ as above.
The following lemma is just \cite[Theorem 1.1]{s10}.
\begin{lem}\label{blm1}
Let $p(\cdot)\in C^{\log}(\rn)$, $q\in [1,\fz]\cap (p_+,\fz]$ and
$s\in(\frac{n}{p_-}-n-1,\fz)\cap{\zz}_+$, where $p_+$ and $p_-$ are as in \eqref{2.1x}.
Then $\hv=\ha$ with equivalent quasi-norms.
\end{lem}
\begin{rem}\label{07-22}
Let $p(\cdot)\in C^{\log}(\rn)$ and $p_+\in(0,1]$. Then, from the proof of
\cite[Theorem 4.5]{ns12}, we deduce that the subspace $H^{p(\cdot)}(\rn)\cap L^2(\rn)$
is dense in $H^{p(\cdot)}(\rn)$.
\end{rem}
Recall that, for any given $\delta\in(0,1]$, a \emph{convolutional $\delta$-type
Calder\'on-Zygmund operator} $T$ means that: $T$ is a linear bounded operator on
$L^2(\rn)$ with kernel $k\in\cs'(\rn)$ coinciding with a locally integrable function on
$\rn\backslash \{\vec0_n\}$ and satisfying that, for all $x$, $y\in\rn$ with $|x|>2|y|$,
$$|k(x-y)-k(x)|\le C\frac{|y|^\delta}{|x|^{n+\delta}}$$
and, for all $f\in L^2(\rn)$, $Tf(x)=k*f(x).$
The first main result of this section reads as follows.
\begin{thm}\label{bdnthm2}
Let $p(\cdot):\ \rn\to(0,1]$ belong to $C^{\log}(\rn)$ and $\delta\in(0,1]$.
Let $T$ be a convolutional $\delta$-type
Calder\'on-Zygmund operator.
If $p_-\in[\frac{n}{n+\delta},1]$ with $p_-$ as in \eqref{2.1x}, then
$T$ has a unique extension on $H^{p(\cdot)}(\rn)$ and, moreover,
for all $f\in\hv$,
$$\|Tf\|_{\whv}\le C\|f\|_{H^{p(\cdot)}(\rn)},$$
where $C$ is a positive constant independent of $f$.
\end{thm}
The proof of Theorem \ref{bdnthm2} is given below.
\begin{rem}\label{3.4.re1}
If $p(\cdot)\equiv p\in(0,1]$, then $\whv=W\!H^p(\rn)$. In this case,
Theorem \ref{bdnthm2} indicates that, if $\delta\in(0,1]$, $p=\frac{n}{n+\delta}$
and $T$ is a convolutional $\delta$-type Calder\'on-Zygmund operator, then
$T$ is bounded from $H^{\frac{n}{n+\delta}}(\rn)$ to $W\!H^{\frac{n}{n+\delta}}(\rn)$,
which is just \cite[Theorem 1]{liu91} (see also \cite[Theorem 5.2]{lyj}).
Here $\frac n{n+\delta}$ is called the \emph{critical index}. Thus,
the boundedness of the Calder\'on-Zygmund operator from $H^{p(\cdot)}(\rn)$ to
$\whv$ obtained in Theorem \ref{bdnthm2} includes the critical case.
\end{rem}
Recall that, for any given $\gamma\in(0,\fz)$, a linear operator $T$ is called
a \emph{$\gamma$-order Calder\'on-Zygmund operator} if $T$ is bounded on $L^2(\rn)$
and its kernel
$$k:\ (\rn\times\rn)\backslash\{(x,x):\ x\in\rn\}\to\mathbb C$$
satisfies that there exists a positive constant $C$ such that,
for any $\az\in\zz^n_+$ with $|\az|\le\lceil\gamma\rceil$
and $x$, $y$, $z\in\rn$ with $|x-y|>2|y-z|$,
\begin{align}\label{3.1.y}
\lf|\partial_x^\az k(x,y)-\partial_x^\az k(x,z)\r|\le
C\frac{|y-z|^{\gamma-\lceil\gamma\rceil}}{|x-y|^{n+\gamma}}
\end{align}
and, for any $f\in L^2(\rn)$ having compact support and $x\notin\supp f$,
$$Tf(x)=\int_{\supp f}k(x,y)f(y)\,dy.$$
Here and hereafter, for any $\gz\in(0,\fz)$, $\lceil \gamma\rceil$ denotes the maximal integer
less than $\gz$.
For $m\in\nn$, an operator $T$ is said to have the \emph{vanishing moment condition up to
order $m$} if, for any $a\in L^2(\rn)$ having compact support and satisfying that,
for all $\beta\in\zz^n_+$ with $|\beta|\le m$, $\int_{\rn}a(x)x^\beta\,dx=0$,
it holds true that $\int_{\rn}x^\beta Ta(x)\,dx=0$.
The second main result of this section is stated as follows.
\begin{thm}\label{bdnthm3}
Let $p(\cdot):\ \rn\to(0,1]$ belong to $C^{\log}(\rn)$ and $\gamma\in(0,\fz)$.
Let $T$ be a $\gamma$-order Calder\'on-Zygmund operator and have
the vanishing moment condition up to order $\lceil \gamma\rceil$.
If $\lceil \gamma\rceil\le n(\frac{1}{p_-}-1)\le\gamma$ with $p_-$
as in \eqref{2.1x}, then $T$ has a unique extension on
$H^{p(\cdot)}(\rn)$ and, moreover, for all $f\in\hv$,
$\|Tf\|_{\whv}\le C\|f\|_{H^{p(\cdot)}(\rn)}$,
where $C$ is a positive constant independent of $f$.
\end{thm}
The proof of Theorem \ref{bdnthm3} is presented below.
\begin{rem}\label{3.4.re2}
Recall that, for $\delta\in(0,1]$, a
\emph{non-convolutional $\delta$-type Calder\'on-Zygmund operator}
$T$ means that: $T$ is a linear bounded operator on $L^2(\rn)$
and there exist a kernel $k$ on $(\rn\times\rn)\backslash\{(x,x):\ x\in\rn\}$
and a positive constant $C$ such that, for any $x$, $y$, $z\in\rn$
with $|x-y|>2|y-z|$,
$$|k(x,y)-k(x,z)|\le C\frac{|y-z|^\delta}{|x-y|^{n+\delta}}$$
and, for any $f\in L^2(\rn)$ having compact support and $x\notin\supp f$,
$$Tf(x)=\int_{\supp f}k(x,y)f(y)\,dy.$$
Notice that, when $\gamma:=\delta\in(0,1]$, the operator $T$
in Theorem \ref{bdnthm3} is just a non-convolutional
$\delta$-type Calder\'on-Zygmund operator.
Thus, the operators of Theorem \ref{bdnthm3} conclude the non-convolutional
$\delta$-type Calder\'on-Zygmund operators as special cases. From this,
it is easy to see that the critical index
of $\gamma$-order Calder\'on-Zygmund operators is $\frac n{n+\gamma}$.
\end{rem}
To prove Theorems \ref{bdnthm2} and \ref{bdnthm3},
we need the following proposition.
\begin{prop}\label{1.14.x3}
Let $r\in(1,\fz)$ and $p(\cdot)\in C^{\log}(\rn)$ with $p_-\in[1,\fz)$,
where $p_-$ is as in \eqref{2.1x}.
Then there exists a positive constant $C$ such that, for any sequence
$\{f_j\}_{j\in\nn}$ of measurable functions and $\az\in(0,\fz)$,
$$\az\lf\|\chi_{\{x\in\rn:\ \{\sum_{j\in\nn}
[\cm(f_j)(x)]^r\}^{\frac1{r}}>\az\}}\r\|_{\lv}
\le C\lf\|\lf(\sum_{j\in\nn}|f_j|^r\r)^{\frac1{r}}\r\|_{\lv}.$$
\end{prop}
The proof of Proposition \ref{1.14.x3} depends on the following
extrapolation theorem corresponding to the {Muckenhoupt weight class} $A_1(\rn)$,
which is just \cite[Theorem 5.24]{cfbook} and a weaker version can also
be found in \cite[Theorem 1.3]{cf06}.
Recall that a locally integrable function $w$,
which is positive almost everywhere on $\rn$, is said to belong to
$A_1(\rn)$ if
$$[w]_{A_1(\rn)}:=\mathop\mathrm{ess\,sup}_{x\in\rn}\frac{\cm(w)(x)}{w(x)}<\fz,$$
where $\cm$ denotes the Hardy-Littlewood maximal function as in \eqref{2.2x}.
\begin{lem}\label{1.14.x2}
Suppose that the family $\cf$ is a set of all pairs of functions
$(F,G)$ satisfying that there exists $p_0\in[1,\fz)$ such that,
for every $w\in A_1(\rn)$,
$$\int_\rn [F(x)]^{p_0}w(x)\,dx\le C_{(p_0,[w]_{A_1(\rn)})}\int_\rn [G(x)]^{p_0}w(x)\,dx,$$
where $C_{(p_0,[w]_{A_1(\rn)})}$ is a positive constant independent of $F$ and $G$,
but may depend on $p_0$ and $[w]_{A_1(\rn)}$.
Let $p(\cdot)\in\cp(\rn)$ be such that
$p_0\le p_-\le p_+<\fz$ with $p_-$ and $p_+$ as in \eqref{2.1x}.
If the maximal operator $\cm$ is bounded on $L^{(p(\cdot)/p_0)^*}(\rn)$,
where, for all $x\in\rn$,
$$\frac{1}{(p(x)/p_0)^*}+\frac{1}{p(x)/p_0}=1,$$
then there exists a positive constant $C$ such that, for all $(F,G)\in\cf$,
$$\|F\|_{\lv}\le C\|G\|_{\lv}.$$
\end{lem}
We also need the following weak-type weighted Fefferman-Stein
vector-valued inequality of the Hardy-Littlewood maximal operator $\cm$ in \eqref{2.2x}
from \cite[Theorem 3.1(a)]{aj80}.
\begin{lem}\label{1.14.x1}
Let $r\in(1,\fz)$ and $w\in A_1(\rn)$. Then there
exists a positive constant $C$, depending on $n$, $r$ and $[w]_{A_1(\rn)}$,
such that, for all $\az\in(0,\fz)$
and sequences $\{f_j\}_{j\in\nn}$ of measurable functions on $\rn$,
$$\az\, w\lf(\lf\{x\in\rn:\ \lf(\sum_{j\in\nn}
[\cm(f_j)(x)]^r\r)^{\frac1{r}}>\az\r\}\r)
\le C\int_{\rn}\lf[\sum_{j\in\nn}|f_j(x)|^r\r]^{\frac1{r}}w(x)\,dx.$$
\end{lem}
\begin{proof}[Proof of Proposition \ref{1.14.x3}]
For any $r\in(1,\fz)$, $\az\in(0,\fz)$ and
any sequence $\{f_j\}_{j\in\nn}$ of measurable functions on $\rn$,
let $\cf_\az$ be the set of all pairs $(F_\az,\ G)$,
where, for all $x\in\rn$,
$$F_\az(x):=\az\chi_{\{x\in\rn:\ \{\sum_{j\in\nn}
[\cm(f_j)(x)]^r\}^{\frac1{r}}>\az\}}(x)
\quad\mbox{and}\quad
G(x):=\lf[\sum_{j\in\nn}|f_j(x)|^r\r]^{\frac1{r}}.$$
Then, by Lemma \ref{1.14.x1}, we know that, for every $w\in A_1(\rn)$,
\begin{align}\label{1.17x}
&\int_{\rn}F_\az(x)w(x)\,dx\noz\\
&\hs=\az\, w\lf(\lf\{x\in\rn:\ \lf(\sum_{j\in\nn}
[\cm(f_j)(x)]^r\r)^{\frac1{r}}>\az\r\}\r)
\ls\int_{\rn}G(x)w(x)\,dx.
\end{align}
From $p(\cdot)\in C^{\log}(\rn)$ and $p_-\in[1,\fz)$, it follows that
$\cm$ is bounded on $L^{p^*(\cdot)}(\rn)$ (see, for example, \cite[Theorem 3.16]{cfbook}).
Thus, by this, \eqref{1.17x} and via applying Lemma \ref{1.14.x2} with $p_0:=1$
and $\cf:=\cf_\az$, we conclude that, for all $\az\in(0,\fz)$,
\begin{align*}
\az\lf\|\chi_{\{x\in\rn:\ \{\sum_{j\in\nn}
[\cm(f_j)(x)]^r\}^{\frac1{r}}>\az\}}\r\|_{L^{p(\cdot)}(\rn)}
=\lf\|F_\az\r\|_{\lv}\ls \|G\|_{\lv}.
\end{align*}
Therefore, we find that, for all $\az\in(0,\fz)$ and sequences $\{f_j\}_{j\in\nn}$
of measurable functions,
$$\az\lf\|\chi_{\{x\in\rn:\ \{\sum_{j\in\nn}
[\cm(f_j)(x)]^r\}^{\frac1{r}}>\az\}}\r\|_{\lv}
\ls\lf\|\lf[\sum_{j\in\nn}|f_j(x)|^r\r]^{\frac1{r}}\r\|_{\lv},$$
which completes the proof of Proposition \ref{1.14.x3}.
\end{proof}
We next prove Theorem \ref{bdnthm2}.
\begin{proof}[Proof of Theorem \ref{bdnthm2}]
Let $p(\cdot)$ and $s$ be as in Lemma \ref{blm1} and $f\in H^{p(\cdot)}(\rn)\cap L^2(\rn)$.
Then, by Lemma \ref{blm1} and its proof (see also the proof of \cite[Theorem 4.5]{ns12}), we know that
there exist sequences $\{\lz_j\}_{j\in\nn}$ of positive constants and
$\{a_j\}_{j\in\nn}$ of $(p(\cdot),2,s)$-atoms supported on balls $\{B_j\}_{j\in\nn}:=\{B(x_j,r_j):\
x_j\in\rn\ \mbox{and}\ r_j\in(0,\fz)\}_{j\in\nn}$ such that
\begin{equation}\label{7.2x}
f=\sum_{j\in\nn}\lz_ja_j\quad {\rm in} \quad L^2(\rn)
\end{equation}
and
$$\lf\|\lf\{\sum_{j\in\nn}\lf[\frac{\lz_j\chi_{B_j}}
{\|\chi_{B_j}\|_{\lv}}\r]
^{\underline{p}}\r\}^{\frac{1}{\underline{p}}}\r\|_{\lv}
\ls\|f\|_{H^{p(\cdot)}(\rn)},$$
where $\underline{p}$ is as in \eqref{2.1y}.
Since the operator $T$ is bounded on $L^2(\rn)$, it follows from \eqref{7.2x} that
\begin{equation*}
Tf=\sum_{j\in\nn}\lz_jTa_j
\end{equation*}
holds true in $L^2(\rn)$, namely, $Tf$ is well defined for any $f\in H^{p(\cdot)}(\rn)\cap
L^2(\rn)$. Let $\phi\in\cs(\rn)$ satisfy $\int_{\rn}\phi(x)\,dx\neq0$.
Then, to prove this theorem, by Theorem \ref{mthm1}, we only need to show that
\begin{align}\label{7.1x}
\lf\|\phi_{+}^\ast(Tf)\r\|_{\wlv}\ls\|f\|_{H^{p(\cdot)}(\rn)},
\end{align}
where $\phi_{+}^\ast(Tf)$ is as in \eqref{3.5x} with $f$ replaced by $Tf$.
For any $\az\in(0,\fz)$, by Remark \ref{r-vlp}, we have
\begin{align*}
\quad&\az\lf\|\chi_{\{x\in\rn:\ \phi_{+}^\ast(Tf)(x)>\az\}}\r\|_{\lv}\\
&\hs\le\az\lf\|\chi_{\{x\in\rn:\ \sum_{j\in\nn}\lz_j
\phi_{+}^\ast(Ta_j)(x)>\az\}}\r\|_{\lv}\\
&\hs\ls\az\lf\|\chi_{\{x\in\rn:\ \sum_{j\in\nn}
\lz_j\phi_{+}^\ast(Ta_j)(x)\chi_{4B_j}(x)
>\frac{\az}2\}}\r\|_{\lv}\\
&\quad\quad+\az\lf\|\chi_{\{x\in\rn:\ \sum_{j\in\nn}\lz_j
\phi_{+}^\ast(Ta_j)(x)\chi_{(4B_j)^{\complement}}(x)
>\frac{\az}2\}}\r\|_{\lv}\\
&\hs\ls\lf\|\sum_{j\in\nn}\lz_j\phi_{+}^\ast(Ta_j)\chi_{4B_j}\r\|_{\lv}
+\az\lf\|\chi_{\{x\in\rn:\ \sum_{j\in\nn}\lz_j
\phi_{+}^\ast(Ta_j)(x)\chi_{(4B_j)^{\complement}}(x)
>\frac{\az}2\}}\r\|_{\lv}\\
&\hs=:{\rm I}+{\rm II}.
\end{align*}
Observe that $\phi_{+}^\ast(Ta_j)\ls\cm(Ta_j)$ and $a_j\in L^2(\rn)$.
From the facts that $\cm$ is bounded on $L^r(\rn)$ with $r\in(1,\fz]$ and that
$T$ is bounded on $L^2(\rn)$, we conclude that
$$\|\phi_{+}^\ast(Ta_j)\|_{L^2(\rn)}\ls\|\cm(Ta_j)\|_{L^2(\rn)}
\ls\|Ta_j\|_{L^2(\rn)}\ls\|a_j\|_{L^2(\rn)}
\ls\frac{|B_j|^{1/2}}{\|\chi_{B_j}\|_{\lv}},$$
which, combined with Lemmas \ref{atlm2} and \ref{mlm1},
implies that
\begin{align}\label{1.26.x1}
{\rm I}&\ls\lf\|\lf\{\sum_{j\in\nn}
\lf[\frac{\lz_j\chi_{4B_j}}{\|\chi_{B_j}\|_{\lv}}\r]
^{\underline{p}}\r\}^{\frac{1}{\underline{p}}}\r\|_{\lv}\noz\\
&\ls\lf\|\lf\{\sum_{j\in\nn}
\lf[\frac{\lz_j\chi_{B_j}}{\|\chi_{B_j}\|_{\lv}}\r]
^{\underline{p}}\r\}^{\frac{1}{\underline{p}}}\r\|_{\lv}
\ls\|f\|_{H^{p(\cdot)}(\rn)}.
\end{align}
Next, we deal with ${\rm II}$. To this end, by an argument similar to that
used in the proof of \cite[(5.4), (5.5),(5.6) and (5.7)]{lyj},
we conclude that for any $j\in\nn$,
\begin{equation*}
\phi_+^\ast(Ta_j)\chi_{(4B_j)^\complement}(x)\ls\frac{{r_j}^{n+\delta}}{|x-x_j|^{n+\delta}}
\frac{1}{\|\chi_{B_j}\|_{\lv}}
\ls\lf[\cm(\chi_{B_j})(x)\r]^{\frac{n+\delta}{n}}
\frac{1}{\|\chi_{B_j}\|_{\lv}}
\end{equation*}
Thus, by Proposition \ref{1.14.x3}, we know that
\begin{align}\label{1.26.x2}
{\rm II}&\ls\az\lf\|\chi_{\{x\in\rn:\ \sum_{j\in\nn}
\frac{\lz_j}{\|\chi_{B_j}\|_{\lv}}
[\cm(\chi_{B_j})(x)]^{\frac{n+\delta}{n}}
>\frac{\az}2\}}\r\|_{\lv}\noz\\
&\ls\frac{\az}{2}\lf\|\chi_{\{x\in\rn:\ [\sum_{j\in\nn}\frac{\lz_j}
{\|\chi_{B_j}\|_{\lv}}\{\cm(\chi_{B_j})(x)\}^{\frac{n+\delta}{n}}]
^{\frac{n}{n+\delta}}>(\frac{\az}{2})^{\frac{n}{n+\delta}}\}}\r\|
_{L^{\frac{(n+\delta)p(\cdot)}{n}}(\rn)}^{\frac{n+\delta}{n}}\noz\\
&\ls\lf\|\lf[\sum_{j\in\nn}\frac{\lz_j\chi_{B_j}}
{\|\chi_{B_j}\|_{\lv}}\r]^{\frac{n}{n+\delta}}\r\|
_{L^{\frac{(n+\delta)p(\cdot)}{n}}(\rn)}^{\frac{n+\delta}{n}}\noz\\
&\ls\lf\|\lf\{\sum_{j\in\nn}
\lf[\frac{\lz_j\chi_{B_j}}{\|\chi_{B_j}\|_{\lv}}\r]
^{\underline{p}}\r\}^{\frac{1}{\underline{p}}}\r\|_{\lv}
\ls\|f\|_{H^{p(\cdot)}(\rn)}.
\end{align}
Finally, combining \eqref{1.26.x1} and \eqref{1.26.x2}, we conclude that,
for any $\az\in(0,\fz)$,
$$\az\lf\|\chi_{\{x\in\rn:\ \phi_{+}^\ast(Tf)(x)>\az\}}\r\|_{\lv}
\ls\|f\|_{H^{p(\cdot)}(\rn)},$$
namely, \eqref{7.1x} holds true.
This, together with Remark \ref{07-22} and a dense argument, finishes the proof of Theorem \ref{bdnthm2}.
\end{proof}
We finally prove Theorem \ref{bdnthm3}.
\begin{proof}[Proof of Theorem \ref{bdnthm3}]
By an argument similar to that used in the proof of Theorem \ref{bdnthm2},
it suffices to show that, for all $\az\in(0,\fz)$ and $f\in\hv\cap L^2(\rn)$,
\begin{align}\label{3.4.x1}
\az\lf\|\chi_{\{x\in\rn:\ \sum_{j\in\nn}\lz_j
\phi_{+}^\ast(Ta_j)(x)\chi_{(4B_j)^{\complement}}(x)
>\frac{\az}2\}}\r\|_{\lv}\ls\|f\|_{\hv},
\end{align}
where $\lz_j$, $a_j$ and $B_j$ are as in the proof of Theorem \ref{bdnthm2}.
To this end, we need some finer estimates about $\phi_{+}^\ast(Ta_j)$.
By the vanishing moment condition of $T$
and the fact that $\lceil \gamma\rceil\le n(\frac{1}{p_-}-1)\le s$,
we know that, for all $j\in\nn$, $t\in(0,\fz)$ and $x\in(4B_j)^{\com}$,
\begin{align}\label{3.1.y2}
\lf|\phi_t*Ta_j(x)\r|
&\le\frac{1}{t^n}\int_{\rn}
\lf|\phi\lf(\frac{x-y}{t}\r)-\sum_{|\beta|\le\lceil\gamma\rceil}
\frac{D^\beta\phi\lf(\frac{x-x_j}{t}\r)}{\beta!}
\lf(\frac{y-x_j}{t}\r)^\beta\r||Ta_j(y)|\,dy\noz\\
&=\frac{1}{t^n}\lf(\int_{|y-x_j|<2r_j}+\int_{2r_j\le|y-x_j|<\frac{|x-x_j|}{2}}
+\int_{|y-x_j|\ge\frac{|x-x_j|}{2}}\r)\noz\\
&\quad\quad\times\lf|\phi\lf(\frac{x-y}{t}\r)-\sum_{|\beta|\le\lceil\gamma\rceil}
\frac{D^\beta\phi\lf(\frac{x-x_j}{t}\r)}{\beta!}
\lf(\frac{y-x_j}{t}\r)^\beta\r||Ta_j(y)|\,dy\noz\\
&=:{\rm A}_1+{\rm A}_2+{\rm A}_3.
\end{align}
For ${\rm A}_1$, by the Taylor remainder theorem, we find that, for any $j\in\nn$ and $y\in\rn$
with $|y-x_j|<2r_j$, there exists $\xi_1(y)\in 2B_j$ such that
\begin{align*}
{\rm A}_1&=\frac{1}{t^n}\int_{|y-x_j|<2r_j}
\lf|\phi\lf(\frac{x-y}{t}\r)-\sum_{|\beta|\le\lceil\gamma\rceil}
\frac{D^\beta\phi\lf(\frac{x-x_j}{t}\r)}{\beta!}
\lf(\frac{y-x_j}{t}\r)^\beta\r||Ta_j(y)|\,dy\\
&\le\frac{1}{t^n}\int_{|y-x_j|<2r_j}
\lf|\sum_{|\beta|=\lceil\gamma\rceil+1}\partial^\beta\phi
\lf(\frac{x-\xi_1(y)}{t}\r)\r|\lf(\frac{|y-x_j|}{t}\r)^{\lceil \gamma\rceil+1}|Ta_j(y)|\,dy,
\end{align*}
which, combined with the H\"older inequality and the fact that $T$ is bounded on $L^2(\rn)$,
implies that, for all $t\in(0,\fz)$ and $x\in(4B_j)^{\com}$,
\begin{align}\label{3.1.y3}
{\rm A}_1
&\ls\frac{1}{t^n}\int_{|y-x_j|<2r_j}
\frac{t^{n+\lceil \gamma\rceil+1}}{|x-x_j|^{n+\lceil \gamma\rceil+1}}
\frac{|y-x_j|^{\lceil \gamma\rceil+1}}{t^{\lceil \gamma\rceil+1}}|Ta_j(y)|\,dy\noz\\
&\ls\frac{r_j^{\lceil \gamma\rceil+1}}
{|x-x_j|^{n+\lceil \gamma\rceil+1}}\|Ta_j\|_{L^2(\rn)}|B_j|^{1/2}
\ls\frac{r_j^{n+\lceil \gamma\rceil+1}}{|x-x_j|^{n+\lceil \gamma\rceil+1}}
\frac{1}{\|\chi_{B_j}\|_{\lv}}.
\end{align}
For ${\rm A}_2$, by the Taylor remainder theorem, the vanishing
moment condition of $a_j$, the fact that
$\lceil \gamma\rceil\le n(\frac{1}{p_-}-1)\le s$,
\eqref{3.1.y} and the H\"older inequality,
we conclude that, for all $z\in B_j$, there exists $\xi_2(z)\in B_j$ such that,
for all $t\in(0,\fz)$ and $x\in(4B_j)^{\com}$,
\begin{align}\label{3.1.y4}
{\rm A}_2&=\frac{1}{t^n}\int_{2r_j\le|y-x_j|<\frac{|x-x_j|}{2}}
\lf|\phi\lf(\frac{x-y}{t}\r)-\sum_{|\beta|\le\lceil \gamma\rceil}
\frac{D^\beta\phi\lf(\frac{x-x_j}{t}\r)}{\beta!}
\lf(\frac{y-x_j}{t}\r)^\beta\r||Ta_j(y)|\,dy\noz\\
&\ls\int_{2r_j\le|y-x_j|<\frac{|x-x_j|}{2}}
\frac{|y-x_j|^{\lceil \gamma\rceil+1}}{|x-x_j|^{n+\lceil \gamma\rceil+1}}\noz\\
&\quad\quad\times\lf[\int_{B_j}|a_j(z)|\lf|k(y,z)-
\sum_{|\beta|\le\lceil \gamma\rceil}\frac{\partial_y^\beta
k(y,x_j)}{\beta!}(z-x_j)^\beta\r|\,dz\r]\,dy\noz\\
&\ls\frac{1}{|x-x_j|^{n+\lceil \gamma\rceil+1}}\int_{2r_j\le|y-x_j|<\frac{|x-x_j|}{2}}
|y-x_j|^{\lceil \gamma\rceil+1}\noz\\
&\quad\quad\times\int_{B_j}|a_j(z)|\lf|\sum_{|\beta|=\lceil \gamma\rceil}\frac{\partial_y^\beta
k(y,x_j)-\partial_y^\beta k(y,\xi_2(z))}{\beta!}(z-x_j)^\beta\r|\,dz\,dy\noz\\
&\ls\frac{1}{|x-x_j|^{n+\lceil \gamma\rceil+1}}\int_{2r_j\le|y-x_j|<\frac{|x-x_j|}{2}}
|y-x_j|^{\lceil \gamma\rceil+1}\int_{B_j}|a_j(z)|\frac{|z-x_j|^\gamma}
{|y-x_j|^{n+\gamma}}\,dz\,dy\noz\\
&\ls\frac{r_j^\gamma}{|x-x_j|^{n+\lceil \gamma\rceil+1}}\int_{2r_j\le|y-x_j|<\frac{|x-x_j|}{2}}
\frac{1}{|y-x_j|^{n+\gamma-\lceil \gamma\rceil-1}}\,dy\|a_j\|_{L^2(\rn)}|B_j|^{1/2}\noz\\
&\ls\frac{r_j^{n+\gamma}}{|x-x_j|^{n+\gamma}}\frac{1}{\|\chi_{B_j}\|_{\lv}}.
\end{align}
For ${\rm A}_3$, by the vanishing moment condition of $a_j$, the fact that
$\lceil \gamma\rceil\le n(\frac{1}{p_-}-1)\le s$,
\eqref{3.1.y} and the H\"older inequality,
we find that, for all $z\in B_j$, there exists $\xi_3(z)\in B_j$ such that,
for all $t\in(0,\fz)$ and $x\in(4B_j)^{\com}$,
\begin{align}\label{3.1.y5}
{\rm A}_3
&\le\int_{|y-x_j|\ge\frac{|x-x_j|}{2}}\lf|\frac{1}{t^n}
\lf[\phi\lf(\frac{x-y}{t}\r)-\sum_{|\beta|\le\lceil \gamma\rceil}
\frac{D^\beta\phi\lf(\frac{x-x_j}{t}\r)}{\beta!}\lf(\frac{y-x_j}{t}\r)^\beta\r]\r|\noz\\
&\quad\quad\times\lf\{\int_{B_j}|a_j(z)|\lf|k(y,z)-\sum_{|\beta|\le\lceil \gamma\rceil}
\frac{\partial_y^\beta k(y,x_j)}{\beta!}(z-x_j)^\beta\r|\,dz\r\}\,dy\noz\\
&\ls\int_{|y-x_j|\ge\frac{|x-x_j|}{2}}\lf|\frac{1}{t^n}
\lf[\phi\lf(\frac{x-y}{t}\r)-\sum_{|\beta|\le\lceil \gamma\rceil}
\frac{D^\beta\phi\lf(\frac{x-x_j}{t}\r)}{\beta!}\lf(\frac{y-x_j}{t}\r)^\beta\r]\r|\noz\\
&\quad\quad\times\int_{B_j}|a_j(z)|\lf|
\sum_{|\beta|=\lceil \gamma\rceil}\frac{\partial_y^\beta
k(y,x_j)-\partial_y^\beta k(y,\xi_3(z))}{\beta!}(z-x_j)^\beta\r|\,dz\,dy\noz\\
&\ls\int_{|y-x_j|\ge\frac{|x-x_j|}{2}}|\phi_t(x-y)|\int_{B_j}|a_j(z)|
\frac{|z-x_j|^\gamma}{|y-x_j|^{n+\gamma}}\,dz\,dy\noz\\
&\quad\quad+\int_{|y-x_j|\ge\frac{|x-x_j|}{2}}\lf|\frac{1}{t^n}
\sum_{|\beta|\le\lceil \gamma\rceil}
\frac{D^\beta\phi\lf(\frac{x-x_j}{t}\r)}{\beta!}
\lf(\frac{y-x_j}{t}\r)^\beta\r|\noz\\
&\quad\quad\times\int_{B_j}|a_j(z)|
\frac{|z-x_j|^\gamma}{|y-x_j|^{n+\gamma}}\,dz\,dy\noz\\
&\ls\frac{r_j^\gamma}{|x-x_j|^{n+\gamma}}\|a_j\|_{L^2(\rn)}|B_j|^{1/2}
\int_{|y-x_j|\ge\frac{|x-x_j|}{2}}\lf|\phi_t(x-y)\r|\,dy\noz\\
&\quad\quad+\sum_{|\beta|\le\lceil \gamma\rceil}
r_j^\gamma\|a_j\|_{L^2(\rn)}|B_j|^{1/2}\noz\\
&\quad\quad\times\int_{|y-x_j|\ge\frac{|x-x_j|}{2}}\frac{1}{t^n}
\frac{t^{n+|\beta|}}{|x-x_j|^{n+|\beta|}}
\frac{|y-x_j|^{|\beta|}}{|t|^{|\beta|}}\frac{1}{|y-x_j|^{n+\gamma}}\,dy\noz\\
&\ls\frac{r_j^{n+\gamma}}{|x-x_j|^{n+\gamma}}\frac{1}{\|\chi_{B_j}\|_{\lv}}.
\end{align}
Combining \eqref{3.1.y2}, \eqref{3.1.y3}, \eqref{3.1.y4} and \eqref{3.1.y5},
we conclude that, for all $x\in(4B_j)^{\com}$,
\begin{align*}
|\phi_{+}^\ast(Ta_j)(x)|&=\sup_{t\in(0,\fz)}\lf|\phi_t*Ta_j(x)\r|
\ls\frac{r_j^{n+\gamma}}{|x-x_j|^{n+\gamma}}\frac{1}{\|\chi_{B_j}\|_{\lv}}\\
&\ls\lf[\cm(\chi_{B_j})(x)\r]^{\frac{n+\gamma}{n}}
\frac{1}{\|\chi_{B_j}\|_{\lv}},
\end{align*}
which implies that
$$\phi_{+}^\ast(Ta_j)(x)\chi_{(4B_j)^{\com}}(x)
\ls\lf[\cm(\chi_{B_j})(x)\r]^{\frac{n+\gamma}{n}}
\frac{1}{\|\chi_{B_j}\|_{\lv}}.$$
Therefore, by Proposition \ref{1.14.x3} and an argument similar to that
used in the proof of \eqref{1.26.x2}, we know that
\begin{align*}
&\az\lf\|\chi_{\{x\in\rn:\ \sum_{j\in\nn}\lz_j
\phi_{+}^\ast(Ta_j)(x)\chi_{(4B_j)^{\complement}}(x)
>\frac{\az}2\}}\r\|_{\lv}\\
&\hs\ls\frac{\az}{2}\lf\|\chi_{\{x\in\rn:\ [\sum_{j\in\nn}\frac{\lz_j}
{\|\chi_{B_j}\|_{\lv}}\{\cm(\chi_{B_j})(x)\}^{\frac{n+\gamma}{n}}]
^{\frac{n}{n+\gamma}}>(\frac{\az}{2})^{\frac{n}{n+\gamma}}\}}\r\|
_{L^{\frac{(n+\gamma)p(\cdot)}{n}}(\rn)}^{\frac{n+\gamma}{n}}\\
&\hs\ls\lf\|\lf[\sum_{j\in\nn}\frac{\lz_j\chi_{B_j}}
{\|\chi_{B_j}\|_{\lv}}\r]^{\frac{n}{n+\gamma}}\r\|
_{L^{\frac{(n+\gamma)p(\cdot)}{n}}(\rn)}^{\frac{n+\gamma}{n}}
\ls\|f\|_{H^{p(\cdot)}(\rn)},
\end{align*}
which shows \eqref{3.4.x1} holds true and hence completes the proof of Theorem \ref{bdnthm3}.
\end{proof}
\textbf{ Acknowledgement.} The authors would like to express their
deep thanks to the referee for his very careful reading and several useful
comments which improve the presentation of this article.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,466 |
.. _Data-QA-QC:
Time Series Data QA/QC
======================
ODM2 Admin can be used as a tool for time series data quality assurance and control. Raw time series can be
copied and then modified with time series result annotations. Portions of time series can be selected and
if values are known to be bad they can be set to not a number 'NaN' while the original values are maintained
in an annotation. The annotation can then also indicate the reason why the value was not good, such as the
instrument was out of calibration.
Let's try it out, we are going to visit the data annotation page for a measurement of water temperature
at the Rio Espiritu Santo Stream House (RESSH), from the `map <http://odm2admin.cuahsi.org/Sandbox/mapdata.html>`_
this is a sampling feature of type 'Stream Gage' labeled RESSH.
.. image:: /images/RioEspirituSantoStreamGage.png
From Here follow the link to the water temperture read by either the pressure transducer or the DO probe, select the
'Annotate Data' link as shown here:
.. image:: /images/SelectWaterTempToAnnotate.png
http://odm2admin.cuahsi.org/Sandbox/graphfa/samplingfeature=776/resultidu=16657/popup=Anno/
You can select some of the points by dragging a selection box around them, it should look like this:
.. image:: /images/RESSH-WaterTemperatureReadings.PNG
With the points selected we can select a new data quality code, enter an annotation, generate a L1, QA/QC level from
a level 0 time series. From a time series that are not raw data (something other then L0) we can set values to Not A
Number (NAN).
.. image:: /images/AnnotatingData.png
Our Annotated data will then look something like this:
.. image:: /images/AnnotatedData.png
You can also add or subtract an offset value to a set of points or drift correct them. Drift correcting points will
apply a linear correction so that the first or last point (depending if you select forward or backward) will have the
value entered. The remaining values you selected will have an offset added to them. This offset is proportional to the
points position in the set of points selected.
.. image:: /images/offsetanddriftcorrect.png
:ref:`5) Data visualization and URL parameters <Data-Visualization>`
* :ref:`ODM2 Admin docs home page<ODM2-Admin>`
* :ref:`Search the docs <search>`
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,260 |
Willie Stratton still remembers one of the first songs he wrote and performed for an audience. It was for his Grade 4 music class and the tune was inspired by his then obsession with vampires and church organs. The sound was unforgettable.
Years later, and after lessons and practice in guitar and songwriting, Stratton has ditched the vampire themes in exchange for a fusion of old and new, folksy tunes with a progressive twist.
With two albums (Willie Stratton and The River) under his belt, one in the works, and a number of gigs at venues around the region, 21-year-old Stratton is getting a name for himself as a young Bedford talent to watch.
His says he wants to create music that blends together the elements of rock and traditional music that has inspired him since he was a kid. He says he's listened to and been inspired by everything from Jimi Hendrix, the Beach Boys, the Doors, Joel Plaskett and Dylan Guthro.
"I just want to create this weird fusion," he says. I've always been interested in the history, the journey behind the music we listen to now." That, he says, it was he calls the "building blocks" of music.
Stratton says his parents have always had his back with his music. "They've always been supportive of my music," he says. "When I was little they had me in music classes." And his sister, Grace, plays bass in the band now, making his music a family affair. Together the band has played at venues like Michaels, The Company House and the Seahorse.
Suzanne is an editor and writer whose work has appeared in The Coast, Lawyers Weekly, Canadian Business, Globe and Mail, Bakers Journal, Our Children, and more. She hosts the radio show Cobequid Magazine on 97.5 Community Radio. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,706 |
{"url":"https:\/\/gateoverflow.in\/382005\/tifr-cse-2022-part-b-question-4","text":"236 views\n\nConsider the following algorithm for computing the factorial of a positive integer $n$, specified in binary:\n\nprod \u2190 1\nfor i from 1 to n\nprod \u2190 prod \u00d7 i\noutput prod\n\nAssume that the number of bit operations required to multiply a $k$-bit positive integer with an $\\ell$-bit positive integer is at least $\\Omega(k+l)$ and at most $O(kl)$. Then, the number of bit operations required by this algorithm is\n\n1. $O(n)$\n2. $O(n \\log n)$ but $\\omega(n)$\n3. $O\\left(n^2\\right)$ but $\\omega(n \\log n)$\n4. $O\\left(n^3\\right)$ but $\\omega\\left(n^2\\right)$\n5. None of the above\n\n### 1 comment\n\n@Sachin Mittal 1\u00a0sir pls provide solution for this question\n\n1 vote","date":"2023-02-06 02:18:53","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\": 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\": 0.9048230648040771, \"perplexity\": 1138.6811295437492}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500303.56\/warc\/CC-MAIN-20230206015710-20230206045710-00249.warc.gz\"}"} | null | null |
Волчко Преслужич (гербу Абданк із відміною; після 1419) — впливовий галицький боярин початку XV століття, засновник Рогатина.
Біографія
Походження та батьки Волчка невідомі, хоча висловлювали припущення, що він міг бути нащадком або родичем руського старости Дмитра Детька, або навіть останніх Романовичів.
Титулувався за зразком польських королів та галицького намісника Владислава Опольчика, як пан та дідич на Рогатині (Wolczko Presszlussicz dominy et heres de Rohatyn).
Вперше згаданий поряд з іншими боярами Руського воєводства серед членів суду, який відбувався поміж польським королем Владиславом II Ягайлом та шляхтянкою Єлизаветою (Ядвігою) Пілецькою в Медиці, що біля Перемишля.
У 1415—1419 роках заснував на місці села Філіповичі (Philippowicze) місто Рогатин (до цього розташовувалося на горі біля сучасного села Підгороддя). Разом із тим Волчко надав Рогатину магдебурзьке право, а також герб, на якому була зображена латинська літера «R» та оленячий ріг.
Відома печатка Волчка Преслужича, вона має незвично великий діаметр — 43 мм, на той час радше притаманний князівським та державним печаткам. Герб Волчка має вигляд латинської літери «W», що зближує його з династичними тризубами Рюриковичів.
Дружина Волчка невідома. Його син, Івашко Богдан Рогатинський, брав участь у Луцькій війні 1431—1432 рр. на стороні великого князя литовського й руського Свидригайла, через що втратив родові володіння на Галичині та переселився на Волинь.
Примітки
Джерела
Вортман Д. Я. Рогатин //
Однороженко О. «Clipearium Teutonicorum» і «Zuricher Wappenrolle» та їх значення для вивчення Руської державної геральдики другої половини ХІІІ — початку ХІV ст. / О. Однороженко // Спеціальні історичні дисципліни: питання теорії та методики: Зб. наук. пр. — 2009. — Вип. 16. — С. 20-34.
Ясінський М. Р. Вплив післявоєнного та нового будівництва на «п'ятий фасад» м. Рогатина / М. Р. Ясінський // Вісник Національного університету «Львівська політехніка». — 2011. — № 716 : Архітектура. Ландшафт дахів історичного центру міста: проблеми збереження і регенерації. — С. 311—316.
Галицькі бояри
Рогатинські | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,795 |
2 May 2017 — 02 May 2017
Girls' Junior School
Boys' Junior School
Sport Matters
Another resounding success at our 17th Annual Horse Trials Championship
The 17th Annual Tintern Interschool Horse Trials was once again a great success. The event was held at the picturesque Wandin Park on Saturday 29 and Sunday 30 April. There were over 270 competitors from 46 schools across Victoria who participated in the competition. Tintern's Interschool Horse Trials is a unique event on the Interschool Equestrian Calendar where competitors participate in three phases of competition over two days – dressage, show jumping and cross country. Dressage being the main event on Saturday and Cross Country and Showjumping on Sunday. In 2016 our event became a qualifying event for the Nationals, which is a notable achievement for an event run by a school.
The weather was kind to us for most of the weekend, aside from a little drizzle on Saturday morning. We were thrilled that Seumas Marwood, whose son Tyler was competing, agreed to walk the Cross Country course with some of our youngest riders. Seumas is a 4* rider who has been competing at National and International level for Australia for over 30 years, so we appreciated his generosity in providing some valuable tips to our Class 5 riders who are quite new to eventing.
Most competitors camp with their horses and team mates at Wandin Park over the course of the weekend. Many of the competitors, teachers and their families enjoyed the dinner and dancing which was organised on Saturday evening whilst others enjoyed staying warm by their camp fires.
The competition is scored individually and as a team event, at the time of writing, we are still awaiting final results from the Event Secretary.
Tintern Grammar had 11 riders in three teams and, whilst our teams didn't place this year, there were some great individual results. Congratulations to all our riders from Tintern Grammar Equestrian Team. There were many personal bests achieved over the course of the weekend and all of our riders were outstanding ambassadors for Tintern Grammar.
Thank you to the parents from the Friends of Equestrian, who all worked together to organise the day with enthusiasm and professionalism. This event would not be possible without the many hundreds of hours also provided by the teams of volunteers who assist the committee during the weekend.
One of our Equestrian Captains, Eliza Harvey (Year 12), was representing Tintern at her 10th Tintern Horse Trials this year. Congratulations, Eliza, on your outstanding contribution to the Equestrian Team during your time at Tintern. We look forward to another successful event in 2018.
by Alison Bezaire, Equestrian Coordinator
Interschool Equestrian State Championships
During the school holidays, we had three riding and two strapping members of the Tintern Equestrian team spend a wet few days camping in Werribee for the Interschool Equestrian State Championships.
This is a very large state event that attracts hundreds of Victorian families from all over the state to compete in a variety of equestrian classes, all gaining points for themselves individually and also as a school team. We were fortunate enough to have three fantastic competitors form a Tintern Team: Lilly Trevorrow, Odette McCallum and Keeley Thomas. These girls along with their support crew of Ella Trevorrow and Darcee McCallum had excellent results. There were over 480 competitors representing 180 schools. The Tintern Team finished 30th with both Lilly and Keeley receiving ribbons for individual achievements.
Tintern Holiday Clinic
The Tintern Equestrian Team members enjoyed a lovely clinic at Wandin Park during the holidays. Every member had two very informative lessons with 4* professional eventers Will Enzinger and Emily Anker in the showjumping and cross country. The weather was perfect, as was the BBQ organised by the parents.
Thank you to the Friends of Equestrian Committee for organising the lessons, we all felt very prepared for the Horse Trials last weekend!
by Eliza Harvey, Equestrian Captain 2017
Women in War Time
Tintern Website
Girls' Junior School
Boys' Junior School
2023 Term Dates
Term 1 (All students)
Commences: Thursday 2 February (full day)
Concludes: Thursday 6 April
Term 2 (All students):
Commences: Wednesday 26 April
Concludes: Friday 16 June
Commences: Tuesday 11 July
Concludes: Thursday 14 September
Commences: Monday 2 October
Concludes (Junior and Middle School students): Monday 11 December | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,443 |
Distance from Massachusetts to Monrovia is 4,127 kilometers. This air travel distance is equal to 2,564 miles.
The air travel (bird fly) shortest distance between Massachusetts and Monrovia is 4,127 km= 2,564 miles.
If you travel with an airplane (which has average speed of 560 miles) from Massachusetts to Monrovia, It takes 4.58 hours to arrive.
Monrovia is located in United States. | {
"redpajama_set_name": "RedPajamaC4"
} | 513 |
A person's wellness can determine a lot about them. Last year I was scoring negative zero on the wellness meter. Ya girl was stressed beyond belief. I was always doing something school related and never took anytime for myself.
My life is always all over the place. The only thing I am consistent with is my skin care. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,199 |
Q: Chrome debugger stops at a function even without a DOM break point Hi I have a angular js application. And when I run my application and hit F12 and open the debugger, for each and every page it continuously calling following function and stops there just like it has a break point.
Here is the stopping point
TagCanvas.NextFrameRAF = function() {
requestAnimationFrame(DrawCanvasRAF);
};
But this function doesn't have any break points.
A: It sounds like you have a Request Animation Frame breakpoint set:
You can find that in dev tools on the Sources tab on the right-hand side under Event Listener Breakpoints > Animation.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,623 |
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var $musicPre = $('#js-music-pre');
var $musicStart = $('#js-music-start');
var $musicNext = $('#js-music-next');
var $sidebarHeader = $('.sidebar-header');
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var currentSrcIndex = 0;
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mouseenter: function() {
$sliderBtn.toggle(600);
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mouseleave: function() {
$sliderBtn.toggle(600);
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// 点击提示出现详情
$sliderBtn.on('click', function() {
$sliderShadow.show(600)
.delay(1000)
.on('click', function() {
$(this).hide(600);
});
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/* 音乐播放功能 */
// 暂停、播放切换
$musicStart.on('click', function() {
if ($music.paused) {
$music.play();
$musicStart.removeClass('music-pause');
} else {
$music.pause();
$musicStart.addClass('music-pause');
}
});
// 上一首
$musicPre.on('click', function() {
--currentSrcIndex < 0 && (currentSrcIndex = $musicList.length - 1);
currentSrc = $musicList.eq(currentSrcIndex).prop("src");
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++currentSrcIndex > $musicList.length - 1 && (currentSrcIndex = 0);
currentSrc = $musicList.eq(currentSrcIndex).prop("src");
$music.src = currentSrc;
$music.play();
});
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/* 侧边栏 */
// 切换效果
$sidebarHeader.on('click', function() {
$sidebarContent.toggle(600);
$(this).toggleClass('sidebar-header-change');
});
/* 侧边栏 end */ | {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"http:\/\/mathandmultimedia.com\/tag\/order-of-operations-problems\/","text":"## Order of Operations: 360 or 354?\n\nLast Saturday, I received \u00a0Facebook message from a student asking help to simplify\u00a0$[5(4)^3 + 6(11-4)] - 36 \/ 9 (2)$. \u00a0He got $354$ but his teacher\u2019s answer was $360$.\n\nAlthough the problem above seems simple, a lot of students get confused by it, and in this case, even the teacher too. The expression above simplifies to \u00bb Read more","date":"2022-08-07 19:40:49","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\": 3, \"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\": 0.6789281368255615, \"perplexity\": 1361.850677928549}, \"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-2022-33\/segments\/1659882570692.22\/warc\/CC-MAIN-20220807181008-20220807211008-00233.warc.gz\"}"} | null | null |
News Security and Defense
Maduro: New Cyber and Electromagnetic Weapons are Being Tested in Venezuela to Achieve "Regime Change" (Video)
April 4, 2019 orinocotribune 1 Comment electric blackout, electric war, EMP, recommence of classes, regime change, Sabotage, US Imperialism
The President of Venezuela, Nicolás Maduro, announced on Wednesday that his country "is experiencing the first war of unconventional dimensions, which aims to impose itself through utilities for a regime change directed by the United States." In a telephone contact with the TV show "Con el Mazo Dando", directed by Diosdado Cabello, he explained that the country is living "the first electric war, with cyber, electromagnetic and physical characteristics, to overthrow a democratic and legitimate government, make a regime change and impose the chaos and domination of a strategic country, with the largest oil reserve in the world, as is Venezuela, who is currently presiding over OPEC with strong leadership in the Third World countries".
"Venezuela is a field for testing new cyber and electromagnetic weapons, and a new war strategy, which is not direct invasion or bombing by missiles and airplanes, but the bombing and destruction of public services, vital for the people, so that people enter into a phase of chaos, people against people, war of dogs, as they call it; and they then enter to dominate the country", explained Maduro.
"It is an electric war unique in the history of Venezuela and the first of these characteristics in the new history of world conflicts from now on," he said.
His statement was issued days after several serious failures at the Guri Hydroelectric Power Plant, which the government has blamed on a cyber attack at its control center, at another similar attack at the Corpoelec control center in Guyana, and at electromagnetic attacks on the transmission lines. In addition, a fire in the transformer yard of Guri on March 25 was initiated by a sniper attack, explained Maduro days ago. There have also been explosions and fires in electrical substations. These serious problems had left more than 80 percent of the country without electricity for several days.
Maduro pointed out that several experts have warned, in analyses made 4 or 5 years ago, that it could attack the electrical system of a country and leave it incapable, to weaken it in a global power dispute.
"They try to take the people to despair, to a situation of being a people blind with rage, so that there is a confrontation between Venezuelans and so that there is violence. But the people have had the awareness of knowing how to handle their anger, how to handle their impotence and their need. "
He said that this occurs "with the complicity of the worst opposition" the country has had in 20 years of revolution: "the most criminal, the most murderous, the most corrupt we ever had". He indicated that "there are things I can not say, that one day I will say, that I know about their criminal plans, those who lead the opposition today, of the plans to kill me, to kill you (Diosdado Cabello), to kill a lot people. On hiring, on plans to attack military units. Maximum intelligence and counterintelligence", he warned.
"And I tell the people: never in 20 years of revolution, have we had a leading nucleus of the opposition with a feeling and a vision so criminal, so murderous, so unethical. They are the main accomplices of this type of war! "
Thursday and Friday schools using flexible schedule
He recalled that "the situation is serious, I will not deceive anyone: the situation is serious, but we are advancing step by step guaranteeing the fundamentals to the people."
He indicated that "we have managed to advance in the resistance phase of this electricity management plan to maintain electric service and water service for the people, and today (Wednesday) we started classes in this resistance phase."
He noted that there was a significant increase in electricity consumption, "that we knew how to get along with the electricity management plan (rationing). Today the orientation has been given, and I ask the educational communities to inform that tomorrow (Thursday) and Friday we continue in classes, with flexible schedules, adapted to the conditions of each educational unit, each locality, each municipality and each town of the country". He congratulated the students and teachers and urged them to continue in classes even if the electricity administration plan (rationing) surprises them.
He urged governors, mayors and authorities to create WhatsApp groups and other mechanisms to keep the people informed on the electric battle, "so that people know when they will have the electrical management plan in their area."
He also demanded that the Clap food boxes be delivered on time and without delays. He approved the resources for the Ministry of Food, in order to promote the protein plan within the Clap (meat, fish and eggs) allowing it to "reach the people this weekend with strength." He also urged the country's farmers and ranchers to continue production.
Source URL: Alba Ciudad
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Chancellor Arreaza Meeting With Al-Assad: Syria an Example of Resistance to Imperialism →
Venezuela & The Mighty Wurlitzer
US-Led Economic War, Not Socialism, Is Tearing Venezuela Apart
December 31, 2018 orinocotribune 0
The Last Round / "From Hell with Gaddafi to the Nightmare Without Him"
March 10, 2019 orinocotribune 2
One thought on "Maduro: New Cyber and Electromagnetic Weapons are Being Tested in Venezuela to Achieve "Regime Change" (Video)"
Pingback:The US Army Cut Off Electricity to its Largest Military Base to Test Reactions on a Cyber Attack – Orinoco Tribune | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,912 |
{"url":"https:\/\/openforecast.org\/adam\/diagnosticsResidualsIIDAuto.html","text":"## 14.5 Residuals are i.i.d.: autocorrelation\n\nOne of the typical characteristics of time series models is the dynamic relation between variables. Even if fundamentally, the sales of ice cream on Monday do not impact sales of the same ice cream on Tuesday, they might impact advertising expenses or sales of a competing product on Tuesday, Wednesday or next week. Missing this structure might lead to the autocorrelation of residuals, influencing the estimates of parameters and final forecasts. This can influence the prediction intervals (causing miscalibration) and in serious cases leads to biased forecasts. Autocorrelations might also arise due to wrong transformations of variables, where the model would systematically underforecast the actuals, producing autocorrelated residuals. In this section, we will see one of the potential ways for the regression diagnostics and try to improve the model stepwise, trying out different orders of the ARIMA model (Section 9).\n\nAs an example, we continue with the same seatbelts data, dropping the dynamic part to see what would happen in this case:\n\nadamSeat09 <- adam(Seatbelts, \"NNN\", lags=12,\nformula=drivers~log(PetrolPrice)+log(kms)+law)\nAICc(adamSeat09)\n## [1] 2651.191\n\nThere are different ways to diagnose this model. We start with a basic plot of residuals over time (Figure 14.16):\n\nplot(adamSeat09, 8, main=\"\")\n\nWe see in Figure 14.16 that on one hand the residuals still contain seasonality and on the other, they do not look stationary. We could conduct ADF and \/ or KPSS tests (which will be discussed in one of the later Chapters of Svetunkov, 2022a) to get a formal answer to the stationarity question:\n\ntseries::kpss.test(resid(adamSeat09))\n## Warning in tseries::kpss.test(resid(adamSeat09)): p-value greater than printed\n## p-value\n##\n## KPSS Test for Level Stationarity\n##\n## KPSS Level = 0.12844, Truncation lag parameter = 4, p-value = 0.1\ntseries::adf.test(resid(adamSeat09))\n## Warning in tseries::adf.test(resid(adamSeat09)): p-value smaller than printed p-\n## value\n##\n## Augmented Dickey-Fuller Test\n##\n## Dickey-Fuller = -6.8851, Lag order = 5, p-value = 0.01\n## alternative hypothesis: stationary\n\nThe tests have opposite null hypotheses, and in our case, we would fail to reject H$$_0$$ on 1% for the KPSS test and reject H$$_0$$ on 1% for the ADF, which means that the residuals look stationary. The main problem in the residuals is the seasonality, which formally makes the residuals non-stationary (their mean changes from month to month), but which cannot be detected by these tests. Yes, there is a Canova-Hansen test, which is implemented in ch.test function in uroot package in R, which tests the seasonal unit root, but I instead of trying it out and coming to conclusions, I will try the model in seasonal differences and see if it is better than the one without it:\n\n# SARIMAX(0,0,0)(0,1,0)_12\nformula=drivers~log(PetrolPrice)+log(kms)+law,\norders=list(i=c(0,1)))\nAICc(adamSeat10)\n## [1] 2547.274\n\nRemark. While in general models in differences are not comparable with the models applied to the original data, adam() allows such comparison, because ARIMA model implemented in it is initialised before the start of the sample and does not loose any observations.\n\nThis leads to an improvement in AICc in comparison with the previous model. The residuals of the model are now also better behaved (Figure 14.17):\n\nplot(adamSeat10, 8, main=\"\")\n\nIn order to see whether there are any other dynamic elements left, we will plot ACF and PACF (discussed in Subsections 8.3.2 and 8.3.3) of residuals (Figure 14.18):\n\npar(mfcol=c(1,2), mar=c(4,4,0,1))\nplot(adamSeat10, 10:11, level=0.99, main=\"\")\n\nIn the case of the adam() objects, these plots, by default, will always have the range for the y-axis from -1 to 1 and will start from lag one on the x-axis. The red horizontal lines represent the \u201cnon-rejection\u201d region. If a point lies inside the region, it is not statistically different from zero on the selected confidence level (the uncertainty around it is so high that it covers zero). The points with numbers are those that are statistically significantly different from zero. So, the ACF \/ PACF analysis might show the statistically significant lags on the selected level (the default one is 0.95). Given that this is a statistical instrument, we expect that approximately (1-level)% (e.g.\u00a01%) of lags lie outside these bounds just due to randomness, even if the null hypothesis is correct. So it is okay if we do not see all points lying inside them. However, we should not see any patterns there, and we might need to investigate the suspicious lags (low orders of up to 3 \u2013 5 and the seasonal lags if they appear). In our example in Figure 14.18, we see that there are spikes in lag 12 for both ACF and PACF, which means that we have missed the seasonal element in the data. There is also a suspicious lag 1 on PACF and lags 1 and 2 on ACF, which could potentially indicate that MA(1) and\/or AR(1), AR(2) elements are needed in the model. While it is not clear what specifically is needed here, we can try out several models and see which one is better to determine the appropriate order of ARIMA. We should start with the seasonal part of the model, as it might obscure the non-seasonal one.\n\n# SARIMAX(0,0,0)(1,1,0)_12\nformula=drivers~log(PetrolPrice)+log(kms)+law,\norders=list(ar=c(0,1),i=c(0,1)))\nAICc(adamSeat11)\n## [1] 2534.39\n# SARIMAX(0,0,0)(0,1,1)_12\nformula=drivers~log(PetrolPrice)+log(kms)+law,\norders=list(i=c(0,1),ma=c(0,1)))\nAICc(adamSeat12)\n## [1] 2426.688\n# SARIMAX(0,0,0)(1,1,1)_12\nformula=drivers~log(PetrolPrice)+log(kms)+law,\norders=list(ar=c(0,1),i=c(0,1),ma=c(0,1)))\nAICc(adamSeat13)\n## [1] 2542.092\n\nBased on this analysis, we would be inclined to include in the model seasonal MA(1) only. Next step in our iterative process \u2013 another ACF \/ PACF plot of the residuals:\n\nIn this case, there is a spike on PACF for lag 1 and a textbook exponential decrease in ACF starting from lag 1, which might mean that we need to include AR(1) component in the model:\n\n# ARIMAX(1,0,0)(0,1,1)_12\nformula=drivers~log(PetrolPrice)+log(kms)+law,\norders=list(ar=c(1,0),i=c(0,1),ma=c(0,1)))\nAICc(adamSeat14)\n## [1] 2386.444\n\nChoosing between the new model and the old one, we should give preference to the model 14, which has a lower AICc than model 12. So, adding MA(1) to the model leads to further improvement in AICc compared to the previous models. Using this iterative procedure, we could continue our investigation to find the most suitable SARIMAX model for the data. Nevertheless, this example should suffice in providing a general idea of how it can be done. What we could do else to simplify the process is to use the automated ARIMA selection algorithm (see discussion in Section 15.2) in adam(), which is built on the principles discussed in this section:\n\nadamSeat15 <-\nformula=drivers~log(PetrolPrice)+log(kms)+law,\norders=list(ar=c(3,2),i=c(2,1),ma=c(3,2),select=TRUE))\nadamSeat15\\$model\n## [1] \"SARIMAX(1,0,0)[1](0,1,1)[12]\"\nAICc(adamSeat15)\n## [1] 2386.444\n\nThis new constructed model is SARIMAX(1,0,0)(0,1,1)$$_{12}$$, which is the model we achieved manually. Note that it is even better than the ETSX(MNM) (model 5) from the previous section in terms of AICc. Its residuals are much better behaved than the ones of model 5 (although we might need to analyse the residuals for the potential outliers, as discussed in Subsection 14.4):\n\npar(mfcol=c(3,1), mar=c(4,4,2,1))\nplot(adamSeat15, c(8,10:11), level=0.99)\n\nSo for the purposes of analytics and forecasting, we would be inclined to use SARIMAX(2,0,0)(0,1,1)$$_{12}$$ rather than ETSX(MNM).\n\nAs a final word for this section, we have focused our discussion on the visual analysis of time series, ignoring the statistical tests (we only used ADF and KPSS). Yes, there is Durbin-Watson test for AR(1) in residuals, and yes, there are Ljung-Box , Box-Pierce and Breusch\u2013Godfrey tests for multiple AR elements. But visual inspection of time series is not less powerful than hypothesis testing. It makes you think and analyse the model and its assumptions, while the tests are the lazy way out that might lead to wrong conclusions because they have the standard limitations of any hypothesis tests (as discussed in Section 5.3 of Svetunkov, 2022a). After all, if you fail to reject H$$_0$$, it does not mean that the effect does not exist. Having said that, the statistical tests become extremely useful when you need to process many time series simultaneously and cannot inspect them manually. So, if you are in that situation, I would recommend reading more about them, but I do not aim to retell the content of Wikipedia in this monograph.\n\n### References\n\n\u2022 Svetunkov, I., 2022a. Statistics for business analytics. https:\/\/openforecast.org\/sba\/ (version: 31.03.2022)\n\u2022 Wikipedia, 2021f. Durbin\u2013Watson statistic. https:\/\/en.wikipedia.org\/wiki\/Durbin%E2%80%93Watson_statistic (version: 2021-05-13)\n\u2022 Wikipedia, 2021g. Ljung\u2013Box test. https:\/\/en.wikipedia.org\/wiki\/Ljung%E2%80%93Box_test (version: 2021-05-13)\n\u2022 Wikipedia, 2021h. Breusch\u2013Godfrey test. https:\/\/en.wikipedia.org\/wiki\/Breusch%E2%80%93Godfrey_test (version: 2021-05-13)","date":"2022-06-27 08:40:44","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\": 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\": 0.6595597267150879, \"perplexity\": 1214.0391597578516}, \"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-2022-27\/segments\/1656103329963.19\/warc\/CC-MAIN-20220627073417-20220627103417-00484.warc.gz\"}"} | null | null |
Q: How can I keep the selected tabs open when page is reloaded This is a side bar menu and until now it closes the tabs each time I reload the page and it's pretty annoying, what should I add to fix this?
here is a code pen snippet
_menu.html.erb
<div id="mySidepanel" class="sidepanel">
<button class="closebtn" onclick="closeNav()"><i class="fas fa-times-circle"></i></button>
<div class="dropdown-container">
<%= link_to "Journal", "#", class: "drop-item" %>
<%= link_to "Stats","#", class: "drop-item" %>
</div>
<li class="dropdown-btn">Suppliers
<i class="fa fa-caret-down"></i>
</li>
<div class="dropdown-container">
<%= link_to "Add a supplier", "#", class: "drop-item" %>
<%= link_to "Suppliers", "#", class: "drop-item" %>
</div>
<%= link_to "Orders", "#", class: "list-item" %>
<%= link_to "Users", "#", class: "list-item" %>
</div>
sidebar.js
edit, added the whole code, I am not using the Bootstrap dropdown
$(document).on('turbolinks:load', function() {
if ($('.dropdown-btn')){
activeTab = document.getElementsByClassName('active')[0].dataset.activeDropdown.myArticles
sessionStorage.setItem('foo', activeTab);
}
if(activeTab){
$('#mySidepanel a[href="' + activeTab + '"]').tab('show');
}
var dropdown = document.getElementsByClassName("dropdown-btn");
var i;
for (i = 0; i < dropdown.length; i++) {
dropdown[i].addEventListener("click", function() {
this.classList.toggle("active");
var dropdownContent = this.nextElementSibling;
if (dropdownContent.style.display === "block") {
dropdownContent.style.display = "none";
} else {
dropdownContent.style.display = "block";
}
});
}
});
function openNav() {
document.getElementById("mySidepanel").style.width = "180px";
}
function closeNav() {
document.getElementById("mySidepanel").style.width = "0";
}
Update
I added this js code but I would need help :
$('.drop-item').on('shown.bs.tab', function (e) {
localStorage.setItem('activeTab', $(e.target).attr('href'));
});
var activeTab = localStorage.getItem('activeTab');
if(activeTab){
$('#mySidepanel a[href="' + activeTab + '"]').tab('show');
}
A: You have to use the sessionStorage (I think that store for the session is enough for what you want to do).
On change on a tab, store the value of your tab (his ID for example):
localStorage.setItem('tabId', 'foo');
Then, on page load, if your key exists, open your tab. With jQuery, should be something like this:
$(function(){
if (localStorage.getItem('tabId') !== null) {
document.getElementById(localStorage.getItem('tabId')).show();
}
}
To give you an idea. It's not the more cleaner as it is. For instance you should rewrite your openNav function to accept a param (the localStorage value)
| {
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Furi is a Perfect Example of How to Open a Game
By Blessing Adeoye Last updated Jun 19, 2018
In my time playing games, I have experienced the beginning hour of many different games. I say this with shame in my voice. The number of games in which I've played the beginning hour exceeds the amount of games in which I've played the final hour by a large number. The difference is exponential. I love to try things out. I also do not have all the time in a world to spend on games that don't immediately grab me. This is why I was surprised while playing the first hour of Furi.
No game has ever given me the same impression that Furi has given me upon entering the game. To be clear, at this point I have only played the first hour of the Furi. However, that hour was so good that I had to pick up my laptop to tell the world about it.
There were three things that struck me upon entering the world of Furi when I first started playing the game. The first thing was the game's style. Furi has a tone that is lost in many games today. The style is like that of a comic book or anime but it is also very dark and brutal in terms of themes. The opening boss's dialogue was menacing to the point where I personally felt threatened. The talk of death and ruthlessness coming from the character immediately lets you know that this guy is playing no games. What's even better is that this brutal, ruthless tone is coupled with a bright, colorful aesthetic. The game looks like an old school game that takes place in the future. It pulls this off in a way that doesn't make it feel dated, but fresh.
What was even more impressive about Furi's opening was the difficulty. The opening began with a boss fight that was fairly brutal off the bat. This boss fight not only gives you an idea of what to expect from the rest of the game, it teaches the mechanics of the game and makes sure that you as a player understands them or else. The first half of the battle is a tutorial. During my own playthrough, I was not paying too much attention seeing as how I expected the game to be simple to understand. That was my downfall. What followed that was a series of deaths and a game over which kicked me back to the tutorial stage of the fight again. It has been a while since I have played a game that punishes the player in the beginning if you are not playing the way you should be. Furi does this unapologetically.
Furi's difficulty is coupled with great gameplay. I had fun during the opening of Furi which is not something I can say about the first hour of most games. Most games spend their first hour setting up mechanics, story, and easing the player in. Furi does all of this, and at the same time makes the gameplay engaging. Furi felt like a fighting game. It was immediately interesting. There was no point at which I wanted to stop playing. That is partly due to its arcade-like feeling and mainly due to the game not being afraid to put the player straight into the action.
Furi is a perfect example of how to open a game. It is quick to the action. It makes an impression. The game also respects the player's ability. My time with Furi during its first hour has made me already appreciate what the game is going for and as I am typing this piece, I'm already excited to pick the controller back up and jump back into the action.
This piece was written by Blessing Adeoye. You can find Blessing on the internet either getting into dance battles or analyzing game culture for the purpose of making the world a better place at @blessingjr on the Twittersphere.
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\section{Introduction and main results}
\subsection*{Dynamical r-matrices and Stokes phenomenon}
In the study of non-commutative Weil algebra \cite{AM}, Alekseev and Meinrenken introduced a particular dynamical $r$-matrix $r_{\rm AM}$, which is an important special case of classical dynamical $r$-matrices (\cite{Felder}, \cite{EV}). Let $\mathfrak g$ be a complex reductive Lie algebra and $t\in S^2(\mathfrak g)^2$ the element corresponding to an invariant inner product on $\mathfrak g$, then $r_{\rm AM}$, as a map from $\mathfrak g^*$ to $\mathfrak g\wedge\mathfrak g$, is defined by
\begin{eqnarray*}
r_{\rm{AM}}(x):=({\rm id}\otimes \phi({\rm ad}_{x^{\vee}}))(t), \ \forall x\in\frak g^*,
\end{eqnarray*}
where $x^{\vee}=(x\otimes \rm{id})(t)$ and $\phi(z):=-\frac{1}{z}+\frac{1}{2}\rm{cotanh}\frac{z}{2},$ $z\in \mathbb{C}\setminus 2\pi i\mathbb{Z}^*$.
Remarkably, this r-matrix came to light naturally in two different applications, i.e., in the context of equivariant cohomology \cite{AM} and in the description of a Poisson structure on the chiral WZNW phase space compatible with classical $G$-symmetry \cite{Feher}.
Let $r\in \mathfrak g\otimes\mathfrak g$ be a classical $r$-matrix such that $r+r^{2,1}=t$ (thus $(\frak g,r)$ is a quasitriangular Lie bialgebra). In \cite{EEM}, Enriquez, Etingof and Marshall constructed formal Poisson isomorphisms between the formal Poisson manifolds $\frak g^*$ and $G^*$ (the dual Poisson Lie group). Here $\frak g^*$ is equipped with its Kostant-Kirillov-Souriau structure, and $G^*$ with its Poisson-Lie structure given by $r$.
Their result relies on constructing a formal map $g:\frak g^*\rightarrow G$ satisfying the following gauge transformation equation (as identity of formal maps $\mathfrak g^*\rightarrow \wedge^2(\mathfrak g)$)
\begin{eqnarray}\label{introequation}
g_1^{-1}d_2(g_1)-g_2^{-1}d_1(g_2)+(\otimes^2{\rm Ad}_g)^{-1}r_0+\langle {\rm id}\otimes {\rm id}\otimes x,[g_1^{-1}d_3(g_1),g_2^{-1}d_3(g_2)]\rangle=r_{\rm{AM}},
\end{eqnarray}
Here $r_0:=\frac{1}{2}(r-r^{2,1})$, $g_1^{-1}d_2(g)(x)=\sum_i g^{-1}\partial_{\varepsilon^{i}}g(x)\otimes e_i$ is viewed as a formal
function $\frak g^*\rightarrow \frak g^{\otimes 2}$, $\{e_i\}$ is a basis of $\frak g$, $\{\xi^i\}$ the corresponding coordinates on $\frak g^*$ and $g_i^{-1}d_j(g_i) = (g_1^{-1}d_2(g_1))^{i,j}$.
Two constructions of solutions of \eqref{introequation} are given: the first one uses the theory of the classical Yang-Baxter equation and gauge transformations; the second one relies on the theory of quantization of Lie bialgebras.
The result in \cite{EEM} may be viewed as a generalization of the formal version of \cite{GW}, in which Ginzburg and Weinstein proved the existence of a Poisson diffeomorphism between the real Poisson manifolds $k^*$ and $K^*$, where $K$ is a compact Lie group and $k$ is its Lie algebra. Different approaches to similar results in the subject of linearization of Poisson structures can be found in \cite{Anton} and \cite{Boalch1}.
The main purpose of the present paper is to give an explicit solution of the above equation (provided $r$ is a standard classical $r$-matrix). This allows us to understand the geometric meaning of equation \eqref{introequation} and clarify its relation with irregular Riemann-Hilbert correspondence. The solutions will be constructed as the monodromy of certain differential equations with irregular types. To be precise, let us consider the meromorphic connection on the trivial holomorphic principal $G$-bundle $P$ on $\mathbb{P}^1$ which has the form
\begin{eqnarray}\label{nabla}
\nabla=d-(\frac{A_0}{z^2}+\frac{x}{z})dz
\end{eqnarray}
where $A_0,x\in \frak g$. We assume that $A_0\in \frak t_{\rm reg}$ and once fixed, the only variable is $x\in\mathfrak g\cong\mathfrak g^*$ (via the inner product on $\mathfrak g$).
Then we consider the monodromy of $\nabla$ from $0$ to $\infty$, known as the connection matrix $C(x)$ of $\nabla$, which is computed as the ratio of two canonical solutions of $\nabla F=0$, one is around $\infty$ and another is on one chosen Stokes sector at $0$. Thus we get a map $C:\mathfrak g^*\rightarrow G$ by mapping $x\in\frak g^*$ to the connection matrix $C(x)$ of $\nabla$. See Section \ref{defineC} for more details. One of the main results of this paper is
\begin{thm}\label{main}
The map $C_{2\pi i}\in {\rm Map}(\frak g^*,G)$, defined by $C_{2\pi i}(x):=C(\frac{1}{2\pi i}x)$ for all $x\in\frak g^*$, is a solution of equation \eqref{introequation}.
\end{thm}
The meromorphic connections $\nabla$ taking the form of \eqref{nabla} were
previously studied by Boalch. In particular, link between the connections $\nabla$ and the dual Poisson Lie group $G^*$ was discovered in \cite{Boalch1}, where the Poisson manifold $G^*$ is proven to be a space of Stokes data, and local analytic isomorphisms $\mathfrak g^*$ to $G^*$ in a neighbourhood of $0$ were constructed. Furthermore, the connection matrix $C$ was used by Boalch to construct the Duistermaat twist \cite{Duistermaat}.
\subsection*{Connection matrices and Drinfeld twists}
Having proved the connection matrix satisfies the gauge transformation equation, we can further discuss its relation with Drinfeld twist. This is based on a series work of Enriquez, Etingof and others. In \cite{EEM}, the gauge transformation equation \eqref{introequation} was interpreted as the classical limit of a vertex-IRF transformation equation (see \cite{EH}) between a dynamical twist $J_d(x)\in {\rm Map}(\mathfrak g^*,U(\mathfrak g)^{\hat{\otimes}2}\left\llbracket\hbar\right\rrbracket)$ and a constant twist $J_c\in U(\mathfrak g)^{\hat{\otimes}2}\left\llbracket\hbar\right\rrbracket$. Here $J_d(x)$ and $J_c$ are respectively the twist quantization of $r_{\rm AM}$ and $r_0$ associated to an admissible associator $\Phi$. As a result, the classical limit of a vertex-IRF transformation
$\rho\in {\rm Map}(\mathfrak g^*,U(\mathfrak g)\left\llbracket\hbar\right\rrbracket)$ gives rise to a solution of \eqref{introequation}. According to \cite{EE}\cite{EEM}, an admissible Drinfeld twist $J\in U(\mathfrak g)^{\hat{\otimes} 2}$ (killing the associator $\Phi$) produces such a vertex-IRF transformation.
Here $J$ kills $\Phi$ means that $J$ satisfies the identity
\begin{equation}\label{twistequation}
\Phi=(J^{2,3} J^{1,23})^{-1} J^{1,2} J^{12,3}.
\end{equation}
Thus in particular, the classical limit of an admissible Drinfeld twist $J$ provides a solution of \eqref{introequation}. Now we have two sources of solutions (as is shown before, another one is from connection matrix $C$), whose relation can be encoded in the following theorem. See section \ref{twist} for more details.
\begin{thm} For each (rescaled) connection matrix $C_{2\pi i}\in {\rm Map}(\mathfrak g^*,G)$, there exists a Drinfeld twist killing the associator $\Phi$ whose classical limit is $C_{2\pi i}$.
\end{thm}
In particular, let $\Phi$ be the Knizhnik-Zamolodchikov (KZ) associator $\Phi_{KZ}$, which is constructed as the monodromy from $1$ to $\infty$ of the KZ equation on $\mathbb{P}^1$ with three simple poles at $0$, $1$, $\infty$. Naively the confluence of two simple poles at $0$ and $1$ in the KZ equation turns the monodromy representing KZ associator to the monodromy $C_{\hbar}$ representing the connection matrix of certain differential equation with one degree two pole. Recall that the connection matrix $C$ is the monodromy from $0$ to $\infty$ of the equation $\nabla F=0$.
Then the above theorem indicates that the monodromy $C_{\hbar}$ is related to certain Drinfeld twist killing $\Phi_{KZ}$. In other words, the confluence of two simple poles in KZ equation may be related to the Drinfeld twist identity \eqref{twistequation}. The precise relation between Stokes phenomenon and the theory of quantum groups is worked out recently by Toledano Laredo in \cite{TL} and in our joint work \cite{TLXu}.
\subsection*{Irregular Riemann-Hilbert correspondence}
In the second part of this paper, we clarify the relation between the gauge transformation equation \eqref{introequation} and certain irregular Riemann-Hilbert correspondence. This is motivated and based on Boalch's works, e.g. \cite{Boalch2} \cite{BoalchG} \cite{Boalch3} \cite{Boalch4}, on the study of the geometry of moduli spaces of meromorphic connections on a trivial holomorphic principal $G$(a complex reductive Lie group)-bundle on Riemann surfaces with divisors. We next present a brief review of these works. In \cite{Boalch2}, natural symplectic structures were found and described on such moduli spaces both explicitly and from an infinite dimensional viewpoint (generalising the Atiyah-Bott approach). Explicitly, the extended moduli space (see Definition 2.6 of \cite{Boalch2}) of meromorphic connections on a trivial $G$-bundle $P$ over $\mathbb{P}^1$, with poles on an effective divisor $D=\sum_{i=1}^mk_i(a_i)$ and a fixed irregular type at each $a_i$, was proven to be isomorphic to the symplectic quotient of the form $\widetilde{O}_1\times\cdot\cdot\cdot\times\widetilde{O}_m/\!\!/ G$, where $\widetilde{O}_i$ is an extended orbit with natural symplectic structure associated to the irregular type at $a_i$. In \cite{Boalch3}, a family of new examples of complex quasi-Hamiltonian $G$-spaces $\widetilde{\mathcal{C}}$ with $G$-valued moment maps was introduced, as generalization of the conjugacy class example of Alekseev–Malkin–Meinrenken \cite{AMM}. It was further shown that given the divisor $D=\sum_{i=1}^mk_i(a_i)$, the symplectic spaces of monodromy data for meromorphic connections on $P$ with poles on $D$ and fixed irregular types is isomorphic to the quasi-Hamiltonian quotient space $\widetilde{\mathcal{C}}_1\circledast\cdot\cdot\cdot \widetilde{\mathcal{C}}_m/\!\!/ G$, where $\widetilde{\mathcal{C}}_i$ is the space of monodromy data at $a_i$ and $\circledast$ denotes the fusion product between quasi Hamiltonian $G$-manifolds \cite{AMM}. In the simple pole case, it recovers the quasi-Hamiltonian description of moduli spaces of flat connections in \cite{AMM}. The main result of \cite{Boalch2} \cite{Boalch3} leads to that the irregular Riemann-Hilbert correspondence
\begin{eqnarray*}
\nu:(\widetilde{O}_1\times \cdot\cdot\cdot \widetilde{O}_m)/\!\!/ \hookrightarrow (\widetilde{\mathcal{C}}_1\times\cdot\cdot\cdot\times\widetilde{\mathcal{C}}_m)/\!\!/ G
\end{eqnarray*}
associating monodromy/Stokes data to a meromorphic connection on $P$ is a symplectic map.
In \cite{Boalch1}, Boalch studied (a $T$-reduction version of) the irregular Riemann-Hilbert correspondence in the case of the meromorphic connections have one simple pole and one order two pole. The key feature of this case is that the correspondence gives rise to a Poisson map from the dual of the Lie algebra $\mathfrak g^*$ to the dual Poisson Lie group $G^*$ associated to the standard classical $r$-matrix on $\mathfrak g$.
\\
\\
Now we give the idea of the proof of the main Theorem \ref{main}. The first step is to find a symplectic geometric interpretation of equation \eqref{introequation}, which turns to be a new geometric framework generalizing the Ginzburg-Weinstein linearization.
For this purpose, we consider a symplectic slice $\Sigma$ of $T^*G$ and its Poisson Lie analogue, a symplectic submanifold $\Sigma'$ of the Lu-Weinstein symplectic double $\Gamma$ (locally isomorphic to $G\times G^*$) \cite{Lu}. See Section \ref{Geometryconstruction} and the appendix for more details. Then associated to any map $g\in {\rm Map}(\frak g^*,G)$, we define a local diffeomorphism $F_g:(\Sigma,\omega)\rightarrow (\Sigma',\omega')$. Then a symplectic geometric interpretation of the gauge transformation equation is as follows.
\begin{thm}
$F_g$ is a local symplectic isomorphism from $(\Sigma,\omega)$ to $(\Sigma',\omega')$ if and only if $g\in {\rm Map}(\frak g^*,G)$ satisfies equation \eqref{introequation}.
\end{thm}
With the help of the above theorem, we only need to prove the expected symplectic geometry property of the connection matrix $C$. This is immediate as long as we consider the irregular Riemann-Hilbert correspondence in the setting of the extended moduli space (see Definition 2.6 in \cite{Boalch2}) of meromorphic connections with one simple pole and one order two pole. Actually, following the discussion above, the corresponding irregular Riemann-Hilbert map is
\begin{eqnarray*}
\nu:(\widetilde{O}_1\times \widetilde{O}_2)/\!\!/ G \hookrightarrow (\widetilde{\mathcal{C}}_1\times\widetilde{\mathcal{C}}_2)/\!\!/ G.
\end{eqnarray*}
On the other hand, the Hamiltonian and quasi-Hamiltonian quotient $(\widetilde{O}_1\times \widetilde{O}_2)/\!\!/ G$ and $(\widetilde{\mathcal{C}}_1\times\widetilde{\mathcal{C}}_2)/\!\!/ G$ are isomorphic to $\Sigma$ and $\Sigma'$ respectively. We thus obtain a symplectic map $\nu:\Sigma\rightarrow \Sigma'$. Next, following the construction of the irregular Riemann-Hilbert map, we prove that $\nu$ can be chosen in such a way that for any $(h,\lambda)\in \Sigma\subset T^*G\cong G\times \frak g^*$ (via left multiplication)
\begin{eqnarray*}
\nu(h, \lambda)=F_C(h,\lambda),
\end{eqnarray*}
where $C(x)$ is the connection matrix of $\nabla$ in \eqref{nabla}. Therefore, combining with Theorem \ref{equivalent}, we prove that the connection matrix $C\in {\rm Map}(\frak g^*,G)$ satisfies the gauge transformation equation \eqref{introequation}. This clarifies the relation between the gauge transformation of dynamical $r$-matrices and certain irregular Riemann-Hilbert problem. As a byproduct, we give a new description of Lu-Weinstein symplectic groupoid via Alekseev-Meinrenken $r$-matrix. We also clarify the meaning of the gauge transformation equation in the framework of generalized classical dynamical $r$-matrix.
\\
\\
The organisation of this paper is as follows. The next section gives the background material and a geometric description of the equation \eqref{introequation}. Section 3 defines the connection matrix $C(x)$ of a meromorphic connection $\nabla$ and states that $C:\frak g^*\rightarrow G$ gives rise to a solution of \eqref{introequation}, i.e., a gauge transformation from $r_0$ to $r_{\rm AM}$. Section 4 discusses the quantum version, i.e., the vertex-IRF transformation equation and formulates a surprising relation between connection matrices and Drinfeld twists. Section 5 gives the background material on the moduli space of meromorphic connections over surfaces and irregular Riemann-Hilbert correspondence in this setting. At the second part of Section 5, we study in details one special case of this correspondence and show that how it gives rise to the equivariant geometric description of the equation \eqref{introequation}. Section 6 describes Lu-Weinstein symplectic groupoid via Alekseev-Meinrenken $r$-matrix. The appendix studies the Poisson structure on the submanifold $\Sigma'$ of Lu-Weinstein symplectic double and gives a proof of the main theorem in Section \eqref{geoconstruction}.
\subsection*{Acknowledgements}
\noindent
I give my warmest thanks to Anton Alekseev for his encouragements as well as inspiring discussions and insightful suggestions. The proof of the main Theorem is based on Philip Boalch's work, to whom I am very grateful for sharing valuable ideas. My sincere thanks is also extended to Benjamin Enriquez, Pavel Etingof, Giovanni Felder, Marco Gualtieri, Anna Lachowska, David Li-Bland, Ian Marshall, Nicolai Reshetikhin, Pavol $\rm{\check{S}}$evera, Valerio Toledano Laredo, Alan Weinstein, Ping Xu and Chenchang Zhu for their useful discussions, suggestions and interests in this paper. This work is partially supported by the Swiss National Science Foundation (SNSF) grants P2GEP2-165118, NCCR SwissMAP and the European Research Council (ERC) project MODFLAT.
\section{Symplectic geometry and gauge transformations of $r$-matrices}\label{Geometryconstruction}
Throughout this paper, let $\frak g$ be a complex reductive Lie algebra and $t\in S^2(\mathfrak g)^{\mathfrak g}$ the element corresponding to an invariant inner product on $\mathfrak g$.
First recall that an element $r\in\frak g\otimes \frak g$ is called a classical $r$-matrix if $r+r^{2,1}\in S^2(\frak g)^{\frak g}$ and $r$ satisfies the classical Yang-Baxter equation:
\begin{eqnarray*}
[r^{1,2},r^{1,3}]+[r^{1,2},r^{2,3}]+[r^{1,3},r^{2,3}]=0.
\end{eqnarray*}
Throughout this paper, we will denote by $r_0:=\frac{1}{2}(r-r^{2,1})$ the skew-symmetric part of a classical $r$-matrix $r$.
A dynamical analog of a classical $r$-matrix is as follows. Let $\eta\subset \frak g$ be a Lie subalgebra. Then a classical dynamical $r$-matrix is an $\eta$-equivariant map $r:\eta^*\rightarrow \frak g\otimes\frak g$ such that $r+r^{2,1}\in S^2(\frak g)^{\frak g}$ and $r$ satisfies the dynamical Yang-Baxter equation (CDYBE):
\begin{eqnarray}\label{CDYBE}
{\rm Alt}(dr)+[r^{1,2},r^{1,3}]+[r^{1,2},r^{2,3}]+[r^{1,3},r^{2,3}]=0,
\end{eqnarray}
where ${\rm Alt}(dr(x))\in \wedge^3\frak g$ is the skew-symmetrization of $dr(x)\in \eta\otimes\frak g\otimes\frak g\subset \frak g\otimes\frak g\otimes\frak g$ for all $x\in\eta^*$.
In the distinguished special case $\eta=\frak g$, the Alekseev-Meinrenken dynamical $r$-matrix $r_{\rm AM}: \mathfrak g^*\rightarrow \mathfrak g\otimes\mathfrak g$ is defined by
\begin{eqnarray*}
r_{\rm{AM}}(x)=({\rm id}\otimes \phi({\rm ad}_{x^{\vee}}))(t), \ \forall x\in\frak g^*,
\end{eqnarray*}
where $x^{\vee}=(x\otimes {\rm id})(t)$ and $\phi(z):=-\frac{1}{z}+\frac{1}{2}{\rm cotanh}\frac{z}{2},$ $z\in \mathbb{C}\setminus 2\pi i\mathbb{Z}^*$. Taking the Taylor expansion of $\phi$ at $0$, we see that $\phi(z)=\frac{z}{12}+\circ(z^2)$, thus $\phi(\rm{ad}_x)$ is well-defined (The maximal domain of definition of $\phi({\rm ad}_x)$ contains all $x\in \frak g^*$ for which the eigenvalues of $\rm{ad}_x$ lie in $\mathbb{C}\setminus 2\pi i \mathbb{Z}^*$).
One can check that $r_{\rm AM}+\frac{t}{2}$ is a classical dynamical $r$-matrix.
Denote by $G$ the formal group with Lie algebra $\mathfrak g$ and by ${\rm Map}_0(\mathfrak g^*,G)$ the space of formal maps $g:\mathfrak g^*\rightarrow G$ such that $g(0)=1$, i.e., the space of maps of the form $e^{u}$, where $u\in\mathfrak g\otimes \hat{S}(\mathfrak g)_{\ge 0}$ ($\hat{S}(\mathfrak g)$ is the degree completion of the symmetric algebra $S(\mathfrak g)$).
The following theorem states the existence of formal solutions of equation \eqref{introequation}.
\begin{thm}{\rm (\cite{EEM})}\label{EEM}
Let $r$ be a classical $r$-matrix with $r+r^{2,1}=t$ and $r_0:=\frac{1}{2}(r-r^{2,1})$. Then there exists a formal map $g\in {\rm Map}_0(\frak g^*,G)$, such that
\begin{eqnarray}\label{gaugeequation}
g_1^{-1}d_2(g_1)-g_2^{-1}d_1(g_2)+(\otimes^2{\rm Ad}_g)^{-1}r_0+\langle {\rm id}\otimes {\rm id}\otimes x,[g_1^{-1}d_3(g_1),g_2^{-1}d_3(g_2)]\rangle=r_{\rm{AM}},
\end{eqnarray}
Here $g_1^{-1}d_2(g)(x):=\sum_i g^{-1}\frac{\partial g}{\partial{\xi^{i}}}(x)\otimes e_i$ is viewed as a formal
function $\frak g^*\rightarrow \frak g^{\otimes 2}$, $\{e_i\}$ is a basis of $\frak g$, $\{\xi^i\}$ the corresponding coordinates on $\frak g^*$ and $g_i^{-1}d_j(g_i) = (g_1^{-1}d_2(g_1))^{i,j}$.
\end{thm}
We will call equation \eqref{gaugeequation} as the gauge transformation equation, and denote its left hand side by $r_0^g\in \rm{Map}(\frak g^*,\frak g\wedge\frak g)$.
In \cite{EEM}, this equation is proven to be the classical limit of vertex-IRF transformation between certain dynamical twists(see section \ref{twist}) and the authors give two constructions of the formal solutions of equation \eqref{gaugeequation} based on formal calculation and quantization of Lie bialgebras respectively.
In the following two sections, we will give a geometric interpretation and construct explicit solutions of equation \eqref{gaugeequation}, where instead of the formal setting, we will work on a local theory.
\subsection{Geometric construction}\label{geoconstruction}
{\bf The symplectic manifold $(\Sigma,\omega)$.} Following the convention from last section, and let $\frak t\subset \frak g$ be a maximal abelian subalgebra and $\frak t'$ the complement of the affine root hyperplanes: $\frak t':= \{\Lambda\in \frak t~|~ \alpha(\Lambda)\notin \mathbb{Z}\}$. In the following, $\frak t'$ is regarded as a subspace of $\mathfrak g^*$ via the isomorphism $\mathfrak g\cong\mathfrak g^*$ induced by inner product. Let $\Sigma$ be a cross-section of $T^*G\cong G\times \mathfrak g^*$ (identification via left multiplication), defined by
\begin{eqnarray*}
\Sigma:=\{(h,\lambda)\in G\times \frak g^*~|~\lambda \in \frak t'\}.
\end{eqnarray*}
Then one can check that $\Sigma$ is a symplectic submanifold of $T^*G$ with the canonical symplectic structure (see \cite{GS} Theorem 26.7). The induced symplectic structure $\omega$ on $\Sigma$ is given for any tangents $v_1=(X_1,R_1),v_2=(X_2,R_2)\in \frak g\times \frak g^*$, where $R_1,R_2\in \frak t^*$, at $(h,\lambda)\in\Sigma$ by
\begin{eqnarray}\label{omegasigma}
\omega(v_1,v_2)=\langle R_1,X_2\rangle-\langle R_2,X_1\rangle+\langle \lambda,[X_1,X_2]\rangle.
\end{eqnarray}
\\
{\bf The Poisson-Lie analogue of $(\Sigma,\omega)$.}
Let $r\in\mathfrak g\otimes\mathfrak g$ be a classical $r$-matrix with $r+r^{2,1}=t$. Let $G^*$ be the simply connected dual Poisson Lie group associated to the quasitriangular Lie biaglebra $(\frak g, r)$ and $D$ the double Lie group with Lie algebra $\frak d=\frak g\Join\frak g^*$ which is locally diffeomorphic to $G\times G^*$ (see e.g \cite{Lu1}). A natural symplectic structure on $D$ is given by the following bivector,
\begin{eqnarray*}
\pi_D=\frac{1}{2}(r_d\pi_0+l_d\pi_0),
\end{eqnarray*}
where $\pi_0\in \frak d\wedge \frak d$ such that $\pi_0(\xi_1+X_1,\xi_2+X_2)=\langle X_1,\xi_2\rangle-\langle X_2,\xi_1\rangle$ for $\xi_1+X_1,\xi_2+X_2\in \frak d^*\cong\frak g^*\oplus \frak g$.
Following \cite{Lu}, the Lu-Weinstein double symplectic groupoid, associated to the Lie bialgebra $(\frak g,r)$, is the set
\begin{eqnarray*}
\Gamma:=\{(h,h^*,u,u^*)~|~h,u\in G,h^*,u^*\in G^*, hh^*=u^*u\in D\}
\end{eqnarray*}
with a unique Poisson structure $\pi_{\Gamma}$ such that the local diffeomorphism $(\Gamma,\pi_{\Gamma})\rightarrow (D,\pi_D)$: $(h,h^*,u,u^*)\mapsto hh^*$ is Poisson. We define a submanifold $\Sigma'$ of $\Gamma$, as a Poisson Lie analogue of $\Sigma$, by
\begin{eqnarray*}
\Sigma':=\{(h,h^*,u,u^*)\in \Gamma~|~h^*\in e^{\frak t'}\subset G^*\}
\end{eqnarray*}
($e$ denotes the exponential map with respect to the Lie algebra $g^*$).
In the appendix, we will prove that $\Sigma'$ is a symplectic submanifold of $(\Gamma,\pi_{\Gamma})$. Now let us take this fact and denote the induced symplectic structure on $\Sigma'$ by $\omega'$.
On the other hand, the map
\begin{eqnarray*}
\Sigma'\rightarrow G\times e^{\frak t'}; \ (h,e^{\lambda},u,u^*)\mapsto (h,e^{\lambda})
\end{eqnarray*}
expresses $\Sigma'$ as a cover of a dense subset of $G\times e^{\frak t'}\subset G\times G^*$. Thus $\Sigma$ and $\Sigma'$ are locally diffeomorphic to each other.
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\\
{\bf Symplectic maps between $(\Sigma,\omega)$ and $(\Sigma',\omega')$.} Associated to any $g\in {\rm Map}_0(\frak g^*,G)$, we define a map $F_g:\Sigma\rightarrow \Sigma'$ by
\begin{eqnarray}\label{DiffFg}
F_g(h,\lambda):=(g({\rm Ad}_h\lambda)h,e^{\lambda},u,u^*), \ \forall (h,\lambda)\in\Sigma,
\end{eqnarray}
where $u\in G, u^*\in G^*$ are determined by the identity $g({\rm Ad}_h\lambda)he^{\lambda}=u^*u$ (understood to hold in the double Lie group $D$). Note that $F_g$ is well-defined for the elements $(h,\lambda)\in \Sigma$ sufficiently near $(e,0)\in G\times\frak t^*$. This is because for these $(h,\lambda)$, $g({\rm Ad}_h\lambda)he^\lambda$ in the double Lie group $D$ is sufficiently near the unit, thus $g({\rm Ad}_h\lambda)he^{\lambda}=u^*u$ uniquely determines $u$ and $u'$. So we can think of $F_g$ defined on a local chart and this is enough for our purpose.
\begin{thm}\label{equivalent}
$F_g$ is a local symplectic isomorphism from $(\Sigma,\omega)$ to $(\Sigma',\omega')$ if and only if $g\in {\rm Map}(\frak g^*,G)$ satisfies the gauge transformation equation \eqref{gaugeequation}, $r_0^{g}=r_{\rm AM}$.
\end{thm}
\noindent{\bf Proof.}\ See the appendix.
\\
\\
{\bf The case when $r$ is a standard $r$-matrix.} Let $T\subset G$ be a maximal torus with Lie algebra $\frak t\subset \frak g$. Let $B_{\pm}$ denote a pair of opposite Borel subgroups with $B_+\cap B_-=T$. For the choice of positive roots $\sfPhi_+$ corresponding to Borel subgroup $B_+$, we take the standard $r$-matrix given by
\begin{eqnarray}\label{rmatrix}
r:=\frac{1}{2}t+\frac{1}{2}\sum_{\alpha\in \sfPhi_+}E_{\alpha}\wedge E_{-\alpha},
\end{eqnarray}
where $t\in S^2(\frak g)^{\frak g}$ is the Casimir element.
In this case, the simply connected dual Poisson Lie group associated to $(\frak g,r)$ is
\begin{eqnarray*}
G^*=\{(b_-,b_+,\Lambda)\in B_-\times B_+\times \frak t~|~\delta(b_-)\delta(b_+)=1,\delta(b_+)=\rm{exp}(\pi i\Lambda)\},
\end{eqnarray*}
where $\delta:\frak g\rightarrow \frak t$ is the projection corresponding to
the root space decomposition.
Thus $\Sigma'$ is a submanifold of the double
\begin{eqnarray*}
\Gamma:\{h,(b_-,b_+,\Lambda_x), u,(c_-,c_+,\Lambda_c)~|~hb_{\pm}=c_{\pm}u\}\subset (G\times G^*)^2,
\end{eqnarray*}
defined by
\begin{eqnarray*}
\Sigma':=\{(h,(e^{-\pi i \Lambda},e^{\pi i \Lambda},\Lambda),u,(c_+,c_-,\Lambda_c))\in\Gamma~|~he^{\pm \pi i \Lambda}=c_{\pm}u, \Lambda\in \frak t'\},
\end{eqnarray*}
where $\frak t'\subset \frak t$ is the complement of the affine root hyperplanes.
To simplify notation, we will write $e^{2\pi i \lambda}\in G^*$ instead of $(e^{\pi i \lambda^{\vee}},e^{-\pi i \lambda^{\vee}},\lambda^{\vee})\in G^*$, where $\lambda\in\mathfrak g^*$, $\lambda^\vee=(\lambda\otimes {\rm id})(t)\in\mathfrak g$ and $e^{2\pi i \lambda}$ (resp. $e^{\pi i \lambda^{\vee}}$) takes the exponential map of the Lie algebra $\frak g^*$ (resp. $\mathfrak g$).
Now given any $g\in {\rm Map}(\frak g^*, G)$, let us consider a local diffeomorphism $F'_g:\Sigma\rightarrow \Sigma'$ (a rescale of $F_g$) which will be more directly involved in the following discussion,
\begin{eqnarray*}
F'_g(h,\lambda):=(g(2\pi i {\rm Ad}_h\lambda)h,e^{2\pi i \lambda},u,u^*), \ \forall (h,\lambda)\in \Sigma
\end{eqnarray*}
where $u\in G, u^*\in G^*$ are uniquely determined by the identity $he^{\lambda}=u^*u$.
It is obvious that the map $(\Sigma,\omega)\rightarrow (\Sigma,\frac{1}{2\pi i}\omega)$, $(h,\lambda)\mapsto (h,2\pi i \lambda)$ is symplectic. Therefore, as a corollary of Theorem \ref{equivalent}, we have
\begin{cor}\label{rescale}
The map $F_{g}'$ is a local symplectic isomorphism from $(\Sigma,\omega)$ to $(\Sigma',\omega')$ (provided the symplectic structure on the right-hand side is divided by $2\pi i$) if and only if $g$ satisfies the gauge transformation equation \eqref{gaugeequation}, $r_0^g=r_{\rm AM}$.
\end{cor}
\section{Gauge transformations via Stokes phenomenon}\label{defineC}
Let $G$ be a complex reductive Lie group, $T\subset G$ a maximal
torus, and $\frak t\subset\mathfrak g$ the Lie algebras of $T$ and $G$ respectively.
Let $\sfPhi\subset\frak t^*$ be the corresponding root system of $\mathfrak g$,
and $\frak t_{\rm reg}$ the set
of regular elements in $\frak t$.
Let $P$ be the holomorphically trivial principal $G$-bundle on $\mathbb{P}^1$. We consider the following meromorphic connection on $P$ of the form
\begin{eqnarray*}
\nabla:=d-(\frac{A_0}{z^2}+\frac{x}{z})dz,
\end{eqnarray*}
where $A_0,x\in\mathfrak g$. We assume henceforth that $A_0\in\frak t_{\rm reg}$ and once fixed, the only variable is $x\in \frak g\cong \frak g^*$ (via the inner product on $\mathfrak g$). Note that the connection $\nabla$ has an order 2 pole at origin and (if $x\neq 0$) a first order pole at $\infty$.
\begin{defi}
The {\it Stokes rays} of the connection $\nabla$ are the rays $\IR_{>0}
\cdot\alpha(A_0)\subset\mathbb{C}^*$, $\alpha\in\sfPhi$. The {\it Stokes sectors}
are the open regions of $\mathbb{C}^*$ bounded by them.
\end{defi}
Let us choose an initial sector ${\rm Sect}_0$ at $0$ bounded by two adjacent Stokes rays and a branch of ${\rm log}(z)$ on ${\rm Sect}_0$.
Then we label the Stokes rays $d_1, d_2,...,d_{2l}$ going in a positive sense and starting on the positive edge of ${\rm Sect}_0$. Set ${\rm Sect}_i={\rm Sect}(d_i,d_{i+1})$ for the open sector bounded by the rays $d_i$ to $d_{i+1}$. (Indices are taken modulo $2l$, so ${\rm Sect}_0={\rm Sect}(d_{2l}, d_1)$).
To each sector ${\rm Sect}_i$, there is a canonical fundamental
solution $F_i$ of $\nabla$ with prescribed asymptotics
in the $i$-th supersector $\widehat{{\rm Sect}_i}:={\rm Sect}(d_i-\frac{\pi}{2},d_i+\frac{\pi}{2})$.
In particular, the following result is proved in \cite{BJL} for $G=GL
_n(\mathbb{C})$, in \cite{BoalchG} for $G$ reductive, and in \cite{BTL1} for an
arbitrary affine algebraic group.
Denote by $\delta(x)$ the projection of $x$ onto $\frak t$ corresponding to
the root space decomposition $\mathfrak g=\frak t\bigoplus_{\alpha\in\sfPhi}\mathfrak g_
\alpha$.
\begin{thm}\label{jurk}
On each sector ${\rm Sect}_i$, there
is a unique holomorphic function $H_i:{\rm Sect}_i\to G$ such that the function
\[F_i=H_i\cdot e^{-\frac{A_0}{z}}\cdot z^{\delta(x)}\]
satisfies $\nabla F_i=0$, and $H_i$ can be analytically continued to $\widehat{{\rm Sect}_i}$ and then $H_i$ is asymptotic to $1$ within $\widehat{{\rm Sect}_i}$.
\end{thm}
\subsection{Connection matrix and dynamical r-matrices}\label{connectiondata}
The meromorphic connetion $\nabla=d-(\frac{A_0}{z^2}+\frac{x}{z})dz$ is said to be {\it non--resonant}
at $z=\infty$ if the eigenvalues of ${\rm ad}(x)$ are not positive integers. The following fact is well-known (see e.g \cite{Wasow} for $G={\rm GL}_n(\mathbb{C})$).
\begin{lem}\label{le:nr dkz}
If $\nabla$ is non--resonant, there is a unique holomorphic function
$H_\infty:\mathbb{P}^1\setminus\{0\}\to G$ such that $H_\infty(\infty)=1$, and the function $F_\infty=H_\infty
\cdot z^{B}$ is a solution of $\nabla F=0$.
\end{lem}
Now let us consider the solutions of $\nabla F=0$:
\begin{eqnarray*}
&&F_0 \ on \ \rm{
Sect}_0,\\
&&F_\infty =H_\infty\cdot z^x \ on \ a \ neighbourhood \ of \ \infty \ slit \ along \ d_1,
\end{eqnarray*}
We define the {\bf connection
matrix} $C(x)\in G$ (with respect to the chosen ${\rm Sect}_0$) by
\[F_\infty=F_0\cdot C(x).\]
Here $F_\infty$ is extended along a path in ${\rm Sect}_0$ then the identity is understood to hold in the domain of definition of $F_0$.
Thus we obtain a map $C_{2\pi i}:\mathfrak g^*_{\rm nr}\to G$ (depends on the choice of $A_0$) which maps any $x\in\frak g^*_{\rm nr}$ to the connection matrix $C(\frac{1}{2\pi i}x)\in G$ of $\nabla=d-(\frac{A_0}{z^2}+\frac{1}{2\pi i}\frac{x}{z})dz$.
Here $\mathfrak g^*_{\rm nr}\subset\mathfrak g^*$ is the dense open set corresponding to the set of elements $x$
such that the eigenvalues of $\frac{1}{2\pi i}\mathrm{ad}(x)$ do not contain positive integers (provided we identify $\mathfrak g\cong\mathfrak g^*$).
Note that the Stokes sector ${\rm Sect}_0$ determines a partition of the root system
$\sfPhi$ of $\mathfrak g$ as follows. Let $\Pi_+$ and $\Pi_-$ be the sets of Stokes
rays which one crosses when going from ${\rm Sect}_0$ to the opposite sector ${\rm Sect}_l$ in the
counterclockwise and clockwise directions respectively. Then $\sfPhi=\sfPhi_+
\sqcup\sfPhi_-$, where
\[\sfPhi_\pm=\{\alpha\in\sfPhi|\,\alpha(A_0)\in\ell, \ell\in\Pi_\pm\}=-\sfPhi_\mp.\]
Now let us consider the equation \eqref{gaugeequation}, in which $r_0$ is the skew-symmetric part of the standard $r$-matrix associated to the positive root system $\sfPhi_+$. Our main theorem states that
\begin{thm}\label{Cequation}
The map $C_{2\pi i}\in {\rm Map}(\frak g^*_{\rm nr},G)$ is a solution of the gauge transformation equation \eqref{gaugeequation}.
\end{thm}
A proof will be given in Section \ref{irregularmap}. The idea is as follows. Following Theorem \ref{equivalent}, to prove $r_0^{C}=r_{\rm AM}$, we only need to verify its symplectic geometric counterpart, i.e., $F_{C}':(\Sigma,\omega)\rightarrow (\Sigma',\omega')$ is a symplectic map. The map $F_C'$ will be realized as certain irregular Riemann-Hilbert map in section \ref{irregularmap}, and then $F_C'$ is symplectic following from the symplectic nature of the irregular Riemann-Hilbert correspondence.
\subsection{Stokes matrices and linearization of $G^*$}\label{ss:Stokes PL}
Given the initial Stokes sector ${\rm Sect}_0$ and the determination of $\log(z)$
with a cut along the Stokes ray $d_1$, the Stokes matrices are essentially the transition matrices between the canonical solutions $F_0$ on ${\rm Sect}_0$ and $F_l$ on the opposite sector ${\rm Sect}_l$, when they are continued along the two possible paths in the punctured disk joining these sectors.
Thus the {\it Stokes matrices} of $\nabla=d-(\frac{A_0}{z^2}+\frac{x}{z})dz$ with respect to
to ${\rm Sect}_0$ are the elements $S_\pm(x)$ of $G$ determined by
\[F_0=F_{l}\cdot S_+(x) e^{2\pi i\delta(x)}, \ \ \ \ \
F_{l}=F_0\cdot S_-(x)
\]
where $\delta(x)$ takes the projection of $\mathfrak g$ onto $\frak t$, and the first (resp. second) identity is understood to hold in ${\rm Sect}_l$
(resp. ${\rm Sect}_0$) after $ F_0$ (resp. $ F_{l}$)
has been analytically continued counterclockwise.
The connection matrix $C(x)$ is related to the Stokes matrices $S_\pm(x)$
by the following monodromy relation (from the fact that a simple positive loop around $0$ is a simple negative loop around $\infty$).
\begin{lem}\label{le:monodromy reln}
The following holds
\[ C(x)^{-1} e^{2\pi i\delta(x)} C(x)=S_-(x) S_+(x) e^{2\pi i\delta(x)}\]
\end{lem}
Recall that the Stokes sector ${\rm Sect}_0$ determines a partition of the root system
$\sfPhi=\sfPhi_+
\sqcup\sfPhi_-$. Let $U_\pm\subset G$
be the unipotent subgroups with Lie algebra $\frak u_\pm=\bigoplus_
{\alpha\in\sfPhi_\pm}\mathfrak g_\alpha$, and $B_\pm$ the corresponding opposite Borel subgroups.
It follows from \cite{BoalchG} that the Stokes matrices
$S_+(x),S_-(x)$ lie in $U_+,U_-$ respectively. Varying $x\in\mathfrak g^*$, we therefore obtain the Stokes map
\[\mu:\mathfrak g^*\to G^*; \ x\mapsto (e^{-\pi i\delta (x)} S_-(x)^{-1},
e^{-\pi i\delta(x)} S_+(x)e^{2\pi i \delta(x)},e^{\pi i \delta(x)}).\]
Here $G^*$ is the dual Poisson Lie group defined in Section \ref{Geometryconstruction}, \begin{eqnarray*}
G^*=\{(b_-,b_+,\Lambda)\in B_-\times B_+\times \frak t~|~\delta(b_-)\delta(b_+)=1,\delta(b_+)=\rm{exp}(\pi i\Lambda)\}.
\end{eqnarray*}
The relation between the Stokes map $\mu$ and the theory of Poisson Lie groups can be shown as follows.
It follows from Theorem \ref{Cequation} and Corrollary \ref{rescale} that the map
\[F'_{C}:\Sigma\rightarrow\Sigma'; (h,\lambda)\mapsto (C({\rm Ad}_h\lambda)h,e^{2\pi i \lambda},u,u^*)\] is a local symplectic isomorphism. This map is equivariant with respect to the symplectic $T$-actions on $\Sigma$ and $\Sigma'$, which are respectively given by
\begin{eqnarray*}
a\cdot (h,\lambda)=(ha,\lambda), \ \ \ \ \ \ \ a\cdot (h,e^{\lambda},u,u^*)=(ha, e^{\lambda},u,u^*), \ \forall a\in T.
\end{eqnarray*}
Define two maps $P:\Sigma\rightarrow \frak g^*$, $P':\Sigma'\rightarrow G^*$ whose fibres are the $T$ orbits
\begin{eqnarray*}
P(h,\lambda)={\rm Ad}^*_h\lambda, \ \forall (h,\lambda)\in \Sigma, \ \ \ \ \ P'(h,e^{\lambda},u,u^*)=d_he^{\lambda}, \ \forall (h,e^{\lambda},u,u^*)\in \Sigma'.
\end{eqnarray*}
Here $d$ denotes the left dressing transformation of $G$ on $G^*$.
By using the monodromy relation \eqref{le:monodromy reln}, we check that the Stokes map $\mu:\frak g^*\rightarrow G^*$ is the unique map such that the following diagram commutes:
$$
\begin{CD}
\Sigma @> F'_{C}>> \Sigma' \\
@V P_1 VV @V P_2 VV \\
\frak g^* @> S_C >> G^*
\end{CD}
$$
One checks that $P$ and $P'$ are Poisson maps (see e.g the appendix), where $\mathfrak g^*$ is equipped with its standard Kirillov--Kostant--Souriau
Poisson structure and $G^*$ the dual Poisson Lie group structure. Therefore, the $T$-reduction of the symplectic isomorphism $F'_{C}$ gives rise to the following remarkable result due to Boalch.
\begin{thm}{\rm\cite{Boalch1}}
The Stokes map $\mu:\frak g^*\rightarrow G^*$ is a local anaytic Poisson isomorphism.
\end{thm}
\section{Vertex-IRF transformations, Drinfeld twists and Stokes phenomenon}\label{twist}
Following \cite{EEM}, the gauge equation \eqref{gaugeequation} is the classical limit of a vertex-IRF transformation equation \cite{EN}.
In particular, a Drinfeld twist killing an admissible associator gives rise to such a vertex-IRF transformation. Therefore the classical limit of such a twist is a solution of equation \eqref{gaugeequation}. Thus we have two sources of solutions (as is shown in last section, another one is from Stokes phenomenon). In this section, we will prove that there exists a Drinfeld twist whose classical limit is the connection matrix $C$ introduced in last section.
Let $(U(\mathfrak g),m,\Delta,\varepsilon)$ denote the universal enveloping algebra of $\mathfrak g$ with the product $m$, the coproduct $\Delta$ and the counit $\varepsilon$. Let $U(\mathfrak g)\llbracket\hbar\rrbracket$ be the corresponding topologically free $\mathbb{C}\llbracket\hbar\rrbracket$-algebra.
\begin{defi}\label{dynamicaltwist}
Let $\Phi=1+\frac{[t^{12},t^{23}]}{24}\hbar^2+O(\hbar^3)\in U(\mathfrak g)^{\hat{\otimes} 3}\llbracket\hbar\rrbracket$ be such that $\Phi$ is $\mathfrak g$-invariant and satisfies the pentagon equation and the counit axiom.
Then a function $J_d: \mathfrak g^*\rightarrow U(\mathfrak g)^{\hat{\otimes} 2}\llbracket\hbar\rrbracket$ is called a dynamical twist associated to $\Phi$ if
$J_d(x)=1+O(\hbar)$ is $\mathfrak{g}$-invariant and
\begin{eqnarray}\label{twist equation}
J_d^{12,3}J_d^{1,2}(x+\hbar h^{(3)})=\Phi^{-1}J_d^{1,23}(x)J_d^{2,3}(x),
\end{eqnarray}
where for $J_d^{1,2}(x+\hbar h^3)$ we use the dynamical convention, i.e.,
\begin{eqnarray*}
J_d^{1,2}(x+{\hbar}^3)=\sum_{N\ge 0}\frac{{\hbar}^N}{N!}\sum_{i_1,...,i_N}^n(\partial_{{\xi}^{i_1}}\cdot\cdot\cdot
\partial_{\xi^{i_N}}{J_d})(x)\otimes(e_{i_1}\cdot\cdot\cdot e_{i_N})
\end{eqnarray*}
where $n={\rm dim}(\mathfrak g)$, and $\{e_i\}_{i=1,...,n}$, $\{\xi^i\}_{i=1,...,n}$ are dual bases of $\mathfrak g$ and $\mathfrak g^*$.
\end{defi}
Assume $(\Phi,J_d(x))$ satisfies the conditions in Definition \ref{dynamicaltwist}. Let $j(x):=(\frac{J_d(x)-1}{\hbar})$ mod $\hbar$, and $r(x):=j(x)-j(x)^{2,1}$. Then following \cite{EE}, $r(x)+\frac{t}{2}$ is a solution of the CDYBE \eqref{CDYBE}, i.e., a classical dynamical $r$-matrix. In this case, ${J_d}(x)$ is called a dynamical twist quantization of $r(x)$.
In particular, a constant twist $J_c\in U(\mathfrak g)^{\hat{\otimes} 2}\llbracket\hbar\rrbracket$ is such that
\begin{eqnarray}\label{constanttwist}
J_c^{12,3}J_c^{1,2}=\Phi^{-1}J_c^{1,23}J_c^{2,3}.
\end{eqnarray}
Similarly, we say $J_c=1+\hbar{\frac{r}{2}}+\circ (\hbar^2)$ is a twist quantization of $r_0:=\frac{1}{2}(r-r^{2,1})$.
Set $U':=U(\hbar \mathfrak g)\llbracket\hbar\rrbracket$, the subalgebra generated by $\hbar x$, $\forall x\in \mathfrak g$. Note that $U'/\hbar U'=\hat{S}(\mathfrak g)$. An associator $\Phi\in U(\mathfrak g)^{\widehat{\otimes}3}\llbracket\hbar\rrbracket$ is called admissible (see \cite{EH}) if
$$\Phi\in 1+\frac{\hbar^2}{24}[t^{1,2},t^{2,3}]+O(\hbar^3),\ \ \ \ \ \hbar {\rm log}(\Phi)\in (U'(\mathfrak g))^{\widehat{\otimes}3}.$$
Given an admissible associator $\Phi\in U(\mathfrak g)^{\hat{\otimes} 3}\llbracket\hbar\rrbracket$, we identify the third component $U(\mathfrak g)$ of this tensor cube with $\mathbb{C}[\mathfrak g^*]$ via the symmetrization (PBW) isomorphism $S^.(\mathfrak g)\rightarrow U(\mathfrak g)$ and use this identification view $\Phi^{-1}$ as a function from $\mathfrak g^*$ to $U(\mathfrak g)^{\hat{\otimes} 2}\llbracket\hbar\rrbracket$, denoted by $\Phi^{-1}(x)$. Then we have $\Phi^{-1}(\hbar^{-1}x)$ is a well-defined element in $U(\mathfrak g)^{\hat{\otimes} 2}\hat{\otimes}\mathbb{C}[\mathfrak g^*]$. Following \cite{EH}, any universal Lie associator gives rise to an admissible associator.
\begin{thm}\rm{\cite{EE}}\label{EEthm}
Assume that $\Phi$ is the image in $U(\mathfrak g)^{\hat{\otimes} 3}\llbracket\hbar\rrbracket$ of a universal Lie associator. Let $J_d(x):=\Phi^{-1}(\hbar^{-1}x)$, where $\Phi^{-1}$ is regarded as an element of $(U(\mathfrak g)^{\hat{\otimes} 2}\otimes \mathbb{C}[\mathfrak g^*])\llbracket\hbar\rrbracket$. Then
$(1)$ $J_d(x)$ is a formal dynamical twist. More precisely, $J_d(x)=1+\hbar j(x)+O(\hbar^2)\in (U(\mathfrak g)^{\hat{\otimes} 2}\hat{\otimes} \hat{S}(\mathfrak g))\llbracket\hbar\rrbracket$, is a series in nonnegative powers of $\hbar$ and satisfies the dynamical twist equation.
$(2)$ $J_d(x)$ is a twist quantization of the Alekseev-Meinrenken dynamical $r$-matrix, that is $r_{\rm AM}=j(x)-j(x)^{2,1}$.
$(3)$ If $\Phi_{\rm KZ}$ is the Knizhnik-Zamolodchikov associator, then $J_d(x)$ is holomorphic on an open set and extends meromorphically to the whole $\mathfrak g^*$.
\end{thm}
\begin{defi}\label{EN}{\rm \cite{EN}}
Let ${J_d}(x): \mathfrak g^*\longrightarrow U(\mathfrak g)^{\hat{\otimes} 2}\llbracket\hbar\rrbracket$ be a function with invertible values and $\rho: \mathfrak g^*\longrightarrow U(\mathfrak g)\llbracket\hbar\rrbracket$ a function with invertible values such that $\varepsilon(\rho(\lambda))=1$ ($\varepsilon$ is the counit). Set
\begin{eqnarray*}
J_d^{\rho}(x)=\Delta(\rho(x)){J_d}(x)\rho^1(x-\hbar h^{(2)})^{-1} \rho^2(x)^{-1},
\end{eqnarray*}
and call $\rho$ a vertex-IRF transformation from ${J_d}(x)$ to $J_d^\rho(x)$, where for $\rho^{-1}(x-\hbar h^{(2)})$ we use the dynamical convention.
\end{defi}
Now let us take an admissible associator $\Phi$. Let $J_c$ (resp. $J_d(x)$) be a (resp. dynamical) twist quantization of $r_0$ (resp. $r_{\rm AM}$). Let $\rho(x)\in (U(\mathfrak g)\otimes \hat{S}^.\mathfrak g)\llbracket\hbar\rrbracket$ be a formal vertex-IRF transformation which maps the $\mathfrak g$-invariant but dynamical twist ${J_d}(x)$ to the constant but non-invariant twist $J_c$.
This is to say
\begin{eqnarray}\label{IRFtransformation}
J_c=\Delta(\rho(x))J_d(x)\rho^1(x-h^{(2)})^{-1} \rho^2(x)^{-1}.
\end{eqnarray}
Then by comparing the coefficients of equation \eqref{IRFtransformation} up to the first order of $\hbar$, we have
\begin{pro}{\rm \cite{EEM}}\label{reduction}
The reduction modulo $\hbar$ of $\rho(x)$, denoted by $g(x)=\rho(x)|_{\hbar=0}$, belongs to ${\rm exp}(\mathfrak g\otimes \hat{S}(\mathfrak g))_{>0}$ (thus a formal map from $\mathfrak g^*$ to ${\rm exp}(\mathfrak g)$) and satisfies the equation $r_0^{g(x)}=r_{\rm AM}.$
\end{pro}
Let $J_d(x)=\Phi(\hbar^{-1}x)$ be the dynamical twist in Theorem \ref{EEthm}, then IRF-transformations satisfying \eqref{IRFtransformation} are constructed in \cite{EEM} as follows. For the admissible associator $\Phi$, there exists a twist killing $\Phi$ (see \cite{Drinfeld}\cite{EK}), and according to \cite{EH}, this twist can be made admissible by a suitable gauge transformation. The resulting twist $J\in U(\mathfrak g)^{\hat{\otimes} 2}\llbracket\hbar\rrbracket$ satisfies $J=1-\hbar\frac{r}{2}+\circ(\hbar)$, $\hbar {\rm log}(J)\in U'^{\widehat{\otimes}2}, (\varepsilon\otimes {\rm id})(J)=({\rm id} \otimes \varepsilon)(J)=1$, and
\begin{equation}\label{Drinfeldtwist}
\Phi=(J^{2,3} J^{1,23})^{-1} J^{1,2} J^{12,3}.
\end{equation}
Let us now identity the second component $U(\mathfrak g)$ of $J$ with $\mathbb{C}(\mathfrak g^*)$ via PBW isomorphism $S^.(\mathfrak g)
\cong U(\mathfrak g)$, and regard $J$ as a formal function from $\mathfrak g^*$ to $U(\mathfrak g)\llbracket\hbar\rrbracket$, denoted by $J(x)$. Let $\rho(x):=J(\hbar^{-1}x)\in {\rm Map}(\mathfrak g^*,U(\mathfrak g)\llbracket\hbar\rrbracket)$ denote the corresponding renormalization by sending $x\in\mathfrak g^*$ to $\hbar^{-1} x$. Then if we identify the third component $U(\mathfrak g)$ of the tensor cube with $\mathbb{C}(\mathfrak g^*)$ in equation \eqref{IRFtransformation} and renormalize the resulting formal maps from $\mathfrak g^*$ to $U(\mathfrak g)^{\hat{\otimes} 2}$ by sending $x\in \mathfrak g^*$ to $\hbar x$, the equation \eqref{Drinfeldtwist} becomes
\begin{eqnarray*}
J^{-1}=\Delta(\rho(x))J_d(x)\rho^1(x-h^{(2)})^{-1} \rho^2(x)^{-1}
\end{eqnarray*}
(Here $J_d(x):=\Phi^{-1}(\hbar^{-1}x)$ is the dynamical twist as in Theorem \ref{EEthm}). One checks that $J_c:=J^{-1}$ satisfies \eqref{constanttwist} (thus a constant twist). Therefore, the admissible Drinfeld twist $J$ gives rise to a vertex-IRF transformation between the dynamical twist $J_d(x)$ and the constant twist $J_c=J^{-1}$. For an admissible Drinfeld twist $J$, regarded as a formal function $J(x):\mathfrak g^*\rightarrow U(\mathfrak g)\llbracket\hbar\rrbracket$ (via PBW), define the renormalized classical limit of $J$ by $g(x):=J(\hbar^{-1}x)|_{\hbar=0}\in {\rm Map}(\mathfrak g^*,U(\mathfrak g)\llbracket\hbar\rrbracket)$ (by Proposition \ref{reduction} $g(x)$ is actually in ${\rm Map}_0(\mathfrak g^*,G)$).
Following Proposition \ref{reduction} and the above discussion, the renormalized classical limit $g(x)\in {\rm Map}(\mathfrak g^*,G)$ of an admissible Drinfeld twist $J$ satisfies equation \eqref{gaugeequation}. For the case $\mathfrak g$ is semisimple, we will prove that the inverse is also true, i.e.,
given any solution
$g(x)$ of \eqref{gaugeequation}, there exists an admissible Drinfeld twist $J$ whose classical limit is $g(x)$. In order to prove this, we need to consider the gauge action on the Drinfeld twist and, as a classical limit, on the space of solutions of \eqref{gaugeequation}.
\\
\\
{\bf Gauge actions on Drinfeld twists.} Recall that $U'=U(\hbar \mathfrak g)\llbracket\hbar\rrbracket$, the subalgebra generated by $\hbar x$, $\forall x\in \mathfrak g$. Let $U'_0:={\rm Ker}(\varepsilon)\cap U'$. Then
$V:=\{u_\hbar\in\hbar^{-1}U'_0\subset U(\mathfrak g)\llbracket\hbar\rrbracket\}~|~u_\hbar=O(\hbar)\}$ is a Lie subalgebra for
the commutator. One checks that $e^{u_\hbar}\ast J:=(e^{u_\hbar})^1(e^{u_\hbar})^2J(\Delta(e^{u_\hbar}))^{-1}$ is a solution of
\eqref{Drinfeldtwist} if $J$ is. Thus $V$ acts on the set of admissible Drinfeld twists by $\delta_{u_\hbar}(J)={u_\hbar}^1J+{u_\hbar}^2J-J{u_\hbar}^{12}$, ${u_\hbar}\in V$. Note that $V/\hbar V=(\hat{S}(\mathfrak g)_{>1},\{-,-\})$. Let $u\in \hat{S}(\mathfrak g)_{>1}$ be the corresponding classical limit of ${u_\hbar}\in V$, and $g\in {\rm exp}(\mathfrak g)\otimes \hat{S}(\mathfrak g)_{>0}$ the classical limit of $J$.
Then the reduction modulo $\hbar$ of this infinitesimal gauge action is given by
\begin{eqnarray}\label{classicalaction}
\delta_u(g)=\{1\otimes u, g\}-g\cdot du,\end{eqnarray}
where $du:=e_i\otimes \frac{\partial u}{\partial\xi^i}\in(\mathfrak g)\otimes \hat{S}(\mathfrak g)$ for an orthogonal basis $\{e_i\}$ of $\mathfrak g$ and the corresponding coordinates $\{\xi^i\}$ on $\mathfrak g^*$.
This infinitesimal gauge action has a geometric description as follows.
\\
\\
{\bf Gauge actions on the space of solutions of \eqref{gaugeequation}.} Let ${\rm Map}_0(\mathfrak g^*,G)$ be the space of formal maps $g:\mathfrak g^*\rightarrow G$ such that $g(0)=1$. Let us introduce a group structure on ${\rm Map}_0(\mathfrak g^*,G)$, defined by $(g_1\ast g_2)(x):=g_2({\mathrm{Ad}}^*_{g_1(x)}x)g_1(x)$. Then there is a natural group homomorphism \[{\rm Map}_0(\mathfrak g^*,G)\rightarrow {\rm Diff}(\mathfrak g^*),\]
which maps $g\in {\rm Map}_0(\mathfrak g^*,G)$ to the obvious diffeomorphism $g\cdot x={\rm Ad}_{g(x)}x, \ \forall x\in\mathfrak g^*$. Let us take the subgroup ${\rm Map}_0^{ham}(\mathfrak g^*,G)$ whose elements, under the above group homomorphism, correspond to Poisson isomorphisms on $\mathfrak g^*$ (equipped with its canonical linear Poisson structure). Explicitly, the elements $g$ of ${\rm Map}_0^{ham}(\mathfrak g^*,G)$ are such that (use the same convention in \eqref{gaugeequation})
\[g_1^{-1}d_2(g_1)-g_2^{-1}d_1(g_2)+\langle {\rm id}\otimes {\rm id}\otimes x,[g_1^{-1}d_3(g_1),g_2^{-1}d_3(g_2)]\rangle=0.\]
Then it is direct to check that ${\rm Map}^{ham}_0(\mathfrak g^*,G)$ is a prounipotent Lie group with Lie algebra $\{\alpha\in \mathfrak g\otimes \hat{S}(\mathfrak g)_{\ge 1}~|~{\rm Alt}(d\alpha)=0\}$. This Lie algebra is isomorphic to $(\hat{S}(\mathfrak g)_{> 1},\{-,-\})$ under the map $d:u\rightarrow du\in\mathfrak g\otimes \hat{S}(\mathfrak g)_{\ge 1}$, for all $f\in\hat{S}(\mathfrak g)_{> 1}$.
The right action of ${\rm Map}_0(\mathfrak g^*,G)$ on itself restricts to an action of ${\rm Map}_0^{ham}(\mathfrak g^*,G)$ on the space of solutions of equation \eqref{gaugeequation}. It induces a natural gauge action of $\alpha\in {\rm Map}_0^{ham}(\mathfrak g^*,G)$ on the space of solutions $g$, $(\alpha\ast g)(x)=g({\mathrm{Ad}}^*_{\alpha(x)}x)\alpha(x)$.
The infinitesimal of this action is that each $u\in (\hat{S}(\mathfrak g)_{> 1},\{-,-\})$ (Lie algebra of ${\rm Map}_0(\mathfrak g^*,G)$) acts by vector fields on the space of solutions by
\begin{eqnarray}\label{classicalaction1}
g^{-1}\delta_u(g)=\langle {\rm id}\otimes {\rm id}\otimes x, [d_3(u_2),g^{-1}_{12}d_3g_{12}]\rangle -du\in \mathfrak g\otimes \hat{S}(\mathfrak g)_{\ge 0}.
\end{eqnarray}
It coincides with the infinitesimal gauge action \eqref{classicalaction}. Therefore we have a commutative diagram of gauge actions and taking classical limit
$$
\begin{CD}
J @> e^{u_\hbar}>> e^{u_\hbar}\ast J \\
@V c.l VV @V c.l VV \\
g @> e^u >> e^u\ast g
\end{CD}
$$
Here we assume the classical limit (c.l) of $J$ (resp. $u_\hbar\in V$) is $g$ (resp. $u\in \hat{S}(\mathfrak g)_{> 1}$), and $e^u$ is seen as an element in ${\rm Map}_0^{ham}(\mathfrak g^*,G)$ under the Lie algebra isomorphism $\hat{S}(\mathfrak g)_{> 1}\cong {\rm Lie}({\rm Map}_0^{ham}(\mathfrak g^*,G))$. This fact enables us to prove the following Proposition.
\begin{pro}
Given any $g\in {\rm Map}_0(\mathfrak g^*,G)$ satisfying $r_0^g=r_{\rm AM}$, there exists an admissible Drinfeld twist $J$ whose renormalized classical limit is $g(x)$, and the identity $r_0^g=r_{\rm AM}$ is the classical limit of the identity $\Phi=(J^{2,3}J^{1,23})^{-1}J^{1,2}J^{12,3}$.
\end{pro}
\noindent{\bf Proof.}\
Let $J'$ be an admissible Drinfeld twist with $g'(x)$ as its renormalization classical limit (thus a solution of
\eqref{gaugeequation}).
Following \cite{EEM}, ${\rm Map}_0^{ham}(\mathfrak g^*,G)$ acts simply and transitively on the space of solutions of \eqref{gaugeequation}.
Therefore, the two solutions $g$ and $g'$ are related by a map $\alpha\in {\rm Map}^{ham}_0(\mathfrak g^*,G)$, i.e., $g(x)=g'\ast \alpha$. Let us assume $u\in \hat{S}(\mathfrak g)_{> 1}$ (Lie algebra of ${\rm
Map}^{ham}_0(\mathfrak g^*,G)$) is such that $e^u=\alpha$, and then take an element $u_\hbar \in V\subset U(\mathfrak g)\llbracket\hbar\rrbracket$ whose reduction modulo $\hbar$ is $u$. The gauge action of $e^{u_\hbar}$ on $J'$ provides a new admissible twist $J:= e^{u_\hbar}\ast J'$. Furthermore the above commutative diagram verifies $g(x)=J(\hbar^{-1}x)|_{\hbar=0}$ (regard $J$ as a formal function from $\mathfrak g^*$ to $U(\mathfrak g)\llbracket\hbar\rrbracket$), i.e., the renormalized classical limit of $J$ is $g(x)$.
\hfill ~\vrule height6pt width6pt depth0pt
\\
In particular, given any connection matrix $C$, $C_{2\pi i}\in {\rm Map}_0(\mathfrak g^*,G)$ is a solution of \eqref{gaugeequation} (see Section \ref{connectiondata}), therefore can be quantized. From the above discussion, it means that if we regard $C_{2\pi i}$ as an element in $U(\mathfrak g)^{\hat{\otimes} 2}$ by taking the Taylor expansion at $0$ and identifying $\hat{S}(\mathfrak g)$ with $U(\mathfrak g)$, then there exists an admissible Drinfeld twist $J\in U(\mathfrak g)^{\hat{\otimes} 2}\llbracket\hbar\rrbracket$ satisfying \eqref{Drinfeldtwist} whose renormalized classical limit is $C_{2\pi i}$.
\\
\begin{thm} Assume $\Phi$ is the image in $U(\mathfrak g)^{\hat{\otimes}3}$ of a universal Lie associator. Then for any connection matrix $C\in {\rm Map}_0(\mathfrak g^*,G)$, there exists an admissible Drinfeld twist $J$ killing the associator $\Phi$ whose renormalized classical limit is $C_{2\pi i}$.
\end{thm}
In particular, let $\Phi$ be the Knizhnik-Zamolodchikov (KZ) associator $\Phi_{KZ}$, which is constructed as the monodromy from $1$ to $\infty$ of the KZ equation on $\mathbb{P}^1$ with three simple poles at $0$, $1$, $\infty$. Naively the confluence of two simple poles at $0$ and $1$ in the KZ equation turns the monodromy representing KZ associator to the monodromy $C_{\hbar}$ representing the connection matrix of certain differential equation with one degree two pole. Recall that the connection matrix $C$ is the monodromy from $0$ to $\infty$ of the equation $\nabla F=0$.
Then the above theorem indicates that the monodromy $C_{\hbar}$ is related to certain Drinfeld twist killing $\Phi_{KZ}$. In other words, the identity $\Phi_{KZ}=(J^{2,3}J^{1,23})^{-1}J^{1,2}J^{12,3}$ may be related to the confluence of simple poles in the KZ equation. The precise relation between Stokes phenomenon and the theory of quantum groups is given in \cite{TL} and \cite{TLXu}.
\section{Irregular Riemann-Hilbert correspondence}\label{irregularmap}
In this section, we will recall symplectic moduli spaces of meromorphic connections on a trivial holomorphic principal $G$-bundle, the corresponding symplectic spaces of monodromy data and the irregular Riemann-Hilbert correspondence between them. We mainly follow the papers \cite{Boalch2}\cite{BoalchG}\cite{Boalch3} of Boalch, in which these symplectic spaces are found and described both explicitly and from an infinite dimensional viewpoint (generalising the Atiyah-Bott approach). After that, we will consider the case of the meromorphic connections with one simple pole and one order two pole, and prove that the irregular Riemann-Hilbert correspondence in this case gives rise to a gauge transformation between $r_0$ and $r_{\rm AM}$.
\subsection{Moduli spaces of meromorphic connections and the spaces of monodromy data}
Let $D=\sum_{i=1}^m k_i(a_i)>0$ be an effective divisor on $\mathbb{P}^1$ and $P$ a holomorphically trivial principal $G$-bundle. Let us consider the meromorphic connections on $P$ with poles on $D$. They can be described explicitly as follows.
Let $z$ be a local coordinate on $\mathbb{P}^1$ vanishing at $a_i$. Then in terms of a local trivialisation of $P$, any meromorphic connection $\nabla$ on $P$ takes the form of $\nabla=d-A$, where
\begin{eqnarray*}
A=\frac{A_{k_i}}{z^{k_i}}dz+\cdot\cdot\cdot \frac{A_1}{z}dz+A_0dz+\cdot\cdot\cdot,
\end{eqnarray*}
and $A_j\in \mathfrak g$, $j\le k_i$.
\begin{defi}
A compatible framing at $a_i$ of a trivial principal $G$-bundle $P$ with a generic connection $\nabla$ is an isomorphism \ $\mathclap{^i}g_0:P_{a_i}\rightarrow G$ between the fibre $P_{a_i}$ and $G$ such that the leading coefficient of $\nabla$ is inside $\frak t_{\rm reg}$ in any local trivialisation of $P$ extending $g_0$.
\end{defi}
Let us choose, at each point $a_i$, an irregular type \[\mathclap{^i}{A}^0:=\mathclap{^i}{A}_{k_i}\frac{dz}{z^{k_i}}+\cdot\cdot\cdot+\mathclap{^i}{A}_{2}\frac{dz}{z^2},\]
where $\mathclap{^i}A_{k_i}\in \frak t_{\rm reg}$ and $\mathclap{^i}A_j\in\frak t$ for $j<k_i$.
Let $\nabla=d-A$ in some local trivialisation (thus a comatible framing is an element in $G$) and $z_i$ a local coordinate vanishing at $a_i$. Then we say $(\nabla,P)$ with compatible framing \ $\mathclap{^i}g_0$ at $a_i$ has irregular type \ $\mathclap{^i}{A}^0$ if there is some formal bundle automorphism $g\in G\llbracket z_i\rrbracket$ with $g(a_i)=\mathclap{^i}g_0$, such that $$g[A]:=gAg^{-1}+dg\cdot g^{-1}=\mathclap{^i}{A}^0+\frac{\mathclap{^i}\Lambda}{z_i}dz_i$$ for some \ $\mathclap{^i}\Lambda\in\frak t$.
We denote by ${\bf a}$ the choice of the effective divisor $D=\sum_{i=1}^m k_i(a_i)$ and all the irregular types \ $\mathclap{^i}A^0$.
\begin{defi}{\rm (\cite{Boalch2})}
The extended moduli space $\widetilde{\mathcal{M}^*}({\bf a})$ is the set of isomorphism classes of triples $(P,\nabla,{\bf g})$, consisting of a generic connection $\nabla$ with poles on $D$ on a trivial holomorphic principal $G$-bundle $P$ on $\mathbb{P}^1$ with compatible framing ${\bf g}=( \ \mathclap{^1}g_0,..., \ \mathclap{^m}g_0)$, such that $(P,\nabla,{\bf g})$ has irregular type \ $\mathclap{^i}A^0$ at each $a_i$.
\end{defi}
Next let us recall (from \cite{Boalch2} Section 2) the building blocks $\widetilde{O}$ of the moduli space $\widetilde{\mathcal{M}^*}({\bf a})$.
Fix an integer $k\neq 2$. Let $G_k:=G(\mathbb{C}[z]/z^k)$ be the group of $(k-1)$-jets of bundle automorphisms, and let $\frak g_k =\rm{Lie}(G_k)$ be its Lie algebra, which contains elements of the form $X=X_0+X_1z+\cdot\cdot\cdot+X_{k-1}z^{k-1}$ with $X_i\in\frak g$. Let $B_k$ be the subgroup of $G_k$ of elements having constant term 1. The group $G_k$ is the semi-direct product $G\ltimes B_k$ (where $G$ acts on $B_k$ by conjugation). Correspondingly the Lie algebra of $G_k$ decomposes as a vector space direct sum and dualising we have: $\frak g^*_k=\frak b^*_k\oplus \frak g^*$. Elements of $\frak g^*_k$ will be written as
\begin{eqnarray*}
A = A_k\frac{dz}{z^k}+\cdot\cdot\cdot +A_{1}\frac{dz}{z}
\end{eqnarray*}
via the pairing with $\frak g_k$ given by $\langle A,X\rangle:= {\rm Res_0}(A,X)=\sum_{j}(A_j,X_{j-1})$. In this way $\frak b^*_k$ is identified with the set of $A$ having zero residue and $\frak g^*$ with those having only a residue term (zero irregular part). Let $\pi_{{\rm res}}:\frak g^*_k\rightarrow\frak g^*$ and $\pi_{{\rm irr}}:\frak g^*_k\rightarrow \frak b^*_k$ denote the corresponding projections.
Now choose an element $A^0=A^0_k\frac{dz}{z^k}+\cdot\cdot\cdot+A^0_{2}\frac{dz}{z^2}$ of $\frak b^*_k$ with $A^0_i\in \frak t$ and $A_0^0\in\frak t_{\rm{reg}}$. Let $O_{A^0}\subset \frak b^*_k$ denote the $B_k$ coadjoint orbit containing $A^0$.
\begin{defi}\rm{(\cite{Boalch2})}
The extended orbit $\widetilde{O}\subset G\times \frak g^*_k$ associated to $O_{A^0}$ is
\begin{eqnarray*}
\widetilde{O}:=\{(g_0,A)\in G\times\frak g^*_k ~|~\pi_{irr}(g_0Ag_0^{-1})\in O_{A^0}\}
\end{eqnarray*}
where $\pi_{irr}:\frak g^*_k\rightarrow\frak b^*_k$ is the natural projection removing the residue.
\end{defi}
$\widetilde{O}$ is naturally a Hamiltonian $G$-manifold. Any tangents $v_1,v_2$ to $\widetilde{O}\in G\times\frak g^*_k$ at $(g_0,A)$ are of the form
\begin{eqnarray*}
v_i=(X_i(0),[A,X_i]+g_0^{-1}{R_i}g_0)\in \frak g\oplus \frak g^*_k
\end{eqnarray*}
for some $X_1,X_2\in \frak g_k$ and ${R_1},{R_2}\in \frak t^*$ (where $\frak g\cong T_{g_0}G$ via left multiplication), and the symplectic structure on $\widetilde{O}$ is given by
\begin{eqnarray}\label{omegaO}
\omega_{\widetilde{O}}(v_1,v_2)=\langle R_1,{\rm Ad}_{g_0}X_2\rangle-\langle R_2,{\rm Ad}_{g_0}X_1\rangle+\langle A,[X_1,X_2]\rangle.
\end{eqnarray}
\begin{pro}\rm{(\cite{Boalch2})}
The $G$ action $h\cdot (g_0,A):=(g_0h^{-1},hAh^{-1})$ on $(\widetilde{O},\omega_{\widetilde{O}})$ is Hamiltonian with moment map $\mu_G:\widetilde{O}\rightarrow \frak g^*$, $\mu(g_0,A)=\pi_{res}(A)$.
\end{pro}
In the simple pole case $k = 1$ we define
\begin{eqnarray*}
\widetilde{O}:=\{(h,x)\in G\times \frak g^* ~|~{\rm Ad}_hx\in\frak t'\}\subset G\times\frak g^*.
\end{eqnarray*}
One checks that the map $\widetilde{O}\rightarrow \Sigma$, $(h,x)\mapsto (h, {\rm Ad}_hx)$ is an isomorphic of $\widetilde{O}$ to the symplectic slice $\Sigma$ defined in section \ref{defineC}.
The spaces $\widetilde{O}$ enable one to construct global symplectic moduli spaces of meromorphic connections on trivial $G$-bundles over $\mathbb{P}^1$ as symplectic quotients of the form $\widetilde{O}_1\times\cdot\cdot\cdot\times\widetilde{O}_m/\!\!/ G$ (the Hamiltonian reduction of the direct product of $m$ Hamiltonian $G$-spaces).
\begin{pro}{\rm (\cite{Boalch2})}
$\widetilde{\mathcal{M}^*}({\bf a})$ is isomorphic to the symplectic quotient
\begin{eqnarray*}
\widetilde{\mathcal{M}^*}({\bf a})\cong \widetilde{O}_1\times\cdot\cdot\cdot\times\widetilde{O}_m/\!\!/ G
\end{eqnarray*}
where $\widetilde{O}_i\subset G\times \frak g^*_{k_i}$ is the extended coadjoint orbit associated to $O_{ \mathclap{^i}A^{0}}\subset \frak b^*_{k_i}$, the $B_{k_i}$ coadjoint orbit containing the element \ $\mathclap{^i}A^{0}\in\frak b^*_k$ (the chosen irregular type at $a_i$).
\end{pro}
{\bf Quasi-Hamiltonian $G$-spaces and symplectic spaces of monodromy data.} Let us recall the quasi-Hamiltonian description of the symplectic structure on the space of monodromy/Stokes data. Let $\theta,\bar{\theta}$ denote the left and right invariant $\mathfrak g$-valued Cartan one-forms on $G$ respectively. Let $\psi$ denote the canonical three-form of $G$, i.e., $\psi:=\frac{1}{6}\langle\theta,[\theta,\theta]\rangle$.
\begin{defi}\rm{(\cite{AMM})}
A complex manifold $M$ is a complex quasi-Hamiltonian $G$-space if there is an action of $G$ on $M$, a $G$-equivariant map $\mu:M\rightarrow G$ (where $G$ acts on itself by conjugation), and a $G$-invariant holomorphic two-form $\omega\in \Omega^2(M)$ such that
\begin{itemize}
\item[(i)] $d\omega=\mu^*(\psi)$, where $\psi$ is the canonical three-form on $G$;
\item[(ii)] $\omega(v_X,\cdot)=\frac{1}{2}\mu^*(\theta+\bar{\theta},X)\in \Omega^1(M)$, for all $X\in\frak g$, where $v_X$ is the fundamental vector field $(v_X)_m=-\frac{d}{dt}(e^{tX}\cdot m)|_{t=0}$.
\item[(iii)] at each point $m\in M$, the kernel of $\omega$ is
\begin{eqnarray*}
{\rm ker} \omega_m=\{(v_X)_m~|~X\in\frak g \ such \ that \ hXh^{-1}=-X, \ where \ h:=\mu(m)\in G\}.
\end{eqnarray*}
\end{itemize}
\end{defi}
These axioms are motivated in terms of Hamiltonian loop group manifolds. See \cite{AMM} for more details.
\begin{ex}
Let $\mathcal{C}\subset G$ be a conjugacy class with the conjugation action of $G$ and moment map $\mu$ given by the inclusion map. Then $\mathcal{C}$ is a quasi-Hamiltonian $G$-space with two-form $\omega$ defined by
\begin{eqnarray*}
\omega_h(v_X,v_Y)=\frac{1}{2}(\langle X, {\rm Ad}_hY \rangle-\langle Y, {\rm Ad}_hX\rangle ),
\end{eqnarray*}
for any $X,Y\in \frak g$ and $v_X,v_Y$ the fundamental vector field with respect to the conjugation action of $G$.
\end{ex}
Similar to the Hamiltonian reduction, we have the following moment map reduction in the quasi-Hamiltonian setting.
\begin{thm}{\rm (\cite{AMM})}
Let $(M,\omega)$ be a quasi-Hamiltonian $G$-space with moment map $\mu:M\rightarrow G$. If the quotient $\mu^{-1}(1)/G$ of the inverse image $\mu^{-1}(1)$ of the identity is a manifold, then the restriction of $\omega$ to $\mu^{-1}(1)$ descends to a symplectic form on the reduced space $M/\!\!/ G:=\mu^{-1}/G$.
\end{thm}
\begin{defi}{\rm (\cite{AMM})}
Let $M_1$ (resp. $M_2$) be a quasi-Hamiltonian $G$-space with moment map $\mu_1$ (resp. $\mu_2$). Their fusion product $M_1\circledast M_2$ is defined to be the quasi-Hamiltonian $G$-space $M_1\times M_2$, where $G$ acts diagonally, with two-form
\begin{eqnarray*}
\widetilde{\omega}=\omega_1+\omega_2-\frac{1}{2}(\mu_1^*\theta,\mu_2^*\bar{\theta})
\end{eqnarray*}
and moment map
\begin{eqnarray*}
\widetilde{\mu}=\mu_1\cdot\mu_2: \ M\rightarrow G.
\end{eqnarray*}
\end{defi}
The quasi-Hamiltonian spaces from conjugacy classes can be seen as the building blocks of moduli spaces of flat connections on the trivial $G$-bundle on $\mathbb{P}^1$. Indeed, following \cite{AMM} let $\Sigma_m$ be a sphere with $m$ boundary components, the quasi-Hamiltonian reduction
\begin{eqnarray*}
\mathcal{C}_1\circledast\cdot\cdot\cdot \circledast \mathcal{C}_m/\!\!/ G
\end{eqnarray*}
of the fusion product of $m$ conjugacy classes $\mathcal{C}_i$ is isomorphic to the moduli space of flat connections on $\Sigma_m$ with the Atiyah-Bott symplectic form.
\\
Let us next recall the building blocks of the monodromy data of meromorphic connections. Let $T$ be a maximal torus of $G$ with Lie algebra $\frak t\subset \frak g$ and $B_{\pm}$ denote a pair of opposite Borel subgroups with $B_+\cap B_-=T$. Let us consider the family of complex manifolds (see \cite{Boalch3} for the geometrical origins of these spaces where their infinite-dimensional counterparts are described)
\begin{eqnarray*}
\widetilde{\mathcal{C}}:=\{(C,{\bf d},{\bf e},\lambda)\in G\times (B_-\times B_+)^{k-1}\times t ~|~\delta(d_j)^{-1}=
e^{\frac{\pi i\lambda}{k-1}}=\delta(e_j) \ for \ all \ j\},
\end{eqnarray*}
parameterised by an integer $k\geq 2$, where ${\bf b}=(d_1,...,d_{k-1})$, ${\bf e}=(e_1,...,e_{k-1})$ with $d_{even},e_{odd}\in B_+$ and
$d_{odd},e_{even}\in B_-$ and $\delta :B_+\rightarrow T$ is the homomorphism with kernel the unipotent subgroups $U_{\pm}$.
\begin{pro}\rm{(\cite{Boalch3})}\label{k=k_i}
The manifold $\widetilde{\mathcal{C}}$
is a complex quasi-Hamiltonian $G$-space with action
\begin{eqnarray*}
g\cdot (C,{\bf d},{\bf e}, \lambda)=(Cg^{-1},{\bf d},{\bf e},\lambda)\in \widetilde{\mathcal{C}}, \ \ \forall g\in G,
\end{eqnarray*}
moment map
\begin{eqnarray*}
\mu:\widetilde{\mathcal{C}}\rightarrow G; \ \ (C,{\bf d},{\bf e},\lambda)\mapsto C^{-1}d_1^{-1}\cdot\cdot\cdot d_{k-1}^{-1}e_{k-1}\cdot\cdot\cdot e_1C,
\end{eqnarray*}
and two-form
\begin{eqnarray*}
\omega=\frac{1}{2}(\bar{\mathcal{D}},\bar{\mathcal{E}})+\frac{1}{2}\sum_{j=1}^{k-1}(\mathcal{D}_j,\mathcal{D}_{j-1})-(\mathcal{E}_j,\mathcal{E}_{j-1}).
\end{eqnarray*}
Here $\bar{\mathcal{D}}=D^*\bar{\theta}$, $\bar{\mathcal{E}}=E^*\bar{\theta}$, $\mathcal{D}_j=D^*_j\theta$, $\mathcal{E}_j=E^*_j\theta\in
\Omega^1(\widetilde{\mathcal{C}},\frak g)$ for maps $D_j$, $E_j:\widetilde{\mathcal{C}}\rightarrow G$ defined by $D_i(C,\mathbb{d},\mathbb{e},\Lambda)=d_i\cdot\cdot\cdot d_1C,$ $E_i=e_i\cdot\cdot\cdot e_1C,$ $D:=D_{k-1},$ $E:=E_{k-1}$, $E_0=D_0:=C$.
\end{pro}
For instance, when $k=2$, $\widetilde{\mathcal{C}}_{k=2}\cong G\times G^*$ and the moment map, two form are given by
\begin{eqnarray}\label{k=2}
\mu=C^{-1}b_-^{-1}b_+C, \ \ \ \omega=\frac{1}{2}(D^*\bar{\theta},E^*\bar{\theta})+\frac{1}{2}(D^*\theta,C^*{\theta})-\frac{1}{2}(E^*\theta,C^*\theta)
\end{eqnarray}
where $D=b_-C,E=b_+C$.
For $k = 1$, we define $\widetilde{\mathcal{C}}_{k=1}:=\{(h,(e^{-\pi i \lambda},e^{\pi i \lambda},\lambda))~|~h\in G,\lambda\in \frak t'\}$ which is a submanifold of $\widetilde{\mathcal{C}}_{k=2}\cong G\times G^*$, and thus inherits a $G$ action. The restriction of the two form and moment map \eqref{k=2} of $\widetilde{\mathcal{C}}_{k=2}$ to $\widetilde{\mathcal{C}}_{k=1}$ makes it into a quasi-Hamiltonian $G$-space.
Given a divisor $D=\sum_{i=1}^mk_i(a_i)$ having each $k_i\geq 1$ at $a_i$ on $\mathbb{P}^1$. Let ${\mathcal{C}}_i$ be the quasi-Hamiltonian $G$-space in Propostion \ref{k=k_i} with $k=k_i$. Then the symplectic space $\widetilde{\mathcal{M}}({\bf a})$ of monodromy data for compatibly meromorphic connections $(V,\nabla,{\bf g})$ with irregular type ${\bf a}$ can be described as follows.
\begin{pro}[Lemma 3.1 \cite{Boalch3}]
The symplectic space $\widetilde{\mathcal{M}}({\bf a})$ is isomorphic to the quasi-Hamiltonian quotient $\widetilde{\mathcal{C}}_1\circledast\cdot\cdot\cdot \circledast\widetilde{\mathcal{C}}_m/\!\!/ G$, where $\circledast$ denotes the fusion product of two quasi-Hamiltonian $G$-manifolds.
\end{pro}
The extension of the Atiyah-Bott symplectic structure to the case of singular $\mathbb{C}^{\infty}$-connections given in \cite{Boalch2} leads to certain Hamiltonian loop group manifolds and $\widetilde{\mathcal{C}}$ is the corresponding quasi-Hamiltonian space.
\subsection{Irregular Riemann-Hilbert correspondence}
Let ${\bf a}$ be the data of a divisor $D=\sum k_i(a_i)$ and irregular types \ $\mathclap{^i}A^0$ at each $a_i$. The irregular Riemann-Hilbert correspondence, which depends on a choice of tentacles $\tau$ (see Definition 3.9 in \cite{Boalch2}), is a map $\nu$ from the global symplectic moduli space of meromorphic connections $\widetilde{\mathcal{M}^*}({\bf a})\cong (\widetilde{O}_1\times \cdot\cdot\cdot \widetilde{O}_m)/\!\!/ G$ to the symplectic space of monodromy data $\widetilde{\mathcal{M}}({\bf a})\cong (\widetilde{\mathcal{C}}_1\times\cdot\cdot\cdot\times\widetilde{\mathcal{C}}_m)/\!\!/ G$. In brief, the map arises as follows.
Let $(P,\nabla,{\bf g})$ be a compatibly framed meromorphic connection on a holomorphic trivial $G$-bundle $P$ with the irregular type ${\bf a}$. The chosen irregular type \ $\mathclap{^i}A^0$ canonically determines some directions at $a_i$ ('anti-Stokes directions'), and we can consider the Stokes sectors at each $a_i$ bounded by these directions (and having some small fixed radius). Then the key fact is that, similar to the discussion in section \ref{defineC}, the framings ${\bf g}$ (and a choice of branch of logarithm at each pole) determine, in a canonical way, a choice
of solutions of the equation $\nabla F=0$ on each Stokes sector at each pole. Then along any path in the punctured sphere $\mathbb{P}^1\setminus \{a_1,... , a_m\}$ between two Stokes sectors, we can extend the two corresponding canonical solutions and obtain an element in $G$ by taking their ratio. The monodromy data of $(P,\nabla,{\bf g})$ is simply the set of all such elements in $G$, plus the exponents of formal monodromy, thus corresponds to a point in the space of monodromy data $\widetilde{\mathcal{C}}_1\times\cdot\cdot\cdot\times\widetilde{\mathcal{C}}_m$. On the other hand, the triple $(P,\nabla,{\bf g})$ represents a point in $\widetilde{O}_1\times\cdot\cdot\cdot\widetilde{O}_m$. Therefore, it produces a map from $\widetilde{O}_1\times\cdot\cdot\cdot\widetilde{O}_m$ to $\widetilde{\mathcal{C}}_1\times\cdot\cdot\cdot\times\widetilde{\mathcal{C}}_m$ by taking the monodromy data of meromorphic connections $(P,\nabla, {\bf g})$. Furthermore, this map is $G$-equivariant and descends to give the irregular Riemann-Hilbert map $\nu$.
The main result of \cite{Boalch2} leads to:
\begin{thm}\label{RiemannHilbert}\rm{(\cite{Boalch2})}
The irregular Riemann-Hilbert map
\begin{eqnarray}\label{monodromymap}
\nu:(\widetilde{O}_1\times \cdot\cdot\cdot \widetilde{O}_m)/\!\!/ G \hookrightarrow
(\widetilde{\mathcal{C}}_1\circledast\cdot\cdot\cdot\circledast\widetilde{\mathcal{C}}_m)/\!\!/ G
\end{eqnarray}
associating monodromy/Stokes data to a meromorphic connection on a trivial $G$-bundle $P$ on $\mathbb{P}^1$ is a symplectic map (provided the symplectic structure on the right-hand side is divided by $2\pi i$).
\end{thm}
We will analyze the case where the meromorphic connections have one pole of order one and one pole of order two, and show that the irregular Riemann-Hilbert map $\nu$ gives rise to a local symplectic isomorphism from $(\Sigma,\omega)$ to $(\Sigma',\omega')$. Furthermore, given a choice of tantacles, the corresponding map $\nu$ can be expressed explicitly by the connection matrix $C\in {\rm Map}(\frak g^*,G)$ defined in section \ref{defineC}. Thus with the help of Theorem \ref{equivalent}, one can prove that $C$ is a solution of the equation \eqref{gaugeequation}, i.e., $r^{C}_0=r_{\rm AM}$.
First we need the following two propositions.
\begin{pro}\label{isoO}
Let $\widetilde{O}_1$ and $\widetilde{O}_2$ be two copies of $\widetilde{O}$ with $k=1$ and $k=2$ respectively. Then the Hamiltonian quotient $\widetilde{O}_1\times\widetilde{O}_2/\!\!/ G$ is symplectic isomorphic to $(\Sigma,\omega)$.
\end{pro}
\noindent{\bf Proof.}\
By definition, $\widetilde{\mathcal{O}}_1=\{(g_1,x_1)\in G\times\frak g^*~|~g_1x_1g_1^{-1}\in \frak t'\}$ and $\widetilde{\mathcal{O}}_2=\{(g_2,A,x_2)\in G\times\frak g^*\times \frak g^*~|~{\rm Ad}_{g_2}A=A_0\}$, where $A_0\in \frak t_{\rm reg}$. Because $A$ is determined by $g_2$, $\widetilde{\mathcal{O}_2}$ is naturally isomorphic to $G\times\frak g^*$ by sending $(g_2,A,x_2)$ to $(g_2,x_2)$. Note that the moment map is
\begin{eqnarray*}
\mu:\widetilde{O}_1\times \widetilde{O}_2\longrightarrow\mathfrak g^*; \ (g_1,x_1,g_2,x_2)\mapsto x_1+x_2.
\end{eqnarray*}
The submanifold $\mu^{-1}(0)$ is defined by $\mu^{-1}(0):=\{(g_1, x_1, g_2, -x_1)\in (G\times\frak g^*)^2~|~{\rm Ad}_{g_1}x_1\in \frak t'\}$. We have a subjective map
\begin{eqnarray*}
\iota:\mu^{-1}(0)\longrightarrow \Sigma; \ (g_1,x_1,g_2,-x_1)\mapsto (g_2g^{-1}_1,-{\rm Ad}^*_{g_1}x_1)
\end{eqnarray*}
whose fibres are the $G$ orbits. Thus it induces an isomorphism from $\widetilde{O}_1\times\widetilde{O}_2/\!\!/ G$ to $(\Sigma,\omega)$. To verify this is actually a symplectic isomorphism, let us take two tangents $v_1,v_2$ to $\mu^{-1}(0)$ which at each point $(g_1,x_1,g_2,-x_1)$ take the forms $v_i=(0,{\rm Ad}_{g_1^{-1}}R_i,{\rm Ad}_{g_2^{-1}}X_i,-{\rm Ad}_{g_1^{-1}}R_i)$ for some $X_i\in\frak g, R_i\in\frak t^*$ and $i=1,2$ ($\frak g\cong T_{g_2}G$ via left multiplication).
Let $\omega_{\mu^{-1}(0)}$ be the restriction of the (direct sum) symplectic structure $\omega_{\widetilde{O}_1\times\widetilde{O}_2}$ on $\mu^{-1}(0)$.
Following the formula \eqref{omegaO}, we have that at $(g_1,x_1,g_2,-x_1)$,
\begin{eqnarray*}
\omega_{\mu^{-1}(0)}(v_1,v_2)&=&\omega_{\widetilde{O}_1}((0,{\rm Ad}_{g_1^{-1}}R_1),(0,{\rm Ad}_{g_1^{-1}}R_2))\nonumber\\
&&+\omega_{\widetilde{O}_2}(({\rm Ad}_{g_2^{-1}}X_1,-{\rm Ad}_{g_1^{-1}}R_1),({\rm Ad}_{g_2^{-1}}X_2,-{\rm Ad}_{g_1^{-1}}R_2))\nonumber\\
&=&\langle R_2,{\rm Ad}_{g_1g_2^{-1}}X_1\rangle-\langle R_1,{\rm Ad}_{g_1g_2^{-1}}X_2\rangle-\langle x_1,{\rm Ad}_{g_2^{-1}}([X_1,X_2])\rangle.
\end{eqnarray*}
On the other hand, a direct computation gives $\iota_*(v_i)=({\rm Ad}_{g_1g_2^{-1}}X_i,-R_i)$ at $(g_2g_1^{-1},Ad_{g_1}x_1)$, here $\frak g\cong T_{g_2g_1^{-1}}G$ via left multiplication. Formula \eqref{omegasigma} makes it transparent that at $(g_2g_1^{-1},-{\rm Ad}^*_{g_1}x_1)\in \Sigma$,
\begin{eqnarray*}
\omega(\iota_*(v_1),\iota_*(v_2))&=&\omega(({\rm Ad}_{g_2g_1^{-1}}X_1,-R_1),({\rm Ad}_{g_2g_1^{-1}}X_2,-R_2))\nonumber\\
&=&\langle R_2,{\rm Ad}_{g_1g^{-1}_2}X_1\rangle-\langle R_1,{\rm Ad}_{g_1g^{-1}_2}X_2\rangle-\langle x_1,Ad_{g^{-1}_2}([X_1,X_2])\rangle.
\end{eqnarray*}
Therefore, we have that $\iota^*\omega=\omega_{\mu^{-1}(0)}$, i.e., $\iota$ induces a symplectic isomorphism between $\widetilde{O}_1\times\widetilde{O}_2/\!\!/ G$ and $(\Sigma,\omega)$.
\hfill ~\vrule height6pt width6pt depth0pt
\\
\\
As for the Poisson Lie counterpart, we have
\begin{pro}\label{isoC}
Let $\widetilde{\mathcal{C}}_1$ and $\widetilde{\mathcal{C}}_2$ be two copies of $\widetilde{\mathcal{C}}$ with $k=1$ and $k=2$ respectively. Then the quasi-Hamiltonian reduction of the fusion of $\widetilde{\mathcal{C}}_1$ and $\widetilde{\mathcal{C}}_2$ is isomorphic as a symplectic manifold to the symplectic submanifold $(\Sigma',\omega')$ of the double $\Gamma$.
\end{pro}
\noindent{\bf Proof.}\
We assume that the Borels chosen at the first pole are opposite to those chosen at the second (which we may since
isomonodromy will give symplectic isomorphisms with the spaces arising from any other
choice of Borels intersecting in $T$). Thus we have,
\begin{eqnarray*}
\widetilde{\mathcal{C}}_1=\{(h,e_-^{2\pi i \lambda})~|~h\in G,\lambda\in {\frak t}'\}, \ \ \ \ \ \widetilde{\mathcal{C}}_2=\{(C,(b_-,b_+,\Lambda))~|~\delta(b_{\pm})=e^{\pm\pi i\Lambda}\},
\end{eqnarray*}
here $e_-^{2\pi i\lambda}=(e^{\pi i\lambda^\vee},e^{-\pi i\lambda^\vee},\lambda^\vee)$ (exponential map of $\frak g^*$ with the opposite Borels chosen). The moment map on $\widetilde{\mathcal{C}}_1\circledast \widetilde{\mathcal{C}}_2$ is $\mu=h^{-1}e^{-2\pi i\lambda^\vee}hC^{-1}b^{-1}_-b_+C$. Therefore the condition
$\mu=1$ becomes $Ce^{2\pi i{\rm Ad}_{h^{-1}}\lambda^\vee}C^{-1}=b_-^{-1}b_+$, where $B:={\rm Ad}_{h^{-1}}(\lambda^\vee)$. Recall that $\Sigma'$ is a submanifold of Lu-Weinstein symplectic double $\Gamma$,
\begin{eqnarray*}
\Sigma':=\{(g_1,e_-^{2\pi i\lambda},g_2,(b_-,b_+,\Lambda))\in\Gamma~|~\delta(b_{\pm})=e^{\pm\pi i\Lambda}, \ g_1e^{\pm \pi i \lambda^\vee}=b_{\pm}g_2\}.
\end{eqnarray*}
We have a surjective map from $\mu^{-1}(1)=\{(h,e_-^{2\pi i \lambda},C,(b_+,b_-,\Lambda))~|~e^{2\pi i {\rm Ad}_{h^{-1}}\lambda^{\vee}}=Cb_-^{-1}b_+C^{-1}\}$ to $\Sigma'$,
\begin{eqnarray*}
(h,e_-^{2\pi i \lambda},C,(b_+,b_-,\Lambda))\rightarrow (Ch^{-1},e^{-2\pi i \lambda},u,(b_-,b_+,\Lambda))
\end{eqnarray*}
whose fibres are precisely the $G$ orbits, where $u:=b_+^{-1}Ch^{-1}e^{\pi i\lambda}\in G$. Therefore, it induces an isomorphism from $\widetilde{\mathcal{C}}_1\circledast\widetilde{\mathcal{C}}_2/\!\!/ G$ to $\Sigma'$. An explicit formula for
the symplectic structure on $\Sigma'$ can be computed by using Theorem 3 of \cite{Anton1}. On the other hand we have
an explicit formula for the symplectic structure on $\widetilde{\mathcal{C}}_1\circledast\widetilde{\mathcal{C}}_2/\!\!/ G$. A straightforward calculation shows these explicit formulae on each side agree.
\hfill ~\vrule height6pt width6pt depth0pt
\\
To specify an irregular Riemann-Hilbert map, we have to make a choice of tentacles (see \cite{Boalch2}). We introduce coordinate $z$ to identify $\mathbb{P}^1$ with $\mathbb{C}\cup \infty$ and assume the divisor $D=1(a_1)+2(a_2)$ where $a_2=0$ and $a_1=\infty$. Then we consider the meromorphic connections $\nabla$ on a trivial holomorphic $G$-bundle $P$ on $\mathbb{P}^1$ with compatible framings ${\bf g}$, such that $(P,\nabla,{\bf g})$ have an irregular type $\frac{A_0}{z^2}$ at $0$, where $A_0\in\frak t_{\rm reg}$. Let us take a prior Stokes sector ${\rm Sect}_0$ between two Stokes rays (only depend on $A_0$) at $0$, and make a choice of tentacles as follows.
\begin{itemize}
\item[\rm(i)] A choice of a point $p_2$ in ${\rm Sect}_0$ at $0$ and a point $p_1$ in ${\rm Sect}_0$ near $\infty$.
\item[\rm(ii)] A lift $\hat{p}_i$ of each $p_i$ to the universal cover of a punctured disc $D_i\backslash \{a_i\}$ containing $p_i$ for $i=1,2$.
\item[\rm(iii)] A base point $p_0$ which coincides with $p_1$.
\item[\rm(iv)] A contractible path $\gamma:[0,1]\rightarrow \mathbb{P}^1\setminus \{0,\infty\}$ in the punctured sphere, from $p_0$ to $p_1$.
\end{itemize}
Note that the chosen point $\hat{p}_2$ determines a branch of ${\rm log}z$ on ${\rm Sect}_0$. According to section \ref{defineC}, let $C\in {\rm Map}(\mathfrak g^*,G)$ be the connection matrix associated to $A_0\in \frak t_{\rm reg}$, the choice of ${\rm Sect}_0$ and the branch of ${\rm log}z$. Then we have
\begin{pro}\label{RHmap}
For the above choice of tentacles, the corresponding irregular Riemann-Hilbert map $\nu:(\widetilde{O}_1\times\widetilde{O}_2)/\!\!/ G\cong\Sigma\rightarrow (\widetilde{\mathcal{C}}_1\times \widetilde{\mathcal{C}}_2)/\!\!/ G\cong \Sigma'$ is given by
\begin{eqnarray*}
\nu(h,\lambda)=(C({\rm Ad}^*_h\lambda)h,e^{2\pi i \lambda},u,u^*), \ \forall (h,\lambda)\in \Sigma,
\end{eqnarray*}
for certain $u\in G, u^*\in G^*$ satisfying $C({\rm Ad}^*_h\lambda)he^{2\pi i \lambda}=u^*u$.
\end{pro}
\noindent{\bf Proof.}\
Let $(P,\nabla,{\bf g}=(g_1,g_2))$ be a compatibly framed meromorphic connection with irregular type $\frac{A_0}{z^2}$ at $a_2$, where $g_1,g_2\in G$ and $A_0\in \frak t_{\rm reg}$.
Upon trivializing $V$, we assume $(P,\nabla,{\bf g})$ represents a point $(g_1,-x,g_2,A,x)\in \widetilde{O}_{1}\times\widetilde{O}_2$, which means that in the trivialization, $\nabla=d-(\frac{A}{z^2}+\frac{x}{z})dz$. Furthermore, the given irregular type $\frac{A_0}{z^2}$ of $\nabla$ in the compatible frame $g_2$ indicates that ${\rm Ad}_{g_2}A=A_0$. Using the convention in Section \ref{connectiondata}, let $F_0$ (resp. $F_\infty$) be the canonical solution of $\nabla_{A_0} F:=dF-(\frac{A_0}{z^2}+\frac{{\rm Ad}_{g_2}x}{z})Fdz=0$ at $0$ (resp. $\infty$). Due to the chosen frame ${\bf g}$, $\Phi_0=g_2^{-1}F_\infty g_2$, $\Phi_1=g_2^{-1}F_\infty g_2g_1^{-1}$ and $\Phi_2=g_2^{-1}F_0$ are the canonical solutions of $\nabla \Phi=0$ on a neighbourhood of $p_0=p_1$ and $p_2$ with respect to the compatible framing $1$, $g_1$ and $g_2$ respectively.
Then the monodromy data of $(P,\nabla,{\bf g})$
\begin{eqnarray*}
(C_1,e_-^{2\pi i\lambda},C_2,(b_-,b_+,\Lambda))\in\widetilde{\mathcal{C}}_1\times\widetilde{\mathcal{C}}_2\cong G\times e_-^{\frak t'}\times G\times G^*,
\end{eqnarray*}
($e_-^{2\pi i\lambda}=(e^{\pi i\lambda^\vee},e^{-\pi i\lambda^\vee},\lambda^\vee)$ is the exponential map of $\frak g^*$ with the opposite Borels chosen) is the set of connection matrices $C_i$ (the ratio of the canonical solutions $\Phi_i$ at $p_i$ with $\Phi_0$ at $p_0$ for $i=1,2$), as well as the Stokes data $(b_-,b_+)$ at $0$ and the formal monodromy at $0$, $\infty$. They are explicitly described as follows.
\\
\\
$\bullet$ along the path $\gamma$ in the punctured sphere $\mathbb{P}^1\setminus \{0, \infty\}$, we extend the two solutions $\Phi_0$ and $\Phi_2$, then $\Phi_2 C_2=\Phi_0$. Therefore we have $C_2=F_0^{-1}F_\infty g_2$. By definition, $F_0^{-1}F_\infty= C({\rm Ad}_{g_2}x)$ the connection matrix of $\nabla_{A_0} F:=dF-(\frac{A_0}{z^2}+\frac{{\rm Ad}_{g_2}x}{z})Fdz=0$;
\\
\\
$\bullet$ $p_0$, $p_1$ can be seen as connected by an identity path, thus $\Phi_1C_1=\Phi_0$. Therefore $C_1$ is equal to $g_1$, the ratio of the frame chosen at $p_0$ and $p_1$;
\\
\\
$\bullet$ $b_-,b_+$ at $0$ are the Stokes matrices of $\nabla_{A_0}$, which are determined by the monodromy relation \eqref{le:monodromy reln}.
\\
Therefore, the chosen tentacle determines a map $\nu':\mu^{-1}(0)\subset \widetilde{O}_1\times\widetilde{O}_2\rightarrow \mu'^{-1}(0)\subset\widetilde{\mathcal{C}}_1\times \widetilde{\mathcal{C}}_2$, which is given by
\begin{eqnarray*}
\nu'(g_1,-x,g_2,x)=(g_1, e_-^{-2\pi i{\rm Ad}_{g_1}x},C({\rm Ad}_{g_2}x)g_2,(b_-,b_+,\Lambda)).
\end{eqnarray*}
This map $\nu'$ is $G$-equivariant and descends to the irregular Riemann-Hilbert map $\nu:\Sigma\rightarrow\Sigma'$
\begin{eqnarray*}
\nu(h,\lambda)=(C({\rm Ad}^*_h\lambda)h,e^{2\pi i \lambda},u,u^*),
\end{eqnarray*}
for any $(h,\lambda)\in \Sigma$. Here $u\in G, u^*\in G^*$ satisfy $C({\rm Ad}^*_h\lambda)he^{2\pi i \lambda}=u^*u$, and we use the isomorphisms $\mu^{-1}(0)/G\cong\Sigma$ and $\mu'^{-1}(1)/G\cong \Sigma'$ constructed in Propostion \ref{isoO} and \ref{isoC} respectively.
\hfill ~\vrule height6pt width6pt depth0pt
\\
\\
Now we can give a proof of our main theorem:
\\
{\bf The proof of Theorem \ref{Cequation}.} Following Proposition \ref{RHmap}, the irregular Riemann-Hilbert map $\nu$ coincides with the local diffeomorphic map $F'_{C}:\Sigma\rightarrow \Sigma'$ defined in Section 2.
According to Corollary \ref{rescale}, $\nu:(\Sigma,\omega)\rightarrow(\Sigma',\omega')$ is a symplectic map, provided the symplectic structure on the right-hand side is divided by $2\pi i$, if and only if $C_{2\pi i}$ is a solution of the equation \eqref{gaugeequation} (Here $C_{2\pi i}(x)=C(\frac{1}{2\pi i}x)$, for all $x\in\mathfrak g^*$). However, the former is guaranteed by Theorem \ref{RiemannHilbert}. As a result, we get a proof of Theorem \ref{Cequation}, i.e., $C_{2\pi i}$ is a solution of the gauge transformation equation \eqref{gaugeequation}.
\hfill ~\vrule height6pt width6pt depth0pt
\section{Lu-Weinstein symplectic groupoids via Alekseev-Meinrenken r-matrices}\label{LuWein}
As an application of the above construction, we describe the Lu-Weinstein symplectic groupoids via Alekseev-Meinrenken $r$-matrices.
Following the construction of section 2.1 in \cite{xu}, for any $r$-matrix with the skew-symmetric part $r_0$,
\begin{eqnarray*}
\pi_{\rm AM}(h,x):=\pi_{\rm KKS}(x)+l_h(\theta)+l_h(r_{AM}(x))-r_h(r_0),
\end{eqnarray*}
defines a symplectic structure on $G\times \frak g^*$, where $\theta:=\frac{\partial}{\partial x^a}\wedge e_a\in\Gamma(\wedge^2(T\mathfrak g^*\oplus \mathfrak g))$ for a base $\{e_a\}$ of $\mathfrak g$ and the corresponding coordinates $\{x^a\}$ on $\mathfrak g^*$. We will prove that $(G\times \frak g^*,\pi_{AM})$ is a natural symplectic groupoid and is locally symplectic isomorphic to Lu-Weinstein double sympelctic groupoid with respect to $(\frak g, r)$.
To do this, let us consider the Semenov-Tian-Shansky (STS) Poisson tensor on $\frak g^*$ defined by
\begin{eqnarray*}
\pi_{\rm STS}(x)(df,dg)=\langle df(x)\otimes dg(x), {\rm ad}_x\otimes\frac{1}{2}{\rm ad}_x{\rm coth}(\frac{1}{2}{\rm ad}_x)(t)-\otimes^2{\rm ad}_x(r_0)\rangle,
\end{eqnarray*}
for any $f, g\in C^{\infty}(\mathfrak g^*)$.
We denote by $L,R$ the group morphisms corresponding to the Lie algebra morphisms $L,R:\frak g^*\rightarrow \frak g$
\begin{eqnarray*}
L(x):=(x\otimes {\rm id})(r), \ \ \ \ \ \ R(x):=-(x\otimes {\rm id})(r^{2,1}) \ \ \ \ \ \ \forall \ x\in \frak g^*.
\end{eqnarray*}
Let $(G^*,\pi_{G^*})$ be the simply connected Poisson Lie group associated to the quasitriangular Lie bialgebra $(\frak g,r)$.
\begin{pro}\rm{\cite{FM}}\label{STSPoisson}
The map $I:(\frak g^*,\pi_{STS})\rightarrow (G^*,\pi_{G^*})$ determined by $e^{x^{\vee}}=L(I(x))^{-1}R(I(x))$ for any $x\in\frak g^*$, is a Poisson map.
\end{pro}
Actually, the STS Poisson structure on $\frak g^*$ is completely determined by the above proposition.
Now let us take any solution $g\in {\rm Map}(\frak g^*,G)$ of the gauge transformation equation \eqref{gaugeequation}.
\begin{thm}
$(G\times\frak g^*,\pi_{{\rm AM}})$ is a Poisson(therefore symplectic) groupoid over $\frak g^*$ with the structure maps given by
\begin{eqnarray*}
&&\alpha(h,x)\rightarrow x\in \frak g^*, \ \beta(h,x)\rightarrow Ad^*_hx\in\frak g^*, \ \forall h\in G, x\in\frak g^*,\nonumber\\
&&\varepsilon:\frak g^*\rightarrow G\times\frak g^*: \ x\mapsto (g(x),Ad_{g(x)}^*x)\in G\times\frak g^*,\\
&&m:\mathcal{G}_2:=\{((h_1,x),(h_2,y))\in\mathcal{G}\times\mathcal{G}~|~\beta(h_1)=\alpha(h_2)\}\rightarrow \mathcal{G}: \nonumber\\ &&(h_1,x), (h_2,y=Ad^*_{h_1}x)\mapsto Ad_{h_1h_2}^*x, \ \forall (h_1,h_2)\in \mathcal{G}_2.\nonumber
\end{eqnarray*}
Furthermore, it is the integration of the STS Poisson structure on $\frak g^*$.
\end{thm}
A straightforward proof can be obtained verifying the following equivalent conditions one by one.
\begin{lem}\rm{(\cite{PingXu})}
$(\mathcal{G}\Longrightarrow M,\pi)$ is a Poisson groupoid if and only if all the following hold.
$(1)$ For all $(x,y)\in\mathcal{G}_2$,
\begin{eqnarray*}
\pi(xy)=R_Y\pi(x)+L_X\pi(y)-R_YL_X\pi(m),
\end{eqnarray*}
where $m=\beta(x)=\alpha(y)$ and $X,Y$ are (local) bisections through $x$ and $y$ respectively.
$(2)$ $M$ is a coisotropic submanifold of $\mathcal{G}$,
$(3)$ For all $x\in\mathcal{G}$, $\alpha_*\pi(x)$ and $\beta_*\pi(x)$ only depend on the base points $\alpha(x)$ and $\beta(y)$ respectively.
$(4)$ For all $f,g\in C^{\infty}(M)$, one has $\{\alpha^*f,\beta^*g\}=0$,
$(5)$ The vector field $X_{\beta^*f}$ is left invariant for all $f\in C^{\infty}(M)$.
\end{lem}
\begin{thm}
The map $v:G\times \frak g^*\rightarrow G\times G^*$, $v(h,x)=(g(x)h,g^*(x))$ for all $(h,x)\in G\times \frak g^*$, gives a local symplectic isomorphism from $(G\times \frak g^*,\pi_{{\rm AM}})$ to Lu-Weinstein symplectic double $\Gamma$,
where $g^*:\frak g^*\rightarrow G^*$ is the unique local
isomorphism defined by the identity
\begin{eqnarray*}
g(x)e^{x^{\vee}}g(x)^{-1} =L(g^*(x))R(g^*(x))^{-1}.
\end{eqnarray*}
\end{thm}
Actually, under the transformation $F:G\times\frak g^*\rightarrow G\times\frak g^*$ given by $F(h,x)=(hg^{-1}(x),Ad^*_{g(x)}x)$, then the Poisson tensor $\pi_{AM}$ becomes
\begin{eqnarray*}
F_*(\pi_{\rm AM})(h,x)=\pi_{\rm STS}(x)+l_h(\theta^{g}(x))+l_h(r_0)-r_h(r_0), \ \forall (h,x)\in G\times\frak g^*
\end{eqnarray*}
where $\theta^{g}\in \Gamma(\wedge^2(T\frak g^*\oplus \frak g))$ is the gauge transformation of $\theta$ under $g\in {\rm Map}(\frak g^*,G)$ (see \cite{xu} for more details about a generalized dynamical $r$-matrix and its gauge transformations). According to \cite{xu}, we have that $(\pi_{\rm STS},\theta^{g},r_0)$ is a gauge transformation of the dynamical $r$-matrix $(\pi_{\rm KKS},\theta,r_{\rm AM})$ under $g\in {\rm Map}(\frak g^*,G)$, thus also a generalized classical dynamical $r$-matrix. It therefore verifies that equation \eqref{gaugeequation} is indeed a gauge transformation equation in the theory of generalized dynamical $r$-matrices.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,488 |
Le Lycée des cancres (Rock 'n' Roll High School) est un film américain réalisé par Allan Arkush et sorti en 1979.
Synopsis
Riff Randell, une lycéenne, rêve de rencontrer Joey Ramone.
Fiche technique
Titre français : Le Lycée des cancres ou Surboum au lycée
Titre original : Rock 'n' Roll High School ou Disco High
Réalisation : Allan Arkush
Scénario : Richard Whitley, Russ Dvonch, Joseph McBride, Allan Arkush et Joe Dante
Photographie : Dean Cundey
Montage : Larry Bock et Gail Werbin
Production : Michael Finnell
Société de production : New World Pictures
Société de distribution : Eurogroup (France) et New World Pictures (États-Unis)
Pays de production :
Langue originale : anglais américain
Format : Couleur Metrocolor - 1,85:1 - 35 mm
Genre : comédie, film musical
Durée : 93 minutes
Dates de sortie :
États-Unis :
France :
Distribution
P. J. Soles : Riff Randell
Vincent Van Patten : Tom Roberts
Clint Howard : Eaglebauer
Dey Young : Kate Rambeau
Mary Woronov : Miss Togar
Paul Bartel : Mr. McGree
Dick Miller : Police Chief
Don Steele : Screamin' Steve Stevens
Alix Elias : Coach Steroid
Loren Lester : Fritz Hansel
Daniel Davies : Fritz Gretel
Lynn Farrell : Angel Dust
Joey Ramone : lui-même
Johnny Ramone : lui-même
Dee Dee Ramone : lui-même
Marky Ramone : lui-même
Production
Accueil
Le film a reçu un accueil positif de la critique. Il obtient un score moyen de 70 % sur Metacritic.
Notes et références
Liens externes
Film américain sorti en 1979
Film musical américain de comédie
Film sur le rock
Ramones
Film musical américain des années 1970 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,866 |
Иваница () — община в Сербии, входит в Моравичский округ.
Население общины составляет 33 878 человек (2007 год), плотность населения составляет 31 чел./км². Занимаемая площадь — 1090 км², из них 47,0 % используется в промышленных целях.
Административный центр общины — город Иваница. Община Иваница состоит из 49 населённых пунктов, средняя площадь населённого пункта — 22,2 км².
Статистика населения общины
Известные уроженцы
Стамболич, Иван (1936—2000) — югославский сербский государственный деятель, председатель Президиума СФРЮ (1986—1987), Премьер-министр Сербии (1978—1982).
Стамболич, Петар (1912—2007) — югославский сербский государственный деятель, председатель Президиума СФРЮ (1982—1983), председатель Союзного Исполнительного Веча СФРЮ (1963—1967).
Примечания
Ссылки
Официальная статистика Сербии
PDF версии издания «Издаје и штампа — Републички завод за статистику Србије»
Общины Сербии | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 330 |
Mr. and Mrs. is a 2012 Nigerian romantic drama film written and produced by Chinwe Egwuagu and directed by Ikechukwu Onyeka, starring Nse Ikpe Etim, Joseph Benjamin, Barbara Soky, Thelma Okoduwa and Paul Apel.
Cast
Nse Ikpe Etim as Susan Abbah
Joseph Benjamin as Kenneth Abbah
Barbara Soky as Mrs Abbah
Thelma Okoduwa-Ojiji as Linda
Paul Apel as Charles
Chioma Nwosu as Mrs Brown
Mpie Mapetla as Monica
Nonye Ike as Kate
Beauty Benson as Maggie
Paul Sambo as Mr Brown
Babajide Bolarinwa as Mr Abbah
Home media
The film was released on DVD on 20 August 2012. According to Chinwe, the performance of Mr. and Mrs. was commendable at the theatres and she's been receiving a lot of requests concerning the DVD release, so she thought the time was right for a DVD release. The film was first released in Ghana a week before the official DVD release and it was well received, with commendable sales in Gold coast the same week.
References
External links
2012 films
English-language Nigerian films
Nigerian romantic drama films
2012 romantic drama films
Films shot in Abuja
Films set in Abuja
2010s English-language films | {
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} | 9,491 |
Никола́й Ефи́мович Мо́мот (26 августа 1928 — 16 марта 1997) — строитель, Герой Социалистического Труда (1985), Заслуженный строитель РСФСР (1978), избирался депутатом Мурманского Областного Совета депутатов.
Образование
Одесский политехнический институт (1955)
Карьера
с 1955 работал мастером, прорабом, начальником участка управления «Севзапэлектромонтаж», затем старший прораб, секретарь партийного комитета треста «Мурманскжилстрой», заведующий отделом строительства Мурманского Областного комитета КПСС.
с 1979 по 1986 работал начальником управления «Главмурманскстрой».
с 1986 сотрудник управления Министерства строительства в северных и западных районах СССР, в октябре 1986 преобразованное в Министерство строительства в северных и западных районах РСФСР.
с августа 1990 работал в Российском государственном концерне по строительству в северных и западных районах РСФСР («Россевзапстрой») преобразованный в 1996 в АО. Занимал должность исполнительный директор Дирекции федеральных и региональных программ и прогнозов, заместитель председателя совета директоров АО .
Являлся руководителем и организатором возведения крупных промышленных предприятий (Ковдорского и Оленегорского комбинатов, комбинатов «Североникель», «Печенганикель» и др.), учреждений быта и культуры в городах Мурманске, Оленегорске, Кировске, Апатиты, Ковдоре, Заполярном, сооружения Кольской АЭС и ГЭС. Был инициатором введения на стройках Мурманской области методов комплексного и бригадного подряда.
Умер в Москве. Похоронен на Троекуровском кладбище.
Награды
Орден «За заслуги перед Отечеством» IV степени (29 декабря 1994)
Орден Ленина и золотая медаль «Серп и Молот» (4 октября 1985)
2 ордена Трудового Красного знамени (11 августа 1966, 5 ноября 1980)
2 ордена «Знак Почёта» (25 августа 1971, 16 марта 1976)
орден «За личное мужество» (28 июля 1989)
Заслуженный строитель РСФСР (1978)
медали
Память
имя Момота Н. Е. присвоено Мурманскому строительному лицею № 14
Ссылки
ПОЧЕМУ МСК НОСИТ ИМЯ ГЕРОЯ СОЦИАЛИСТИЧЕСКОГО ТРУДА НИКОЛАЯ ЕФИМОВИЧА МОМОТА?
Память Мурмана
Похороненные на Троекуровском кладбище
Выпускники Одесского национального политехнического университета
Депутаты Мурманского областного совета | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,617 |
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Book Review: Jerusalem: The Spatial Politics of a Divided Metropolis by Anne B. Shlay and Gillad Rosen
In Jerusalem: The Spatial Politics of a Divided Metropolis, Anne B. Shlay and Gillad Rosen outline the geographic dynamics of contemporary Jerusalem. While the book is occasionally simplistic in some areas of its analysis, Kenny Schmitt praises the authors for navigating complex terrain with skill and clarity to produce an approachable introduction to the spatial politics of the city.
Jerusalem: The Spatial Politics of a Divided Metropolis. Anne B. Shlay and Gillad Rosen. Polity Press. 2015.
Violence has come to Jerusalem, again. It erupted during the Jewish and Muslim holidays, which virtually coincided this year. Since the beginning of October, at least 44 Palestinians and eight Israelis have lost their lives. From my home in East Jerusalem, the tension is palpable and the fear is pervasive. How can one move past the shocking headlines to an engaged and thoughtful analysis of the city?
In Jerusalem: The Spatial Politics of a Divided Metropolis, Anne B. Shlay and Gillad Rosen have written a book that attempts to convey the complexity of the city, whilst remaining accessible to a wide audience. This book is about the politics of space and the 'constellation of competing interests' over it (13). Shlay and Rosen, a sociologist and geographer respectively, explore the various geographic dynamics of Jerusalem and how the conflict plays out in specific locations. Their goal is not to 'inflame or incite but to analyze and inform' (15). It is a worthy goal. In this review I argue that the authors accomplish it, mostly.
The chapters are well organised and the content is accessible. Chapter titles, such as 'What is Jerusalem?', 'Who is Jerusalem?' and 'The Palestinian Challenge and Resistance in Arab Jerusalem', give the reader clear signposts about the presentation and organisation of the book's content. The narrative flows and readers have a sense of momentum throughout. The concluding chapter is particularly engaging as the authors take readers on a virtual tour of Jerusalem, pointing out key locations and sites of struggle along the way. They navigate complex terrain with skill and clarity.
One point I found astute – and particularly helpful for a wide audience – was that much of the conflict occurs behind closed doors. Masses may be drawn into the spectacle of stabbings and burning tyres; yet, in reality, the bulk of the conflict is played out in policy decisions, budget allocations and the ways that people tell stories about the past. The authors illustrate this point vividly in their discussion of the City of David and Silwan community (5-7, 79, 83-85). As the book highlights this sensitive issue, and others like it, readers are subtly challenged to consider the implications of their own ideologies and narrative histories. Working through this process is essential for anyone wanting to understand the city of Jerusalem.
Image Credit: Jerusalem, The Temple Mount/Haram al-Sharif (Aleksander Miler)
In the chapter 'Who is Jerusalem?', the authors discuss the various populations of the city at length. This is very helpful, and written to a level appropriate for a wide audience. Jewish residents are analysed geographically, politically and religiously. Explanations are helpful. The authors also discuss Palestinians in the city. Here, the analysis is primarily geographical. When Palestinian religion and politics are mentioned, the discussion is too simplistic. The Palestinian community of Jerusalem is much more complex and varied than the authors lead their readers to believe. For one, they fail to mention the Islamic Movement in Israel led by Sheykh Raid Salah. This group has explicit religious and political agendas for the city, which directly shape spatial politics at the Temple Mount/Haram al-Sharif, among other locales. As Israel has marginalised traditional Palestinian political parties, the Islamic movement has arguably filled the gap. Their role in the city must be understood – even in a mass market publication.
Another shortfall of the book is the neglect of Jerusalem localities behind the barrier/fence/wall [sic]. Readers are given basic information and informed that 'a significant portion' of the population live behind the barrier/fence/wall (74-79). But without numbers, readers are left wondering just how 'significant' that portion is. Estimates from 2012 put the number around 90,000 (see also Candace Graff). With a total Arab population of approximately 300,000, this means around 25% of Arab Jerusalemites live behind the barrier/fence/wall. This is astounding! The communities of Qafr 'Aqab, Shuafat Camp, Ras Khamis, Ras al-Shahada and Dahiyat al-Salam are all behind the barrier/fence/wall. They are the most densely populated and impoverished locations in the city, lacking basic infrastructure and municipality services. They have virtually no city planning or law enforcement. Beyond this, the residents of Shuafat Camp are doubly displaced refugees, which makes a difficult living situation more complicated (the refugees in Shuafat camp were expelled from their homes in 1948 and relocated to the Old City of Jerusalem. In 1967, at the completion of the Six-Day War, they were displaced a second time to the Shuafat Refugee camp). Since the barrier/fence/wall was built, the population in these communities has skyrocketed. Beyond this, residents of these communities make up a significant portion of the wage labour force in West Jerusalem (see also Nir Hasson). It comes as little surprise that much of the violence plaguing Jerusalem today originates from behind this wall. How can a book on the spatial politics of the city only mention these communities in passing?
Initially, I was troubled by the authors' claim that East Jerusalem is an 'Israeli construction […] an artifact of Israeli power and political domination' (139). This seemed to qualify, minimise and negate Palestinian claims to the city. However, upon grasping their concept of 'Arab Jerusalem', my concerns were assuaged. Arab Jerusalem is not just the status quo boundaries established by the Oslo process; it is the very essence of what Palestinians consider their city to be: the East Jerusalem of today and the West Jerusalem of memory and history. Arab Jerusalem directly challenges the dominant Israeli discourse. The authors are correct in explaining that resistance to Israeli dominance occurs on the ground level among Arab Jerusalemites. What they have missed, however, is the role of the international community in challenging this dominance as well. One concrete expression of this is the US refusal to relocate its embassy to Jerusalem. Another is the international funding that supports Arab Jerusalem from a great number of nations and international organisations. These demonstrate that Israel's dominance of Jerusalem is not as 'common sense' as the authors would suggest (142).
Despite these criticisms, I believe that Shlay and Rosen have crafted an approachable and provocative introduction to the spatial politics of the city of Jerusalem. They have navigated a complex and contentious terrain skillfully. The reader who wants to push beyond headlines will find themselves highly engaged by this book.
Kenny Schmitt is a PhD Candidate at the Institute of Arab and Islamic Studies, Exeter University. His research examines Muslim religious practices in Jerusalem and how they are influenced and transformed by ongoing conflict in the city. He has nine years of experience in the Middle East, four of those based full-time in East Jerusalem. He is fluent in Modern Standard Arabic and Palestinian dialect.
Note: This review gives the views of the author, and not the position of the LSE Review of Books blog, or of the London School of Economics.
November 25th, 2015|Africa and the Middle East, Architecture and Urban Studies book reviews, Featured, History book reviews, International Relations book reviews, Politics book reviews|1 Comment
Impact of Social Sciences – Book Review: Jerusalem: The Spatial Politics of a Divided Metropolis by Anne B. Shlay and Gillad Rosen 11/28/2015 at 10:00 am - Reply
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This work by LSE Review of Books is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 2.0 UK: England & Wales. | {
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Are you searching for an eco-friendly bed mattress for your bedroom? Then you might wish to think about PlushBed's Botanical Bliss bed mattress. Developed with organic latex, this bed mattress aims to serve the very best for your sleep health and for the environment.
Plushbeds are constructed with 3 3-inch layers of organic Dunlop Latex. This isn't like the Talalay Latex which often has an airy and light feel. Dunlop has more density and exhibits a firmer structure.
Plushbeds' three Dunlop areas are classified as Soft, Medium or Firm. These three layers can likewise be set up in whatever order it finest fits you.
PlushBed says that for each mattress, there are an overall of 6 unique firmnesses. There is likewise a split firmness alternatives readily available for sizes King and Cal King. Are you fretted that configuring the layers to your needs would be challenging to find out? No worries!
PlushBed provides a series of infographics on their website detailing the numerous ways to position the latex section. In the event that you're disappointed with the feel, PlushBeds is willing to include another layer of firm latex for included assistance.
For the function of the PlushBeds evaluation, we'll be house on the Botanical Bliss in its "Soft" orientation. This mattress includes a soft layer as its first layer, medium layer in the middle, and company layer as its base.
Latex Layers: Again, there are three sections after the cover. These layers consist of 100% all-natural Dunlop Latex. Also, the products used to built these layers are thick and reacts rapidly to pressure. This develops a bed that is extremely bouncy.Unlike memory foam, latex does not feature much besides contouring. So, you're probably not going to feel stuck in the mattress.In addition, you will likely experience sleeping cooly on latex. This is certainly a fantastic choice if you tend to sleep hot throughout the night.Regarding the particular configuration, putting together the Soft layer on top enables some preliminary deep sinkage into the bed mattress. This is is then a little safeguarded by the Medium latex area as you transition into the Firm section/base listed below.
Qualified Organic: Rejoice, eco-warriors! The building of PlushBed's Botanical Bliss is considerable since it's 100% licensed natural. This implies that the materials utilized to produce the mattress have actually been accredited by the Global Organic Textile Standard and Global Organic Latex Standard. In addition, the bed mattress is GREENGUARD Gold Certified.This ensures that the bed mattress fulfilled most of the world's most precise and inclusive requirements for low emissions of unpredictable organic compounds into the indoor air.
Pressure Map: Pressure is another necessary part for feeling the bed mattress specifically understanding where the pressure points might form whilst resting on top of a new mattress.
Back: Lying on your back, you will really feel yourself sinking into the bed, with its soft layer of latex foam working to complete the area at your back area. Even for all that sinkage, the responsiveness of the latex makes it simple for you to roll and change positions.
Edge Support: Scooting as near the side, there is noticeable deep compression through the top layers. While it doesn't feel as though you simply rolled out of bed, you wouldn't state that this bed mattress has an especially strong edge support.
Side: If you're a side sleeper, you most likely like softer mattresses. Remarkably, you will feel fantastic pressure relief in this position. There won't be any tension to the hips or shoulders, which is typically an issue location for those who sleep mainly on their side.
Stomach: If you sleep on your stomach, you might feel a little pain as the soft nature of the bed causes your hips to sink out of alignment with your shoulders. Stomach sleepers also tend to choose firm beds as it helps keep their spinal column in a good, even line. Therefore, those who sleep solely on their stomachs may wish to opt for a firmer setup of the Botanical Bliss.
Will you be sharing the PlushBed mattress with a partner? If that's a case, then you'll wish to consider the mattress' movement transfer. This describes the quantity of disturbance you and your partner will likely feel from one side of the bed mattress to another.Basically, evaluating the mattress' motion transfer will let you know just how troubled you'll be by your bed mate's movements in the night.Professionals showed this motion transfer test by dropping a 10 lb steel ball from various heights of 4, 8 and 12 inches. Then, they determined the disruption brought on by the ball and reached to the conclusion that the larger the lines, the bigger the disturbance.Each of the ball's drop represents a distinct motion which you will probably experience on the bed mattress from tossing and turning, rising, and complete on leaping. Furthermore, the test suggested that you will feel your partner's every toss and turn. In another note, this stepping in bounce is a common discontentment associated with the product and is taken part in other all-latex beds.
One of the most significant advantage of this mattress is its bouncy feel which can be an enticing function for sleepers who look for buoyant assistance.
The Botanical Bliss helps you sleep cool. This is significant professional given that it is a natural latex and is great at temperature level policy. This makes sure that your body will not overheat throughout the night.
As pointed out above, the PlushBed bed mattress doesn't have remarkable motion transfer outcomes. This means that you will likely be a bit disrupted by movements throughout the night.
Further, bear in mind that you will not be experiencing lots of contour here. This is an extremely common grievance about latex which could be a dealbreaker for those who need deep pressure relief. | {
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{"url":"https:\/\/eng.libretexts.org\/Bookshelves\/Computer_Science\/Databases_and_Data_Structures\/Book%3A_Open_Data_Structures_-_An_Introduction_(Morin)\/04%3A_Skiplists\/4.02%3A_SkiplistSSet_-_An_Efficient_SSet","text":"# 4.2: SkiplistSSet - An Efficient SSet\n\n\nA SkiplistSSet uses a skiplist structure to implement the SSet interface. When used in this way, the list $$L_0$$ stores the elements of the SSet in sorted order. The $$\\mathtt{find(x)}$$ method works by following the search path for the smallest value $$\\mathtt{y}$$ such that $$\\mathtt{y}\\ge\\mathtt{x}$$:\n\n Node<T> findPredNode(T x) {\nNode<T> u = sentinel;\nint r = h;\nwhile (r >= 0) {\nwhile (u.next[r] != null && compare(u.next[r].x,x) < 0)\nu = u.next[r]; \/\/ go right in list r\nr--; \/\/ go down into list r-1\n}\nreturn u;\n}\nT find(T x) {\nNode<T> u = findPredNode(x);\nreturn u.next[0] == null ? null : u.next[0].x;\n}\n\n\nFollowing the search path for $$\\mathtt{y}$$ is easy: when situated at some node, $$\\mathtt{u}$$, in $$L_{\\mathtt{r}}$$, we look right to $$\\texttt{u.next[r].x}$$. If $$\\mathtt{x}>\\texttt{u.next[r].x}$$, then we take a step to the right in $$L_{\\mathtt{r}}$$; otherwise, we move down into $$L_{\\mathtt{r}-1}$$. Each step (right or down) in this search takes only constant time; thus, by Lemma 4.1.1, the expected running time of $$\\mathtt{find(x)}$$ is $$O(\\log \\mathtt{n})$$.\n\nBefore we can add an element to a SkipListSSet, we need a method to simulate tossing coins to determine the height, $$\\mathtt{k}$$, of a new node. We do so by picking a random integer, $$\\mathtt{z}$$, and counting the number of trailing $$1$$s in the binary representation of $$\\mathtt{z}$$:1\n\n int pickHeight() {\nint z = rand.nextInt();\nint k = 0;\nint m = 1;\nwhile ((z & m) != 0) {\nk++;\nm <<= 1;\n}\nreturn k;\n}\n\n\nTo implement the $$\\mathtt{add(x)}$$ method in a SkiplistSSet we search for $$\\mathtt{x}$$ and then splice $$\\mathtt{x}$$ into a few lists $$L_0$$,..., $$L_{\\mathtt{k}}$$, where $$\\mathtt{k}$$ is selected using the $$\\mathtt{pickHeight()}$$ method. The easiest way to do this is to use an array, $$\\mathtt{stack}$$, that keeps track of the nodes at which the search path goes down from some list $$L_{\\mathtt{r}}$$ into $$L_{\\mathtt{r}-1}$$. More precisely, $$\\mathtt{stack[r]}$$ is the node in $$L_{\\mathtt{r}}$$ where the search path proceeded down into $$L_{\\mathtt{r}-1}$$. The nodes that we modify to insert $$\\mathtt{x}$$ are precisely the nodes $$\\mathtt{stack[0]},\\ldots,\\mathtt{stack[k]}$$. The following code implements this algorithm for $$\\mathtt{add(x)}$$:\n\n boolean add(T x) {\nNode<T> u = sentinel;\nint r = h;\nint comp = 0;\nwhile (r >= 0) {\nwhile (u.next[r] != null\n&& (comp = compare(u.next[r].x,x)) < 0)\nu = u.next[r];\nif (u.next[r] != null && comp == 0) return false;\nstack[r--] = u; \/\/ going down, store u\n}\nNode<T> w = new Node<T>(x, pickHeight());\nwhile (h < w.height())\nstack[++h] = sentinel; \/\/ height increased\nfor (int i = 0; i < w.next.length; i++) {\nw.next[i] = stack[i].next[i];\nstack[i].next[i] = w;\n}\nn++;\nreturn true;\n}\n\n\nRemoving an element, $$\\mathtt{x}$$, is done in a similar way, except that there is no need for $$\\mathtt{stack}$$ to keep track of the search path. The removal can be done as we are following the search path. We search for $$\\mathtt{x}$$ and each time the search moves downward from a node $$\\mathtt{u}$$, we check if $$\\texttt{u.next.x}=\\mathtt{x}$$ and if so, we splice $$\\mathtt{u}$$ out of the list:\n\n boolean remove(T x) {\nboolean removed = false;\nNode<T> u = sentinel;\nint r = h;\nint comp = 0;\nwhile (r >= 0) {\nwhile (u.next[r] != null\n&& (comp = compare(u.next[r].x, x)) < 0) {\nu = u.next[r];\n}\nif (u.next[r] != null && comp == 0) {\nremoved = true;\nu.next[r] = u.next[r].next[r];\nif (u == sentinel && u.next[r] == null)\nh--; \/\/ height has gone down\n}\nr--;\n}\nif (removed) n--;\nreturn removed;\n}\n\n\n## $$\\PageIndex{1}$$ Summary\n\nThe following theorem summarizes the performance of skiplists when used to implement sorted sets:\n\nTheorem $$\\PageIndex{1}$$.\n\nSkiplistSSet implements the SSet interface. A SkiplistSSet supports the operations $$\\mathtt{add(x)}$$, $$\\mathtt{remove(x)}$$, and $$\\mathtt{find(x)}$$ in $$O(\\log \\mathtt{n})$$ expected time per operation.\n\n#### Footnotes\n\n1This method does not exactly replicate the coin-tossing experiment since the value of $$\\mathtt{k}$$ will always be less than the number of bits in an $$\\mathtt{int}$$. However, this will have negligible impact unless the number of elements in the structure is much greater than $$2^{32}=4294967296$$.\n\n4.2: SkiplistSSet - An Efficient SSet is shared under a CC BY license and was authored, remixed, and\/or curated by Pat Morin.","date":"2022-05-20 16:25:46","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\": 0, \"mathjax_display_tex\": 1, \"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\": 0.47472843527793884, \"perplexity\": 1471.9661657431964}, \"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\/1652662533972.17\/warc\/CC-MAIN-20220520160139-20220520190139-00368.warc.gz\"}"} | null | null |
It's a tough world for young people today: so much pressure to succeed, against narrow definitions of success that quite frankly just aren't all that relevant to the world they're growing up into.
1) You can always be more than one thing.
There's no obligation to define yourself against one set of criteria or one definition of success. Today, I'm a marketing strategist; a personal branding consultant, and the co-founder of a trail running community. Also a mum and a runner. Occasionally a writer. Definitely an introvert. The point is, you don't have to choose one path and follow it to the bitter end. Go exploring.
2) Don't give up on what you love, just because it doesn't seem like a 'career option' … yet.
Today my work-life balance is exactly that: a balance. Made up of the things I need, a) to be happy and b) to get by. Time and money; thinking and doing; navigating tricky business problems and using bloody-minded determination to get myself from point A to point B. Family, work and play. It's a dream come true – and it's proof of concept; you CAN live on what you love doing.
I've always remembered my senior school PE teacher telling me I couldn't run – and for years I believed him. But then I decided to believe in myself instead. Now, I might not be the most elegant runner on the planet, but my legs have carried me 44 miles in one go so far and this month at Maxi Race, I fully intend to take that up to 50. Don't let other people tell you what you can't do.
4) Be generous in your dealings with others.
The smartest and most successful people I know, are those who give credit where it's due; listen at least as much as they speak, help others when they can, and treat other people as they would wish to be treated themselves. And that goes for all people.
5) Be patient with yourself.
If you don't know what you want to be when you grow up, don't force it. Go with it – and in the meantime, keep doing what you love. I never knew what I wanted to do when I grew up. Sometimes, I think I still don't know. But I know that pretty much everything is a transferrable skill: the thinking skills I learned through two degrees in English Literature, I use now to deconstruct business situations and extract marketing insight. Focus on what you know feels right for you, and its time will come.
6) Opening doors is never a bad thing.
Passing exams isn't the be-all and end-all in this life, but on the other hand, there's no good reason not to take the chances that come your way. If you're going to have to be in the exam hall anyway, and you've got the chance to gain a qualification that might just get your foot through a door somewhere way down the line, then there's no reason not to give it your best shot. But don't lose your sense of proportion. Exams aren't the whole picture. They're just one piece of the jigsaw.
7) Don't value tomorrow more highly than today.
Life is about the journey – not the destination, and that's particularly true when you're at school and all the talk is about where you're going next and what you're going to do. Yes, it matters: but today matters too. Never fall into the trap of valuing where you think you're going, over where you know you are. | {
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} | 1,318 |
Inequality and insecurity in UK households
Measurements of, and debates about, economic recovery in the UK have tended to overlook deepening inequality along the lines of class, gender, race, ability, age and sexuality.
Johnna Montgomerie Daniela Tepe-Belfrage Genevieve LeBaron
This is the first in a series of ten articles on the theme of rethinking recovery.
Whether the UK has 'achieved' recovery or not depends on whom you ask, how you phrase the question, and what is at stake for them in providing a particular answer. To say that the very nature of recovery is contested is a rather large understatement. For some, recovery has been achieved because the growth rate of GDP has been restored, although even here less sanguine commentators have pointed out that growth still relies heavily on debt-driven consumer spending, house price inflation and asset bubbles.
For others, a continued dependence on finance-led growth is not recovery because there has not been a rebalancing of the UK economy away from its dependence on finance and services toward manufacturing, nor a revitalisation of sectors of the economy dominated by small and medium-sized enterprises. Indeed, the sheer variety of perspectives and variables used to measure recovery highlights the difficulty of discerning whether the UK is moving out of – or into – a prolonged stagnation.
Largely ignored even by such critical accounts is the reality that political and economic policies associated with 'recovery' in the UK have deepened inequality and exclusion along the overlapping lines of class, gender, race, ability, age and sexuality. Sweeping welfare reforms, for instance, are disproportionately targeting women and low-income couples with children, with particularly dire consequences for single mothers. The newly imposed 'bedroom tax' – which has reduced housing benefits for thousands of tenants, while requiring many thousands more to transfer to smaller homes – has had especially devastating consequences for disabled tenants, who have lost homes adapted to support their disability.
One key reason that these social, financial, and emotional costs of recovery remain hidden is that the typically narrow focus on national level of debts, deficits, taxes and expenditure tends to overlook changes at the micro-level of the household and daily life. It is here that the human costs and broader social challenges of recovery become apparent. For many, rising hunger forces individuals and families to choose between heating and eating; employment has become so precarious and poorly paid that even those with jobs are struggling to pay bills; and still others are trapped in abusive forced labour relations, which have become endemic in certain UK industries.
In short, the UK's economic 'recovery' has come at a high social, emotional and financial cost for those who can least afford it, while leaving the wealthy to stockpile ever-larger sums of cash. We can expect these tendencies to become even more pronounced under the deepening austerity agenda of the new Conservative government, unless a counter-narrative can elucidate the true human costs of this growth model and inspire action towards an alternative.
We believe that an important starting point in developing a coherent critique of austerity lies in documenting and analysing the growing insecurity and inequality spurred by policies designed to achieve economic recovery. This series of posts will contribute to this urgent task. Grounded in an analysis of recovery's hidden costs at the level of everyday life, it will hone in on the profound shifts that austerity is sparking in UK households and in the UK labour market. It is linked to, and supported by, an Economic and Social Research Council seminar series grant, ''From REcovery to DIScovery: Opening the debate on alternatives to financialisation'.
The first half of the series – launched today, and proceeding via four weekly instalments to be published over the next month or so – draws together posts investigating the impact of austerity on households, including pieces on the gendered impact of universal credit, intergenerational inequality, and how the companies in the 'recovery industry' are profiting from deepening poverty and inequality.
The second half of the series will resume in June 2016, again with five weekly instalments, and will tackle the impact of recovery policies on the UK labour market, including posts on forced labour, the impact of recovery policy on wages and working conditions, and deepening gendered and racial inequality within paid and unpaid labour.
The overall aim of the series is to move the political project of economic re-imaginations forward by capturing and analysing everyday consequences of austerity, exposing differentiated experiences of it, and drawing attention to the driving forces of these hidden costs in national and global economic and political policies and logics. The series will enable us not only to understand the scale of developing economic and policy contradictions but also the crisis dynamics themselves and the social and political struggle over the direction of recovery.
Finally, we hope that the series of posts will begin to chart new ways of thinking and even develop some new mid-level policies that move us beyond the failures of austerity. These will include policies that address corporate and labour market governance, as well as wealth redistribution, and will seek to stimulate UK prosperity in a more equitable way.
This article originally appeared on SPERI Comment, the Sheffield Political Economy Research Institute blog.
Austerity and Recovery | {
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Sarah: Are these photos useful, or are they just part of a big happy family charade we're playing or is this technology actually just a way for us to run away from real experience, real despair, real loss?
Posted on June 17, 2015, in Movie Quote of the Day and tagged A Year With Women, Anna Margaret Hollyman, Annie Howell, Lisa Robinson, Small Beautifully Moving Parts. Bookmark the permalink. 1 Comment. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,620 |
HPV vaccine: How to talk to your kids
January 2017 : HPV vaccine: How to talk to your kids
If you're wondering about how to talk about the HPV vaccine with your kids, our expert says treat it like any other vaccine. The HPV vaccine is a safe and effective shot that protects males and females from several types of cancer over their lifetime.
BY Kellie Bramlet
The HPV vaccine prevents infection from strains of HPV that cause cervical, vulvar, vaginal, penile, anal and oropharyngeal (throat) cancers. It also protects against strains that cause genital warts.
All males and females ages 9–26 should get the HPV vaccine. It is most effective when given at ages 11–12. Unvaccinated men and women ages 27–45 can also get the HPV vaccine and should talk to their doctor about the benefits of the vaccine.
But some parents are hesitant to get their children vaccinated because they associate the HPV vaccine with sexual activity or are scared of potential side effects.
"Getting them the HPV vaccine doesn't mean you're encouraging your child to become sexually promiscuous or even sexually active," says Lois Ramondetta, M.D., professor in Gynecologic Oncology and Reproductive Medicine at MD Anderson. "Getting the HPV vaccine is a no-brainer."
She recommends that parents follow these tips when talking to their kids about the HPV vaccine.
Leave sex out of the conversation if you're not comfortable talking about it.
"A conversation about the vaccine doesn't have to be a conversation about sex," Ramondetta says.
HPV is a sexually transmitted infection. It is so common that almost 80% of men and women get the disease at some point. It can be spread through vaginal or oral sex as well as intimate skin-to-skin contact and kissing. Although most cases of HPV clear up on their own, some can stay in the system and cause genital warts or cancer.
When discussing the HPV vaccine with your children, it's important to stress that the vaccine is for cancer prevention over the course of their lifetime.
"Tell your kids that the shot will keep them from getting an HPV infection and is expected to help protect them from up to six types of cancer," she says.
The HPV vaccine should be treated just like any other vaccine.
Lois Ramondetta, M.D.
Tell your children this is just one more shot they need to get along with their other vaccines.
"The HPV vaccine should be treated just like any other vaccine," Ramondetta says.
Talk to your child's doctor about including the shot with the regular vaccines for your child at this age. If it's not offered, bring it up.
The Centers for Disease Control and Prevention, the Advisory Committee on Immunization Practices, the American Academy of Pediatrics and many others recommend the shot be given along with the Tdap vaccination (which protects against tetanus, diphtheria and whooping cough) and the meningococcal B vaccine (the vaccine for meningitis) at age 11-12 in all boys and girls.
Remind your children that the shot is safe and effective.
"Some parents and their children may worry about the possibility of side effects," says Ramondetta. "But the HPV vaccine has been proven safe and effective."
The most commonly reported side effects from the vaccine are mild redness and swelling at the site of the shot. There is no evidence that links the vaccine to adverse or severe reactions.
"Those very minor risks associated with the vaccine are well worth it," Ramondetta says. "This vaccine could help eliminate certain typse of cancer, starting with cervical cancer."
HPV vaccine: Help your kids prevent cancer
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Phytochemicals are compounds in plant foods that can help prevent chronic diseases like cancer.
When setting diet and exercise goals, be SMART
A good goal-setting strategy is the SMART goal checklist. SMART goals are specific, measurable, attainable, realistic and time-bound. | {
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} | 8,564 |
{"url":"https:\/\/quant.stackexchange.com\/questions\/57608\/how-to-compute-par-yield-from-zero-rate-curve","text":"# How to compute par yield from zero rate curve?\n\nHow does one calculate the below two-year par yield given the zero rate curve: Assume the following two-year zero rate curve, with continuous compounding:\n\n2.0% @ 0.5 year\n2.5% @ 1.0 year\n3.0% @ 1.5 years\n3.5% @ 2.0 years\n\n\nWhat is the two year par yield?\n\nI know we can use the zero rate curve to derive a theoretical price of the bond but don't know how that can be used for calculating the par yield.\n\nFor simplicity, let us assume continuously compounded zero rates and periodically compounded par yields. If you have to work with continuous rates, you may adapt the formulas accordingly.\n\nUsing the zero rate discount factors $$D(T) \\equiv e^{-r(T)T}$$, the present value of a coupon bearing bond is\n\n$$$$PV=\\sum_i^N c D(t_i) + D(t_N)$$$$\n\nFor a coupon bearing bond, we can relate the coupon rate of a par bond (!) to the yield structure as:\n\n\\begin{align} 100\\%=&\\sum_i^N \\frac{c}{(1+y_T)^{t_i}} + \\frac{1}{(1+y_T)^{t_N}}\\\\ \\Leftrightarrow c=&y_T \\end{align}\n\nIf your bond pays at semi-annual frequency, then $$y_T$$ is the corresponding semi-annual yield rate, and your annual yield would of course be $$\\tilde{y}=(1+y)^2-1$$.\n\nWe have thus established that the coupon rate of a par bond reflects the yield information. Thus, all you now need to do is to find coupon rates such that your hypothetical bonds are priced at par:\n\n\\begin{align} 1&=0.5c_T\\sum_i^N D(t_i) + D(t_N)\\\\ \\Leftrightarrow c_T&=2\\frac{1-D(t_N)}{\\sum_i^N D(t_i)} \\end{align}\n\nIn your case, the discount factors are \\begin{align} D(t_{0.5})=e^{-0.5*0.020}&=0.9900498\\\\ D(t_{1.0})=e^{-1.0*0.025}&=0.9753099\\\\ D(t_{1.5})=e^{-1.5*0.030}&=0.9559975\\\\ D(t_{2.0})=e^{-2.0*0.035}&=0.9323938\\\\ \\end{align}\n\nAnd hence your semi-annual coupons are $$y_{0.5}=0.02010033$$, $$y_{1.0}=0.02512526$$, $$y_{1.5}=0.03012471$$, $$y_{2.0}=0.03508591$$\n\nFor the annualised yields, we then obtain\n\n$$$$y_{ann,i}=(1+0.5*c_i)^2-1$$$$\n\nAs a sanity check, you may want to compute the present value of a 2-year bond using the coupon of 3.508591% (annual) and the corresponding yield of 3.539366% (annualised).\n\nHTH\n\nPS: ultimatively, you can also invert the \u201astandard\u2018 boot strapping equation:\n\n$$$$\\frac{1-D(t_N)}{\\sum_i^N D(t_i)}$$$$\n\nin order to quickly arrive at the par rates.","date":"2021-04-20 01:35:01","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\": 13, \"wp-katex-eq\": 0, \"align\": 3, \"equation\": 3, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9233703017234802, \"perplexity\": 3267.5545922062047}, \"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-2021-17\/segments\/1618038921860.72\/warc\/CC-MAIN-20210419235235-20210420025235-00058.warc.gz\"}"} | null | null |
The Gillespie Street-Clinton River Bridge is a bridge carrying Gillespie Street over the Clinton River in Pontiac, Michigan. It is a relatively early example of a rigid-frame bridge in Michigan. The bridge was listed on the National Register of Historic Places in 2000.
History
This bridge was apparently the first at this location. In 1936, the city of Pontiac deepened the Clinton River and extended Gillespie Street. They contracted with consulting engineer Harold Hawley Corson to design this bridge, who was serving as Birmingham's city engineer after a stint with the Michigan State Highway Department. Corson designed this rigid frame bridge, then a relatively new type of bridge which had been introduced in Michigan in the early 1930s.
Description
The Gillespie Street bridge is a rigid-frame bridge with shallow spandrels ornamented with recessed panels. The bridge is 34 feet long, spanning a 33-foot-wide channel, and 50.5 feet wide. The railings are simple metal panels, ending in chain link fencing. The roadway is 36.5 feet wide, with sidewalks on each side.
References
External links
Gillespie Street-Clinton River Bridge at Historic Bridges.org
National Register of Historic Places in Oakland County, Michigan
Bridges completed in 1936 | {
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Q: auto hides keyboard after scrolling ListView on android Im new on android please help me to auto hide after scrolling the listview here is my code but could not get right solution
xml file :
<ListView
android:id="@+id/offline_list"
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:background="#dde1e3"
android:clickable="true"
android:focusable="true"
>
</ListView>
code:
lvCustomList.setOnFocusChangeListener(new View.OnFocusChangeListener() {
@Override
public void onFocusChange(View v, boolean hasFocus) {
// TODO Auto-generated method stub
if(!hasFocus)
hideKeyboard(v);
}
private void hideKeyboard(View view) {
// TODO Auto-generated method stub
InputMethodManager inputMethodManger = (InputMethodManager)getSystemService(Activity
.INPUT_METHOD_SERVICE);
inputMethodManger.hideSoftInputFromWindow(view.getWindowToken(), 0);
}
});
A: Try this..
why don't you use OnTouchListener for ListView like below
lvCustomList.setOnTouchListener(new OnTouchListener() {
@Override
public boolean onTouch(View v, MotionEvent event) {
InputMethodManager imm = (InputMethodManager) getSystemService(Context.INPUT_METHOD_SERVICE);
imm.hideSoftInputFromWindow(edittext.getWindowToken(), 0);
return false;
}
});
A: It is better to use onScrollStateChanged instead of onScroll and by using scrollState == 0. So keyboard will hide when user is really scrolling.
listview.setOnScrollListener(new AbsListView.OnScrollListener() {
@Override
public void onScrollStateChanged(AbsListView view,
int scrollState) {
if (scrollState == 0) {
InputMethodManager inputMethodManger = (InputMethodManager) getActivity().getSystemService(Activity
.INPUT_METHOD_SERVICE);
inputMethodManger.hideSoftInputFromWindow(view.getWindowToken(), 0);
}
}
@Override
public void onScroll(AbsListView view, int firstVisibleItem,
int visibleItemCount, int totalItemCount) {
}
});
A: Recyclerview,to hide the keyboard on scrolling - SCROLL_STATE_DRAGGING ( Mentioned by @doubleA )
public void onScrollStateChanged(@NonNull RecyclerView recyclerView, int newState) {
super.onScrollStateChanged(recyclerView, newState);
if(newState == RecyclerView.SCROLL_STATE_DRAGGING){
InputMethodManager imm = (InputMethodManager) recyclerView.getContext().getSystemService(Context.INPUT_METHOD_SERVICE);
imm.hideSoftInputFromWindow(recyclerView.getWindowToken(), 0);
}
}
A: recyclerview.addOnScrollListener(new RecyclerView.OnScrollListener() {
@Override
public void onScrolled(@NonNull RecyclerView recyclerView, int dx, int dy) {
super.onScrolled(recyclerView, dx, dy);
//Hide keyboard code
InputMethodManager imm = (InputMethodManager) activity.getSystemService(Activity.INPUT_METHOD_SERVICE);
//Find the currently focused view, so we can grab the correct window token from it.
View view = activity.getCurrentFocus();
//If no view currently has focus, create a new one, just so we can grab a window token from it
if (view == null) {
view = new View(activity);
}
imm.hideSoftInputFromWindow(view.getWindowToken(), 0);
}
});
A: Try this one.
listview.setOnScrollListener(new OnScrollListener() {
@Override
public void onScrollStateChanged(AbsListView view,
int scrollState) { // TODO Auto-generated method stub
}
@Override
public void onScroll(AbsListView view, int firstVisibleItem,
int visibleItemCount, int totalItemCount) {
InputMethodManager inputMethodManger = (InputMethodManager)getSystemService(Activity
.INPUT_METHOD_SERVICE);
inputMethodManger.hideSoftInputFromWindow(view.getWindowToken(), 0);
}
});
Hope it helps. Cheers!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,009 |
Alfred Bütow (* 28. Juli 1902 in Berlin; † 3. April 1986 in Herrsching am Ammersee) war ein deutscher Szenenbildner bei Film und Fernsehen.
Leben und Wirken
Bütow hatte eine Rundumausbildung zum Bühnenbildner, Gebrauchsgraphiker, Schaufenster- und Innendekorateur erhalten. Von 1916 bis 1919 durchlief er eine Lehrzeit als Theatermaler, besuchte dann die Kunstgewerbeschule und Fachschulkurse für perspektivische Architektur und Stilkunde. Zwischen 1919 und 1924 war er beim Theaterausstatter Hugo Baruch & Cie. als Theatermaler angestellt, anschließend ging Bütow als Montagearbeiter in die Niederlande. Von 1927 bis 1931 besaß er sein eigenes Atelier für Theaterkulissen. 1932 wechselte Bütow als Bühnenbildner nach Berlin, wo er unter anderem für das Lessing-Theater, das Metropol-Theater und die Ostdeutschen Landestheater tätig werden sollte.
Im Winter 1933/1934 sammelte er erste filmische Erfahrungen als Kunstmaler, 1934 debütierte er als Filmarchitekt. Zunächst entwarf Alfred Bütow Kulissen im Auftrage kleinerer Produktionsfirmen. 1936 designte er erstmals die Dekorationen für einen Film des Produzenten und Regisseurs Richard Eichberg. Mit dessen Zweiteiler Der Tiger von Eschnapur und Das indische Grabmal im Jahr darauf wurde Bütow mit einer aufwendigen, ausstattungsprächtigen Großproduktion betraut, gleich danach realisierte er die Ausstattung für einige Rühmann-Lustspiele. Von 1939 bis 1945 kooperierte er häufig mit dem Kollegen Heinrich Beisenherz. Bütows Nachkriegsfilme, deren Designs er mehrfach in Zusammenarbeit mit Ernst Schomer realisierte, sind überwiegend bedeutungslos. 1957 verabschiedete sich Alfred Bütow von der Kinofilmdekoration und begann sich dem Fernsehen zuzuwenden. Bereits seine erste, 1958 gedrehte Großproduktion, der stark beachtete Mehrteiler So weit die Füße tragen, wurde ein enormer Publikumserfolg.
Filmografie
1934: Herz ist Trumpf
1935: Petersburger Nächte. Walzer an der Newa
1935: Der Vogelhändler
1936: Der Kurier des Zaren
1936: Inkognito
1936: Moskau – Shanghai
1936: Annemarie
1937: Alarm in Peking
1937: Der Tiger von Eschnapur
1937: Das indische Grabmal
1938: Fünf Millionen suchen einen Erben
1938: Diskretion Ehrensache
1938: In geheimer Mission
1938: Liebelei und Liebe
1938: Nanu, Sie kennen Korff noch nicht?
1939: Ehe in Dosen
1939: Dein Leben gehört mir
1939: Hurra! Ich bin Papa!
1940: Lauter Liebe
1940: Die lustigen Vagabunden
1940: Wunschkonzert
1940: Spähtrupp Hallgarten
1941: Frau Luna
1941: Was geschah in dieser Nacht?
1942: Maske in Blau
1943: Karneval der Liebe
1943: Die beiden Schwestern
1943: Die Affäre Roedern
1944: Das fremde Leben
1945: Heidesommer (unvollendet)
1948: Eine reizende Familie
1949: Gesucht wird Majora
1949: Madonna in Ketten
1949: Hochzeit mit Erika
1951: Der Tiger Akbar
1952: Das Bankett der Schmuggler (Le Banquet des fraudeurs)
1952: Die Diebin von Bagdad
1952: Fritz und Friederike
1952: Ich hab' mein Herz in Heidelberg verloren
1953: Wenn der weiße Flieder wieder blüht
1953: Weg ohne Umkehr
1954: Staatsanwältin Corda
1954: Ball der Nationen
1954: Das zweite Leben
1954: Die goldene Pest
1955: Vom Himmel gefallen (Special Delivery)
1955: Rumpelstilzchen
1955: Reifende Jugend
1955: Viele kamen vorbei
1956: Von der Liebe bewegt
1956: Der Mustergatte
1957: Zwei Herzen voller Seligkeit
1957: Die Freundin meines Mannes
1957: Rosalinde (TV)
1959: So weit die Füße tragen (TV-Mehrteiler)
1960: Am grünen Strand der Spree (TV-Mehrteiler)
1960: Langusten (TV)
1961: Inspektor Hornleigh greift ein… (TV-Serie)
1961: Das Kartenspiel (TV)
Literatur
Kay Weniger: Das große Personenlexikon des Films. Die Schauspieler, Regisseure, Kameraleute, Produzenten, Komponisten, Drehbuchautoren, Filmarchitekten, Ausstatter, Kostümbildner, Cutter, Tontechniker, Maskenbildner und Special Effects Designer des 20. Jahrhunderts. Band 1: A – C. Erik Aaes – Jack Carson. Schwarzkopf & Schwarzkopf, Berlin 2001, ISBN 3-89602-340-3, S. 627.
Weblinks
Szenenbildner
Deutscher
Geboren 1902
Gestorben 1986
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,299 |
{"url":"https:\/\/eaforum.issarice.com\/posts\/ahr8k42ZMTvTmTdwm\/how-good-is-the-humane-league-compared-to-the-against","text":"# How good is The Humane League compared to the Against Malaria Foundation?\n\npost by smclare, AidanGoth \u00b7 2020-04-29T13:40:38.361Z \u00b7 score: 63 (27 votes) \u00b7 EA \u00b7 GW \u00b7 25 comments\n\n## Contents\n\n 1. Summary\n2. Motivation\n3. Theoretical Framework\nWhy this is hard\nMoral weight vs. moral status\nCandidates for key variables that determine moral weight\n4. Methodology\nInputs\n5. Results\nWhere are the inflection points?\nWhat do our median and expected value estimates suggest?\n6. Discussion\nLearnings\nWould future work be useful?\nNone\n\n\nThanks to the experts with whom we spoke while researching this report!\n\nThis was written by Stephen Clare and Aidan Goth.\n\n# 1. Summary\n\n\u2022 In order to help us prioritize Founders Pledge\u2019s research and advising efforts across cause areas, we created a rough model that tries to compare an animal welfare intervention (The Human League\u2019s cage-free campaigns) to a global health intervention (AMF\u2019s bednet distribution).\n\u2022 In assessing the impact of THL\u2019s intervention, we identified three main sources of uncertainty:\n\u2022 How many animal years are affected by the intervention\n\u2022 How much the intervention improves each animal\u2019s subjective well-being\n\u2022 How much an animal\u2019s well-being matters compared to a human\u2019s (the moral weight)\n\u2022 The last factor is really tough to figure out. There are good reasons to think the weight might be quite high, and good reasons to think it might be very low. That means the range of our moral weight estimates spans multiple orders of magnitude.\n\u2022 For this report, we made spreadsheet and Guesstimate models that compare The Humane League to the Against Malaria Foundation for a range of different assumptions about the above uncertain factors.\n\u2022 Importantly, we assumed hedonism (sentient experience is all that matters morally), that chickens have moral status (their experience matters morally), and anti-speciesism (the value of an experience is independent of the species of animal that is experiencing it). Accordingly, this analysis does not offer an all-things-considered view on the relative goodness of THL and AMF \u2013 it assumes a particular worldview that is relatively favourable to THL.\n\u2022 In this model, in most of the most plausible scenarios, THL appears better than AMF. The difference in cost-effectiveness is usually within 1 or 2 orders of magnitude. Under some sets of reasonable assumptions, AMF looks better than THL. Because we have so much uncertainty, one could reasonably believe that AMF is more cost-effective than THL or one could reasonably believe that THL is more cost-effective than AMF.\n\u2022 In general, if you value human well-being >10,000 times more than chicken well-being, AMF looks better. If you value human well-being <300 times more than chicken well-being, THL looks better. But between these moral weights the ranking is less clear. We think there\u2019s a good chance (at least 50%) that the moral weight falls between these bounds, where factors like THL\u2019s effectiveness and the badness of battery cages are more important.\n\u2022 It\u2019s very likely that we\u2019re missing key considerations that could change our estimates by orders of magnitude. For example, we haven\u2019t tried to account for moral uncertainty, indirect effects of the interventions or longtermist considerations.\n\n# 2. Motivation\n\nWe have to decide how to allocate limited resources between the best human charities and the best animal charities. We work out the trade-offs involved explicitly or make them implicitly, but we have to somehow decide.\n\n# 3. Theoretical Framework\n\n### Why this is hard\n\n\u2022 Our epistemic status is weak. It\u2019s hard to know what the relevant variables are, much less how they vary among different animals. We do not have access to the internal experience of other animal species. We have to try and approximate what it\u2019s like based on observations about their behavior, reflecting on our own experience, and considering the different biological and philosophical factors that seem likely to shape consciousness\n\u2022 Animal-focused interventions are less well studied than human-focused interventions, so we know much less about how effective interventions to improve animal welfare are.\n\u2022 So there are (at least) three sources of uncertainty:\n\u2022 How many animal years are affected by the interventions\n\u2022 How much do the interventions improve each animal\u2019s subjective well-being\n\u2022 How much does an animal\u2019s well-being matter compared to a human\u2019s?\n\n### Moral weight vs. moral status\n\n\u2022 A moral weight suggests how much we should value the welfare of an animal in a given species relative to an animal from a different species. These are usually defined relative to humans, i.e. humans are given a moral weight of 1.\n\u2022 Moral weights are different to an animal\u2019s moral status, i.e. whether its welfare matters to us. Here we focus on moral weights, which relate more to a creature\u2019s \u2018capacity for welfare.\u2019 For farm animals, Muehlhauser\u2019s 2017 report, the most in-depth treatment of this question we found, assigns chickens an 80% chance of moral status\n\n### Candidates for key variables that determine moral weight\n\nSome of the variables we came across in researching moral weights are:\n\n\u2022 Clock speed of consciousness (suggested by Muehlhauser [LW \u00b7 GW])\n\u2022 Smaller animals have faster reaction times (e.g. imagine trying to swat a fly).\n\u2022 To the extent those reactions are under conscious control, smaller animals would experience more subjective moments per unit of objective time\n\u2022 When measuring well-being, it seems very likely we should care about the subjective length of experience more\n\u2022 This updates us towards valuing smaller animals more\n\u2022 Experience intensity (Muehlhauser)\n\u2022 More \u2018intense\u2019 experiences seem like they should matter more\n\u2022 Human experiences seem like they\u2019re probably more intense. But experiences of other species may be more intense - for example, it\u2019s not clear whether our \u2018linguistic thoughts\u2019 make our experience more or less intense\n\u2022 Unity of consciousness (Muehlhauser)\n\u2022 Refers broadly to how various different conscious inputs are combined into a single experience. E.g. conscious states over time all belong to some persistent \u201cself\u201d (subject unity) and contents of conscious states are unified (representational unity and phenomenal unity)\n\u2022 Debatable how much this influences moral weight, but subject and phenomenal unity relate to subjective experience which seems relevant to moral weight. Humans probably have more unity of consciousness than animals.\n\u2022 Brain size\/complexity (Tomasik)\n\u2022 Brian Tomasik is very uncertain, but weakly suggests a view where moral weight scales non-linearly with brain size. As a rough approximation he suggests scaling by N^(2\/5), where N is the number of neurons the animals has.\n\u2022 However some evidence suggests [EA \u00b7 GW] that the correlation between cognitive sophistication and neuron count seems weak.\n\n# 4. Methodology\n\nWe made a spreadsheet model that generates cost-effectiveness estimates for many different plausible values and noted where the key considerations lie. We only estimated a model to compare our most popular global health intervention (AMF) to our top animal welfare recommendation (The Humane League\u2019s campaigns for aviaries instead of battery cages for egg-laying hens). Our model is:\n\nwhere\n\nnumber of 'hen-years' spent in aviaries rather than battery cages\n\nbenefits of moving from a battery cage to an aviary\n\nmoral weight of chickens\n\nhow many DALYs are averted by $1 to AMF ### Inputs Effectiveness of THL campaigns in changing corporate behaviour The FP report on animal welfare (published 2018) estimates that THL moves 10 hen-years from battery cages to aviaries per dollar donated. This number assumes there is a 60% probability that companies honour their cage-free commitments and that THL\u2019s advocacy brought these pledges forward by between half a year and one year on average. Simcikas (2019) [EA \u00b7 GW]\u2019s estimates of corporate campaign effectiveness are higher, though not all corporate campaigns relate to battery cages and aviaries specifically. His upper bound is that 160 hen-years are affected per dollar, with a median estimate of 54. Similarly, Bollard (2016) suggests corporate campaigns spare at least 38 hen years from battery cages per dollar. On the other hand, Bollard (2019) documents that some companies have delayed or reneged on their pledges. While he remains optimistic, this indicates the need for ongoing campaigning to ensure pledges are fulfilled which would raise the expected cost. We think the 2018 FP estimate of 10 hen-years\/$ is likely a slight underestimate. Across the different tabs on the spreadsheet, we model four scenarios: 1, 10, 30 and 100 hen-years affected per dollar.\n\nBenefits of moving from battery cages to aviaries\n\nWe measure the benefits of moving from battery cages to aviaries by estimating how bad battery cages are and then estimating how bad aviaries are as a proportion of the badness of battery cages.\n\nBattery cages are plausibly extremely bad. Hens are kept in tiny spaces, live on wire racks, are unable to move around, and are kept from engaging in natural behaviors such as rooting, preening, and socializing. Pages 20 through 23 of our report have more detail. In aviaries, birds are still kept in quite cramped conditions. However, birds have up to 80% more space, access to litter and perches, can move around, and can engage in more of their preferred behaviors. However, there is some evidence that the rate of hen mortality is higher in aviaries. Due to this, the FP report estimates there is a 5-10% chance that aviaries are worse than battery cages. OpenPhil\u2019s \u201ccurrent \u2013 though uncertain \u2013 best guess is that even without additional reforms, the U.S. transition to cage-free housing systems will on net reduce hen suffering once mortality rates have stabilized.\u201d\n\nThere have been a couple other attempts at quantifying this welfare change. Charity Entrepreneurship\u2019s weighted animal welfare index gives battery cages a score of -57 out of -100. As of April 2020, this is the worst score on their scale. In Compassion, by the Pound, F. Bailey Norwood gives caged egg-laying hens a welfare score of -8 (again, the worst score on his scale) and cage-free egg-laying hens a score of +2 (see Table 8.2 on p. 229).\n\nWe think that life in a battery cage is very likely to have a negative value - i.e. we conceptualize their badness as a negative multiplier x of 1 unit of healthy time. Plausible estimates of x could vary across several orders of magnitude. Battery cages could be subjectively unpleasant, in which case x would be close to 0. Battery cages could also be truly horrific, with hens spending their entire lives in extreme distress and pain. In that case we think battery cages could be -100 or -1000, meaning 100 to 1000 weeks of cage-free life would be morally cancelled-out by 1 week or 1 year in a cage.\n\n(The existence of a unit of \u201chealthy life\u201d as an upper bound to human welfare is common in analyses based on Quality and\/or Disability Adjusted Life Years. However, we recognise that there are varying levels of \u201chealthy life\u201d, some of which are better than others, and that well-being might not be bounded in this way. Defining a unit of positive well-being is important to our model because the negative units of well-being are defined in terms of trade-offs with positive units of well-being. We suggest interpreting \u201chealthy life\u201d for a chicken as living with all needs met, no or minimal fear of predation and disease-free (e.g. perhaps the best moments on a very good farm animal sanctuary) and defining one unit of well-being as one instant of healthy life, understood in this way. We\u2019re uncertain about whether this is the best way to formulate our model).\n\nOur median estimates, made independently, were between -10 and -30, but the bounds of our spreadsheet model are -0.1 to -1000. Since battery cages could plausibly be extremely bad (e.g. well-being level of around -1000), we think that the expected well-being of life in a battery cage is lower than the median.\n\nThe spreadsheet model is not intended to have a probabilistic interpretation, so these bounds do not represent a specific confidence interval. In the first instance, the spreadsheet model shows how much better or worse donating to THL is than AMF given various assumptions without committing to a judgement about how likely it is that those assumptions are true. In particular, the model can help to identify points at small changes in assumptions change which charity looks more cost-effective. This can sometimes be sufficient for making decisions.\n\nWhile we are concerned by the data showing increased mortality rates in aviaries, we do not model the scenario in which aviaries are worse for hens than battery cages in our non-probabilistic model because the cost-effectiveness would, of course, be infinitely worse than AMF. In our probabilistic model, we use a probability density function (pdf) that puts some weight (5-10%) on the possibility that aviaries are worse than battery cages.\n\nMoral weight: how much does a hen\u2019s suffering matter compared to a human\u2019s?\n\nThese estimates assume hedonism (all that matters morally is conscious experience of pleasure and suffering), that chickens have moral status (their experience matters morally), and anti-speciesism (the moral value of an experience depends only on the quality of the experience, not on the species of animal who experiences it).\n\nNot much research has been done on moral weights and most people are reluctant to give even speculative estimates, so we rely pretty heavily on Luke Muehlhauser\u2019s work [LW \u00b7 GW] (while recognizing its limitations). Extreme uncertainty entails extremely wide bounds, especially because some considerations push in different directions. Some factors suggest animals have more weight while others suggest the opposite. This is why we consider what would happen throughout the range of plausible estimates.\n\nOur model has moral weights ranging from to (i.e. from 1 human = 1 million chickens to 1 human = 1 chicken). We think it\u2019s very likely that if animals have moral status, their moral weight is not vanishingly small (say, less 1 in 1 million).\n\nBenefits of AMF\n\nWe use GiveWell\u2019s updated estimate of AMF\u2019s cost-effectiveness as a point estimate, i.e. ~$1,700 per outcome as good as saving the life of a child under 5. We assume that saving a life is worth about 50 DALYs. These estimates have a margin of error, but since they\u2019re unlikely to be wrong by an order of magnitude that shouldn\u2019t affect our findings too much. # 5. Results In our spreadsheet model, the numbers in the cells show how many orders of magnitude better THL is than AMF. So if the number is black (positive), THL is better. If the number is white (negative), AMF is better. The columns are moral weight values, and show how chicken experience is valued relative to human experience. The lower bound is , i.e. 1 million chickens to 1 human; the upper bound is 1. The rows measure how good moving from a battery cage to an aviary is for a chicken. This has two layers. First, there\u2019s a range for the badness of battery cages, from -0.1 (close to indifference between life and death), to -1,000 (extreme torture, 1000 days of battery cage life outweighs 1 day of healthy life). Second, there\u2019s a range for how much better aviaries are, ranging from 0.7 (70% as bad as battery cages) to -.3 (a life worth living, 30% as good as battery cages are bad). In the different tabs, we replicate this spreadsheet for different estimates of THL\u2019s effectiveness. We also translated these inputs to Guesstimate to get an expected value. We describe the Guesstimate model at the end of this section. ## Where are the inflection points? \u2022 If you think the moral weight of chickens is less than 1\/10,000, then AMF is better than THL unless battery cages are extremely bad \u2022 If you think the moral weight of chickens is more than 1\/100, then THL is better than AMF unless THL is very ineffective and battery cages aren\u2019t that bad \u2022 If you think battery cages are very bad (-100 or worse), then THL is better than AMF unless chickens have a very low moral weight (<1\/10,000) or aviaries are as bad or worse than battery cages \u2022 If the FP estimate of THL is a little bit pessimistic and THL\u2019s effectiveness is closer to Saulius\u2019 estimates, then THL is usually better than AMF unless battery cages are not that bad (>-1) or chickens have an extremely low moral weight (<1\/100,000) \u2022 It rarely matters how much better aviaries are than battery cages, assuming aviaries are at least 30% better than battery cages For each scenario, we have graphed the line along which AMF and THL are equally cost-effective given various assumptions. Above this line, our model suggests a donation to AMF is better; below the line, THL is better. The y-axis is the inverse of the moral weight, i.e. the number of chickens equal to one human, and the x-axis shows the (negative) momentary well-being of life in a battery cage. We plotted graphs separately for different assumptions about how many hen-years THL affects per dollar, with two graphs per scenario. One plot includes values of battery cage badness from -1 to -1000 with logarithmic axes and one plot that zooms in on 0 to -30 with linear axes. The different lines represent different assumptions about how bad life in an aviary is compared to life in a battery cage. Here we include the plots for the scenario in which 30 hen-years are affected per dollar donated to THL. We have plots for other scenarios of THL\u2019s effectiveness in this ibb album. ## What do our median and expected value estimates suggest? Caveats: \u2022 These estimates are speculative and not stable \u2022 A combination of median estimates is not the same as the overall best guess, e.g. , so combine estimates with care \u2022 There are potentially important factors for which our model doesn\u2019t account (e.g. rich meat eaters, non-hedonistic considerations, variable moral status by species, effects of corporate campaigns on the number of chickens that exist) Our low-confidence median parameter estimates: \u2022 THL affects 30 hen-years per$\n\u2022 Life in battery cages is worth around -10 to -30 units\n\u2022 Life in aviaries is probably a bit better than battery cages, but with lots of uncertainty. When we translate our credence to Guesstimate, the model gives an expected value of aviaries being ~50% as bad as battery cages\n\u2022 Human experience is about 300-500 times more valuable than chicken experience. Our distribution would place >10% probability on each order of magnitude between 10x and 10,000x, with tails stretching out to 1M and 1\/10 (i.e. chicken experience worth more than humans, due to clock speed and experience intensity)\n\nA Guesstimate model with these estimates suggests that:\n\n\u2022 In expectation, THL is >100x better than AMF\n\u2022 In the median scenario, THL is about 2-4x more cost-effective than AMF\n\u2022 A 71% chance that THL is more cost-effective than AMF (calculations here)\n\nThe expected moral weight in our guesstimate model is about 0.03 (chicken experience is ~30x less valuable than human experience), which might seem very high. However, note that (1) we assume moral status and (2) if one thinks there is some probability that the moral weight is one, then there is a lower bound to the expected moral weight. If, as we do here, one assumes hedonism and that chickens have moral status, then we think that it is difficult to rule out the chance that humans and chickens have equal moral weights. As a result, we would expect a relatively high moral weight in expectation. There may be other reasons for caring about human experience more than chicken experience such that an all things considered view would be less favourable to chickens. We have not taken such considerations into account in this analysis.\n\nWe should note some other limitations of the Guesstimate model:\n\n\u2022 It does not seem stable (e.g. the numbers change if you refresh the page and get a new sample)\n\u2022 The bounds and distribution you choose for the badness of battery cages and the moral weight have big effects\n\u2022 Specifying probability distributions that accurately represent our credences for many of these variables is really hard\n\u2022 Our pdf for the moral weight is neither lognormal nor normal and we weren\u2019t sure how best to represent it, so we\u2019ve had to fudge the bounds a bit to get a reasonable approximation\n\u2022 Our pdf for how bad aviaries are has some weight that they\u2019re worse than battery cages, but also some weight that they\u2019re much better. We\u2019ve fudged the bounds to get an ~8% chance aviaries are worse than battery cages\n\u2022 We\u2019re not sure if we should be calculating the ratio of the expected cost-effectiveness of each charity, or the expected ratio of the cost-effectiveness of each charity. But this doesn\u2019t seem to matter much\n\u2022 Nevertheless, we\u2019ve played around with the bounds and distributions for key parameters and under most reasonable assumptions, THL is expected to be >100 times better than AMF\n\u2022 Although according to our model THL is much more cost-effective than AMF in expectation, the probability that THL is more cost-effective than AMF is relatively low, at 0.71\n\n# 6. Discussion\n\n### Learnings\n\nWe think THL is more cost-effective in expectation than AMF given certain reasonable assumptions, but due to high uncertainty we don\u2019t think that our models offer strong evidence for this claim in general.\n\nGiven our current (lack of) understanding of animal sentience and suffering, one could reasonably believe that AMF is more cost-effective than THL or one could reasonably believe that THL is more cost-effective than AMF, even given our THL-friendly assumptions.\n\n### Would future work be useful?\n\n\u2022 The effect of AMF on animals - i.e. the Meat Eater Problem.\n\u2022 How to introduce moral uncertainty into the model. We assume hedonism, but there are other moral views on which we have some probability that would produce very different estimates (e.g. deny chickens moral status).\n\nIt seems unlikely that we\u2019ll learn about more variables that contribute to moral weights without scientific advances in our ability to understand consciousness. However, a better understanding of how bad aviaries are relative to battery cages could be valuable.\n\nDespite the significant limitations of this work, we still think it's important to try and make cross-cause comparisons. We'd welcome any feedback on how to interpret the results or improve our approach to shed more light on this question!\n\ncomment by saulius \u00b7 2020-04-30T15:04:44.391Z \u00b7 score: 21 (8 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThis is another nitpick but I just want to say this to prevent slightly incorrect information from spreading.\n\nWe use GiveWell\u2019s updated estimate of AMF\u2019s cost-effectiveness as a point estimate, i.e. ~$1,700 to save the life of a child under 5. When you look at GiveWell\u2019s latest estimates, you can see that the cost per outcome as good as averting the death of a child under 5 is ~$1,700. It costs around $3,710 to avert a death of a child under 5. comment by smclare \u00b7 2020-05-01T09:09:50.902Z \u00b7 score: 3 (2 votes) \u00b7 EA(p) \u00b7 GW(p) Nice catch, thanks for the careful read Saulius. I think this is especially important because it means that moral weight considerations creep into our measure of AMF's cost-efficiency even before we try to compare them to THL. GW currently assigns the same value to averting under-5 and age 5+ deaths (100 units), so that's convenient. I'd guess the \"Cost per outcome as good as\" cell also factors in other benefits from reduced morbidity? comment by saulius \u00b7 2020-05-01T11:18:59.164Z \u00b7 score: 8 (4 votes) \u00b7 EA(p) \u00b7 GW(p) I don\u2019t fully understand GiveWell\u2019s spreadsheet myself but I\u2019ll try to answer. By default, \"Cost per outcome as good as\" cell seems to factor in averting under-5 deaths (46% of the total benefit), averting age 5+ deaths (27%) and development effects (28%). Developmental effects here seem to refer to the fact that reducing the burden of malaria may have a lasting impact on children's development, and thus on their ability to be productive and successful throughout life. In the \u2018results\u2019 tab, you see that by default, the estimation doesn\u2019t include additional adjustments. If you change that, then the estimate takes into account the effects listed in the \u201cInclusion\/Exclusion\u201d sheet (see below) It also takes into account something but I haven\u2019t figured out what. In the end including additional adjustments changes \"Cost per outcome as good as\" very modestly, from$1,690 to $1,678. comment by saulius \u00b7 2020-05-01T11:40:51.626Z \u00b7 score: 16 (7 votes) \u00b7 EA(p) \u00b7 GW(p) Note that according to WHO, in 2018 there were 228 million cases of malaria worldwide resulting in an estimated 405,000 deaths. So for every lethal case, there were 405,000 \/ 228 million = 563 non-lethal cases. AMF founder said that bednets prevent these non lethal cases as well. I don\u2019t know how much suffering an average case of malaria causes but the combined effect is probably significant. Especially when we take into account some of the complications that sometimes arise from malaria. For example, GiveWell claims that \u201cIt is also believed that malaria can cause permanent disability (hearing impairment, visual impairment, epilepsy, etc.)\u201d. An old Giving What We Can report says \u201cour model suggests that the distribution of long-lasting insecticide treated bednets averts one case of epilepsy for about$25,000.\u201d Note that it is not only difficult to live with epilepsy, but it\u2019s also difficult and stressful to raise a child that has epilepsy (see this video).\n\nHow much to weigh these effects and effects of other diseases AMF may prevent (e.g. dengue, yellow fever, zika, encephalitis) depends on the subjective trade-off between preventing deaths and preventing suffering. I feel that my personal trade-off would give much more relative weight to the suffering than GiveWell does. Although I\u2019m sure that GiveWell has solid reasons for making their estimates in the way that they did.\n\nFinally, GiveWell\u2019s estimate doesn\u2019t seem to take into account many other effects. E.g.:\n\n\u2022 Preventing deaths prevents grief of parents, siblings and friends\n\u2022 Preventing morbidity also prevents a lot of additional trouble associated with it\n\u2022 Bednets prevent mosquito bites which we all know are annoying\n\u2022 On the other hand, I remember reading somewhere that people feel discomfort when sleeping under bednets\n\u2022 Malaria has a high economic burden and bednets reduce that as well.\n\u2022 Relatedly, bednets can empower some struggling people. An old GiveWell blog says: \u201ca substantial part of the good that one does may be indirect: the people that one helps directly (by e.g. funding distribution of bednets) become more empowered to contribute to society, and this in turn may empower others, etc. If one believes that, on average, people tend to accomplish good when they become more empowered, it\u2019s conceivable that the indirect benefits of one\u2019s giving swamp the first-order effects.\u201d\n\u2022 Fighting malaria impacts the size of the human populations which has many different consequences\n\u2022 FInally, AMF has an impact on mosquitos. The only analysis of that that I know of is the one by Brian Tomasik but it is from a negative utilitarian point of view. Also, I don\u2019t know if killed mosquitos mean that some other animals also don\u2019t get mosquito bites and associated problems.\n\nNote that I'm not at all an expert on any of these problems so don't put too much weight on what I say.\n\ncomment by Brian_Tomasik \u00b7 2020-05-03T19:02:58.036Z \u00b7 score: 3 (2 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nGood points. :) That post of mine isn't really about the mosquitoes themselves but more about the impacts that a larger human population would have on invertebrates (assuming AMF does increase the size of the human population, which is a question I also mention briefly).\n\ncomment by saulius \u00b7 2020-04-29T17:58:50.258Z \u00b7 score: 16 (8 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThis doesn't change the bottom line much, but for the technical correctness sake, I feel it should be noted that not all cage-free commitments that THL wins shift hens from battery cages to aviaries. For example, some THL-funded campaigns are in the EU or UK where battery cages have been banned since 2012. You can see in this graph that in the EU, caged hens are in enriched cages, not battery cages. Enriched cages are better than battery cages. Charity Entrepreneurship\u2019s weighted animal welfare index gives battery cages a score of -57 and enriched cages a score of -46. That said, looking at THL\u2019s 2020 room for more funding report, it seems that a lot of the cage-free focus will be on countries where battery cages (rather than enriched cages) are used.\n\nLess importantly, aviary is not the only cage-free system that producers may switch to after converting. E.g., some producers may switch to barn, free-range, or organic systems.\n\ncomment by AidanGoth \u00b7 2020-05-01T14:08:52.499Z \u00b7 score: 7 (4 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThanks for this. I think this stems from the same issue as your nitpick about AMF bringing about outcomes as good as saving lives of children under 5. The Founders Pledge Animal Welfare Report estimates that THL historically brought about outcomes as good as moving 10 hen-years from battery cages to aviaries per dollar, so we took this as our starting point and that's why this is framed in terms of moving hens from battery cages to aviaries. We should have been clearer about this though, to avoid suggesting that the only outcomes of THL are shifts from battery cages to aviaries.\n\ncomment by saulius \u00b7 2020-05-01T14:32:53.359Z \u00b7 score: 9 (5 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nNote that (unless I missed something) your animal welfare report commits this same minor mistake of assuming that all hens used by companies that made cage-free commitments were in battery cages. While I think that's true for the majority of hens, some of them were already in cage-free systems, and some were in enriched cages. But this is more than outweighed by some very conservative assumptions. E.g., that THL's work only moved policies forward by 1 year or something like that. So it's no big deal :)\n\ncomment by saulius \u00b7 2020-05-01T14:35:03.684Z \u00b7 score: 10 (4 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nWhat is way more important is all the indirect effects and other factors that I list in the \"Ways this estimate could be misleading\" section of my corporate campaigns CEA here [EA \u00b7 GW]. I think that they might be more important than direct effects. The same could also be true about AMF [EA \u00b7 GW].\n\ncomment by MichaelStJules \u00b7 2020-04-29T21:32:00.126Z \u00b7 score: 13 (6 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nA bit of a nitpick, but \"reasonable assumptions\" maybe not be reasonable to others, so what's considered reasonable is really personal. You could say assumptions close to what you think are representative of EAs.\n\nPersonally, I found your assumptions about tradeoffs between chickens and humans reasonable, but I also don't find that death in itself (or the loss of pleasure to the individual because of it) is bad. Others might have opposite intuitions.\n\ncomment by Jason Schukraft \u00b7 2020-04-29T14:38:25.967Z \u00b7 score: 9 (7 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThanks Stephen and Aidan for this great report! These sorts of questions are super difficult but plausibly quite important. I appreciate how transparent you are about your uncertainty. Rethink Priorities has been doing some work on moral weight that will point to ways to hopefully reduce some key uncertainties. Stay tuned in the next few weeks as we begin to release our reports!\n\ncomment by smclare \u00b7 2020-05-01T09:10:44.057Z \u00b7 score: 1 (1 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThanks Jason! Looking forward to reading the new research.\n\ncomment by tylermjohn \u00b7 2020-05-04T14:38:18.671Z \u00b7 score: 6 (4 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThanks for doing this! Though it seems like you kinda buried the lede. Why isn't this in the top level summary?\n\n\u2022 In expectation, THL is >100x better than AMF\n\u2022 In the median scenario, THL is about 2-4x more cost-effective than AMF\n\u2022 A 71% chance that THL is more cost-effective than AMF\ncomment by AidanGoth \u00b7 2020-05-05T13:33:19.734Z \u00b7 score: 9 (7 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThanks for raising this. It's a fair question but I think I disagree that the numbers you quote should be in the top level summary.\n\nI'm wary of overemphasising precise numbers. We're really uncertain about many parts of this question and we arrived at these numbers by making many strong assumptions, so these numbers don't represent our all-things-considered-view and it might be misleading to state them without a lot of context. In particular, the numbers you quote came from the Guesstimate model, which isn't where the bulk of the work on this project was focused (though we could have acknowledged that more). To my mind, the upshot of this investigation is better described by this bullet in the summary than by the numbers you quote:\n\n\u2022 In this model, in most of the most plausible scenarios, THL appears better than AMF. The difference in cost-effectiveness is usually within 1 or 2 orders of magnitude. Under some sets of reasonable assumptions, AMF looks better than THL. Because we have so much uncertainty, one could reasonably believe that AMF is more cost-effective than THL or one could reasonably believe that THL is more cost-effective than AMF.\ncomment by tylermjohn \u00b7 2020-05-06T18:48:56.961Z \u00b7 score: 7 (2 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThanks! I appreciate your wariness of overemphasizing precise numbers and I agree that it is important to hedge your estimates in this way.\n\nHowever, none of the claims in the bullet you cite give us any indication of the expected value of each intervention. For two interventions A and B, all of the following is consistent with the expected value of A being astronomically higher than the expected value of B:\n\n\u2022 B is better than A in most of the most plausible scenarios\n\u2022 On most models the difference in cost-effectiveness is small (within 1 or 2 orders of magnitude)\n\u2022 One could reasonably believe that B is better than A or that B is better than A\n\nExtremely little information is communicated about the relative expected value of A and B by the above points, and what information is communicated misleadingly suggests that both interventions are quite close in expected value. Because EAs are concerned with the expected value of interventions, I think you ought to communicate more about the relative expected value of the interventions and frame your summary of the interventions in a way that is less likely to mislead people about the relative expected value of each intervention.\n\nI think the ideally informative way to both communicate the relative expected value of the interventions and hedge on your model uncertainty in the summary is to (1) provide your expected value estimate, (2) explain that you have high model uncertainty and one could arrive at a different expected value estimate with different assumptions, and (3) invite participants to adjust the Guesstimate and generate their own predictions.\n\ncomment by AidanGoth \u00b7 2020-05-12T16:23:12.962Z \u00b7 score: 1 (1 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThanks, this is a good criticism. I think I agree with the main thrust of your comment but in a bit of a roundabout way.\n\nI agree that focusing on expected value is important and that ideally we should communicate how arguments and results affect expected values. I think it's helpful to distinguish between (1) expected value estimates that our models output and (2) the overall expected value of an action\/intervention, which is informed by our models and arguments etc. The guesstimate model is so speculative that it doesn't actually do that much work in my overall expected value, so I don't want to overemphasise it. Perhaps we under-emphasised it though.\n\nThe non-probabilistic model is also speculative of course, but I think this offers stronger evidence about the relative cost-effectiveness than the output of the guesstimate model. It doesn't offer a precise number in the same way that the guesstimate model does but the guesstimate model only does that by making arbitrary distributional assumptions, so I don't think it adds much information. I think that the non-probabilistic model offers evidence of greater cost-effectiveness of THL relative to AMF (given hedonism, anti-speciesism) because THL tends to come out better and sometimes comes out much, much better. I also think this isn't super strong evidence but that you're right that our summary is overly agnostic, in light of this.\n\nIn case it's helpful, here's a possible explanation for why we communicated the findings in this way. We actually came into this project expecting THL to be much more cost-effective, given a wide range of assumptions about the parameters of our model (and assuming hedonism, anti-speciesism) and we were surprised to see that AMF could plausibly be more cost-effective. So for me, this project gave an update slightly in favour of AMF in terms of expected cost-effectiveness (though I was probably previously overconfident in THL). For many priors, this project should update the other way and for even more priors, this project should leave you expecting THL to be more cost-effective. I expect we were a bit torn in communicating how we updated and what the project showed and didn't have the time to think this through and write this down explicitly, given other projects competing for our time and energy. It's been helpful to clarify a few things through this discussion though :)\n\ncomment by MichaelStJules \u00b7 2020-04-29T20:23:07.838Z \u00b7 score: 6 (3 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nSomething I've been wondering about lately is the supply and demand effects of cage-free reforms. If cage-free costs more, then presumably production should decrease, and fewer hens will be raised for eggs. Maybe this makes up for some of the concerns about higher mortalities in cage-free systems?\n\ncomment by Jason Schukraft \u00b7 2020-04-30T01:24:05.970Z \u00b7 score: 7 (2 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nIn Compassion, by the Pound, Norwood and Lusk estimate that a transition from cage to cage-free eggs would increase prices 21% which would decrease consumption 4% (350-351). But cage-free eggs also require more chickens. Norwood and Lusk estimate that one cage egg requires 0.003212204 chickens and one cage-free egg requires 0.003229267 chickens (233).\n\ncomment by saulius \u00b7 2020-04-30T13:32:52.081Z \u00b7 score: 23 (7 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nNorwood and Lusk estimate that one cage egg requires 0.003212204 chickens and one cage-free egg requires 0.003229267 chickens (233).\n\nThere is actually a typo in the table 8.4 in page 233 on which you are basing this. If you read the text closely, you can see that the value for \"Number of non breeder animals associated with one cage egg\" should be be 1\/509 = 0.001964637, not 1\/314 = 0.003184713. The book does not make the same mistake in a very similar table 8.7.\n\nHowever, in my opinion, what matters more is how many chicken-years are required per egg. And since cage-free hens seem to live shorter lives, the difference in chicken-years required per egg is not as big as chickens required per egg.\n\nBut according to a person with more knowledge, the bigger problem is that the book is comparing industrial cage systems with small-scale cage-free systems that are not using optimal genetics. That is not the relevant comparison for the current situation where large-scale producers are switching to cage-free systems. Numbers that the book uses differ quite a lot from numbers in other sources that are discussing industrial systems.\n\nI have spent two or three weeks looking into these issues and have quite neat document about it that I decided not to publish. If somebody thinks that the information in the document could be action-relevant to them, you can email me at saulius@rethinkpriorities.org and I will send you the document.\n\ncomment by MichaelStJules \u00b7 2020-04-30T02:08:23.514Z \u00b7 score: 5 (3 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nAwesome, thanks! Looks like the difference in number of chickens required per egg is basically dominated by the 4% change in demand, working out to about 3.5% fewer chickens. It seems plausible to me that the roughly 3.5% fewer chickens raised might even dominate the changes in average welfare, assuming their lives are very bad either way.\n\nThere are also recent analyses of ballot initiatives in California, both ex ante and ex post that might tell us about this, too, e.g.: http:\/\/www.zachgroff.com\/2017\/11\/animal-welfare-reforms-are-looking.html?m=1\n\ncomment by smclare \u00b7 2020-05-01T09:16:55.811Z \u00b7 score: 1 (1 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nI agree that this seems important. It also makes me worry about the equilibrium effects. If producer A switches to a more expensive system and producer B doesn't, then I wonder how many consumers just end up buying more cheap eggs from B.\n\ncomment by saulius \u00b7 2020-05-01T16:33:21.432Z \u00b7 score: 6 (3 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nCommitments are usually made by grocers, restaurants, hotels, etc., not producers. You can see in this document by USDA that at least in the U.S., most important companies that made commitments are retailers, followed by restaurants. I think it's somewhat unlikely that many people will go to another grocer just to save a little bit of money on eggs. Similarly, I don't think that it will impact people's choice of restaurants much because egg prices probably won't influence meal prices that much. Also, some animal advocates believe that eventually all the production in some countries\/regions like the U.S. will be cage-free because egg producers won't want to invest in new caged facilities when there is a risk that further corporate campaigns or law changes will take away the few remaining customers that buy caged eggs.\n\ncomment by MichaelStJules \u00b7 2020-04-29T21:21:48.768Z \u00b7 score: 5 (3 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nIt's not clear to me that DALYs or QALYs track hedonistic welfare that well. Although life satisfaction isn't hedonistic either, QALYs give relatively less weight to mental pain (anxiety and depression) and ability to perform usual activities compared to life satisfaction [EA \u00b7 GW]. Michael Plant also argues in favour of using life satisfaction over QALYs in that post. DALYs are calculated slightly differently, based on the judgements of experts rather than something closer to a random sample from the general population, but they may also have no personal experience living with the conditions.\n\nI think your sensitivity analysis might be broad enough for this to not matter, though, since from that link, the difference seems to be at most a factor of about 3.\n\nOn the other hand, there's the question about whether the kinds of tradeoffs people make between pleasure and suffering, or different levels of suffering or different levels of pleasure for different durations actually track hedonistic value. Often there are too many different factors involved to isolate the hedonistic value (and when they try to, like with the experience machine or wireheading, many people seem to reject hedonism and experientialism outright, so the kinds of tradeoffs people make normally might not refer much to the value of experiences; then again, see this). It seems unlikely that there's a one-size-fits-all, but maybe the average responses are good enough, or the best we can do.\n\ncomment by AidanGoth \u00b7 2020-05-01T13:57:26.056Z \u00b7 score: 5 (3 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nThanks for this comment, you raise a number of important points. I agree with everything you've written about QALYs and DALYs. We decided to frame this in terms of DALYs for simplicity and familiarity. This was probably just a bit confusing though, especially as we wanted to consider values of well-being (much) less than 0 and, in principle, greater than 1. So maybe a generic unit of hedonistic well-being would have been better. I think you're right that this doesn't matter a huge amount because we're uncertain over many orders of magnitude for other variables, such as the moral weight of chickens.\n\nThe trade-off problem is really tricky. I share your scepticism about people's actual preferences tracking hedonistic value. We just took it for granted that there is a single, privileged way to make such trade-offs but I agree that it's far from obvious that this is true. I had in mind something like \"a given experience has well-being -1 if an idealised agent\/an agent with the experiencer's idealised preferences would be indifferent between non-existence and a life consisting of that experience as well as an experience of well-being 1\". There are a number of problems with this conception, including the issue that there might not be a single idealised set of preferences for these trade-offs, as you suggest. I think we needed to make some kind of assumption like this to get this project off the ground but I'd be really interested to hear thoughts\/see future discussion on this topic!\n\ncomment by Milan_Griffes \u00b7 2020-04-30T00:26:38.958Z \u00b7 score: 4 (2 votes) \u00b7 EA(p) \u00b7 GW(p)\n\nSome more discussion of welfare metrics here: Why does EA use QALYs instead of experience sampling? 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module Hbc::Staged
def info_plist_file(index = 0)
index = 0 if index == :first
index = 1 if index == :second
index = -1 if index == :last
Hbc.appdir.join(@cask.artifacts[:app].to_a.at(index).first, "Contents", "Info.plist")
end
def plist_exec(cmd)
@command.run!("/usr/libexec/PlistBuddy", args: ["-c", cmd, info_plist_file])
end
def plist_set(key, value)
plist_exec("Set #{key} #{value}")
rescue StandardError => e
raise Hbc::CaskError, "#{@cask.token}: 'plist_set' failed with: #{e}"
end
def bundle_identifier
plist_exec("Print CFBundleIdentifier").stdout.chomp
rescue StandardError => e
raise Hbc::CaskError, "#{@cask.token}: 'bundle_identifier' failed with: #{e}"
end
def set_permissions(paths, permissions_str)
full_paths = remove_nonexistent(paths)
return if full_paths.empty?
@command.run!("/bin/chmod", args: ["-R", "--", permissions_str] + full_paths,
sudo: true)
end
def set_ownership(paths, user: current_user, group: "staff")
full_paths = remove_nonexistent(paths)
return if full_paths.empty?
@command.run!("/usr/sbin/chown", args: ["-R", "--", "#{user}:#{group}"] + full_paths,
sudo: true)
end
def current_user
Hbc::Utils.current_user
end
private
def remove_nonexistent(paths)
Array(paths).map { |p| Pathname(p).expand_path }.select(&:exist?)
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 267 |
\section{Introduction and main results}
We consider the following problems:
\begin{equation}\label{B.Khiar}
\left\{\begin{array}{lll} u_{tt}-u_{xx}=0 \;\;\;\;
\mbox{for} \;\;\;\; 0<x<a(t), \, t > 0,\\
u(x,0)=\phi(x), u_t(x,0)=\psi(x), \;\; 0<x<a(0), \\
\end{array}\right.
\end{equation}
$(\phi,\psi)\in H^1((0,a(0)))\times L^2((0,a(0)))$, with Dirichlet
boundary conditions
\begin{equation}\label{dir}
u(0,t)=0 \;\;\;\; \mbox{and} \;\;\;\; u(a(t),t)= 0, \, t > 0,
\end{equation}
or with mixed boundary conditions for which (\ref{dir}) is replaced
by
\begin{equation}\label{neu}
u(0,t)=0 \;\;\;\; \mbox{and} \;\;\;\; u_x(a(t),t)=0, \, t > 0,
\end{equation}
the subscripts denote partial differentiations, here $a$ is a
strictly positive real function which is continuous, periodic,
piecewise linear.\\
Our major concern will
be to find the associated curves $a(t)$ for which,
\begin{equation}\label{obd}
\int_{0}^{T} \left|u_x(a(t),t)\right|^2 \, dt \geq C^{*} \,
\left(\|\phi\|^2_{H^1_0(0,a(0))} + \|\psi\|^2_ {L^{2}(0,a(0))} \right),
\end{equation}
for (\ref{B.Khiar})-(\ref{dir}) and
\begin{equation}\label{obn}
\int_{0}^{T} \left|u_t(a(t),t)\right|^2 \, dt \geq C^{*} \,
\left(\|\phi\|^2_{H^1_l(0,a(0))} + \|\psi\|^2_ {L^{2}(0,a(0))} \right),
\end{equation}
for (\ref{B.Khiar})-(\ref{neu}) are valid, where $H_l^1(0,a(0)) = \left\{ f \in
H^1(0,a(0)) \mbox{ such that } f(0)=0\right\}.$
Note that if $a$ is a constant, the
observability inequality
\begin{equation}\label{Obser1}
\int_{0}^{T} \left|u_x(a,t)\right|^2 \, dt \geq C^{*} \,
\left(\|\phi\|^2_{H^1_0(0,a)} + \|\psi\|^2_ {L^{2}(0,a)} \right)\;
\mbox{for some positive constant} \; C^{*},
\end{equation}
holds if $T\geq 2a$ for the Dirichlet problem. Also,
\begin{equation}\label{Obser2}
\int_{0}^{T} \left|u_t(a,t)\right|^2 \, dt \geq C^{*} \,
\left(\|\phi\|^2_{H^1_l(0,a)} + \|\psi\|^2_ {L^{2}(0,a)} \right),
\end{equation}
holds for the mixed problem.
\bigskip
In \cite{C} the author consider the system
\begin{equation*}
\left\{\begin{array}{lll} \varphi_{tt}-\varphi_{xx}=0 \;\;\;\;
\mbox{for} \;\;\;\; 0<x<1, \, t > 0,\\
\varphi(0,t) = \varphi(1,t) = 0, \, t > 0, \\
\varphi(x,0)=\phi(x), \varphi_t(x,0)=\psi(x), \;\; 0<x<1. \\
\end{array}\right.
\end{equation*}
For a suitable class of curves $a(t)$, which are $a : [0,T] \rightarrow (0,L)$ in the class $C^1([0, T])$ piecewise, i.e. there exists a partition of $[0, T], 0 = t_0 < t_1 < \cdots < t_n = T,$ such that $a\in C^1([t_i, t_{i+1}])$ for all $i = 0, \cdots, n-1.$ Assume also that this partition can be chosen in such a way that $1-|a'(t)|$ does not change the sign in $t\in [t_i, t_{i+1}],$ for all $i = 0, 1, \cdots , n-1.$ Also he makes the following hypothesis:
\begin{enumerate}
\item There exists constants $c_1, c_2 > 0$ and a finite number of open subintervals $I_j \subset [0, T]$ with
$j = 0, \cdots , J$ such that, for each subinterval $I_j$, $a\in C^1(I_j)$ and it satisfies the following two
conditions :\\
$\bullet$ $c_1\leq |a'(t)| \leq c_2$ for all $t \in I_j$,\\
$\bullet$ $1 -|a'(t)|$ does not change the sign in $t\in I_j$. \\
We assume, without loss of generality, that there exists $j_1$ with $-1 \leq j_1 \leq J$ such that $a(t)$ is decreasing in $I_j$ for $0 \leq j \leq j_1$, and $a(t)$ is increasing in $I_j$ for $j_1 < j \leq J.$\\
The case $j_1 = -1$ corresponds to that where $a(t)$ is increasing in all the subintervals $I_j$. Analogously, $j_1 = J$ corresponds to the case where $a(t)$ is decreasing in all the subintervals $I_j$.
\item For each $j = 0, \cdots , J$, let $U_j$ be the subintervals defined as follows:
\begin{equation*}
U_j = \left\{ \begin{array}{ll} \{ s-a(s) \mbox{ with } s\in I_j \} \mbox{ if } j\leq j_1 \\
\{ s+a(s) \mbox{ with } s\in I_j \} \mbox{ if } j> j_1 . \end{array} \right.
\end{equation*}
Then, there exists an interval $W_1$ with length $(W_1) > 2L$ such that
$$
W_1 \subset \overline{\overset{J}{\underset{j=0}{\cup}}U_j.}
$$
\item For each $j = 0, \cdots , J$, let $V_j$ be the subintervals defined as follows:
\begin{equation*}
V_j = \left\{ \begin{array}{ll} \{ s+a(s) \mbox{ with } s\in I_j \} \mbox{ if } j\leq j_1 \\
\{ s-a(s) \mbox{ with } s\in I_j \} \mbox{ if } j> j_1 . \end{array} \right.
\end{equation*}
\end{enumerate}
He gets the following observability estimate:
\begin{equation}\label{cas}
\int_{0}^{T} \left|\frac{d}{dt} [\varphi(a(t),t)]\right|^2 \, dt \geq C^{*} \,
\left(\|\phi\|^2_{H^1_0(0,1)} + \|\psi\|^2_ {L^{2}(0,1)} \right),
\end{equation}
where $T$ is given by an optical geometric condition requiring that any ray, starting anywhere in the
domain and with any initial direction, must meet the dissipation zone before the time $T$.
Also, he gives several examples of curves for exact controllability related to the following system:
\begin{equation*}
\left\{\begin{array}{lll} u_{tt}-u_{xx}=f(t)\delta_{a(t)} \;\;\;\;
\mbox{for} \;\;\;\; 0<x<1, \, t > 0,\\
u(0,t) = u(1,t) = 0, \, t > 0, \\
u(x,0)=\phi(x), u_t(x,0)=\psi(x), \;\; 0<x<1, \\
\end{array}\right.
\end{equation*}
via (\ref{cas}).
\bigskip
Here we introduce a new approach that provides (\ref{obd}) and (\ref{obn}) for another class of curves $a(t)$ with less of regularity.
We start with some notations and known results.
\bigskip
Let $\mbox{Lip} (\mbox{\sr R})$ be the space
of Lipschitz continuous functions on $\mbox{\sr R}$. We shall denote the
Lipschitz constant of a function $F$ by
$$
L(F):=\sup_{x,y\in \mbox{\sr R},x\neq y} \left | \frac{F(x)-F(y)}{x-y}\right
|.
$$
Furthermore, denote by $D_{p}$ the set of functions continuous and
strictly increasing of the form $x+g(x)$, where $g(x)$ is a periodic
continuous function.
\begin{prop}
Let $a$ be a periodic function. Then
\begin{equation}\label{Fexpr}
F:=(I+a)\circ (I-a)^{-1}
\end{equation}
belongs to $D_{p}$. Moreover, the rotation number $\rho(F)$ defined
by
$$
\rho(F)= \displaystyle \lim_{n \rightarrow \infty} \frac{F^n(x)-x}{n}
$$
exists, and the limit is equal for all $x \in \mbox{\sr R}$.
\end{prop}
Rigorous studies pointing out the use of rotation numbers has led to
fruitful contributions, one of which is an elegant and important result
(see \cite[section II]{Her} for more details):
\bigskip
Assume that $a(t)$ is a periodic function, $a(t)>0$, $a \in
\mbox{Lip}(\mbox{\sr R})$ such that $L(a)\in [0,1)$. Assume also that
$|a'(t)|<1$ for all $t \in \mbox{\sr R}$ and $\rho (F)\in\mbox{\sr R}\setminus\mbox{\sr Q}$ such
that there exists a function $H \in D_{p}$ and
\begin{equation}\label{redF}
H^{-1} \circ F \circ H(\xi) = \xi + \rho(F).
\end{equation}
Before stating our main results, let us specify some hypotheses on
$H$.
\begin{assumptions}\label{a1}
There exist $\lambda_1>0$ and $\lambda_2>0$ such that
\begin{equation}
\lambda_1 \leq H^\prime (t) \leq \lambda_2, \, t \in \mathbb{R}.
\end{equation}
\end{assumptions}
\begin{assumptions}\label{a2}
The function $\displaystyle
b(t):=\frac{H'(a(t)+t)-H'(-a(t)+t)}{H'(a(t)+t)+H'(-a(t)+t)}$
satisfies
\begin{equation}
c_1 \leq b(t) \leq c_2, \;\; c_1,c_2>0, \mbox{ for all } t\in \mathbb{R}.
\end{equation}
\end{assumptions}
\begin{rmk} We make less assumptions and get on the occasion a larger class of functions $a(t)$ in connexion with the work of Castro \cite{C}.
\end{rmk}
We give an example where assumptions \ref{a1} and \ref{a2} are
guaranteed as in \cite{Gon1}.
\smallskip
Let $a$ be continuous and
periodic on $\mbox{\sr R}$, $a>0$, be such that
\[
a(t):=\left\{\begin{array}{ll} \alpha
t+\frac{\alpha(1-\alpha)(1+\beta)}{2(\alpha-\beta)} & \mbox{if
$\frac{\alpha(1+\beta)}{2(\alpha-\beta)}\leq t\leq
\frac{\alpha(1+\beta)-2\beta}{2(\alpha-\beta)},$}\\
\beta t-\beta+\frac{\alpha(1-\beta^2)}{2(\alpha-\beta)} & \mbox{if
$\frac{\alpha(1+\beta)-2\beta}{2(\alpha-\beta)} \leq
t\leq\frac{\alpha(3+\beta)-2\beta}{2(\alpha-\beta)}$,}
\end{array}
\right.
\]
with $\alpha$, $\beta\in (-1,1)$. Let
$l_1:=\frac{1+\alpha}{1-\alpha}$, $l_2:=\frac{1+\beta}{1-\beta}$.
The definition of $a$ is choosen such that $F$ is directly given on
$[0,1)$ and we extend $F$ through the formula: $F(x+1)=F(x)+1$ for
any $x\in\mbox{\sr R}$. The function $F$ is defined by :
\[
F(x):=(I+a)\circ (I-a)^{-1}(x)=\left\{\begin{array}{ll}l_1x+F_0
& \mbox{if $0\leq x\leq x_0$}\\
l_2x+F_0+1-l_2 & \mbox{if $x_0<x<1,$}
\end{array}
\right.
\]
with $F_0:=\frac{l_2(l_1-1)}{l_1-l_2}$,
$x_0:=\frac{1-l_2}{l_1-l_2}$.
\smallskip
Also the rotation number is given by the expression:
\begin{equation}
\label{formrho} \rho (F)=\frac{\ln l_1}{\ln
\left(\frac{l_1}{l_2}\right)},
\end{equation}
and the function $H$ given by (\ref{redF}) is done
by
$$H(x)=h_0\ln \left(|x+h_1| \right) + h_2,$$ where
$h_0=\frac{1}{\ln\left(\frac{l_1}{l_2}\right)}$, $h_1=\frac{l_2}{l_1-l_2}$ and
$h_2=-\ln \left(|h_1| \right),$ and satisfies the following inequalities: if $l_1>l_2$,
\begin{equation}\label{l2l1}
\frac{1}{\ln( \frac{l_1}{l_2})}\frac{l_1-l_2}{l_1}\leq H'(x)\leq
\frac{1}{\ln( \frac{l_1}{l_2})}\frac{l_1-l_2}{l_2},
\end{equation}
and if $l_1<l_2$,
\begin{equation}\label{l1l2}
\frac{1}{\ln( \frac{l_1}{l_2})}\frac{l_1-l_2}{l_2}\leq H'(x)\leq
\frac{1}{\ln( \frac{l_1}{l_2})}\frac{l_1-l_2}{l_1}\cdot
\end{equation}
The function $b$ which is 1-periodic is defined on $[0,1)$ by
$$\displaystyle{b(t)=-\frac{a(t)}{t+\frac{l_2}{l_1-l_2}}\cdot}$$
Assuming that $l_1<l_2,$ this function satisfies for all $t \in \mathbb{R}$
\begin{equation}\label{btau}
\frac{a_{min}(l_2-l_1)}{l_2}\leq\frac{a(t)(l_2-l_1)}{l_2}\leq
b(t)\leq\frac{a(t)(l_2-l_1)}{l_1}\leq\frac{a_{max}(l_2-l_1)}{l_1}.
\end{equation}
On the existence of solutions to the Dirichlet or the mixed problem,
we refer the reader to \cite{gon}. We have the following
proposition:
\begin{prop}
If $a\in\mbox{Lip} (\mbox{\sr R})$, $L(a)\in [0,1)$, $a>0$ and
$$
(\varphi_0,\varphi_1)\in H_0^1((0,a(0)))\times L^2((0,a(0))),\; \hbox{or in} \; H_l^1((0,a(0)))\times L^2((0,a(0))),
$$
denote by $Q:=(0,a(t))\times \mbox{\sr R}_{+}$ and $Q_{\tau}:=(0,a(t))\times
(0,\tau), \tau \in \mbox{\sr R}_{+}$. There exists a unique weak solution\footnote{$u\in H^1(Q_{\tau})$ is called a weak solution of either the Dirichlet or the mixed problem if $u_{tt}-u_{xx}=0$ in ${\mathcal D}^\prime(Q)$ and the boundary conditions are satisfied.}$u$
of either the Dirichlet or the mixed problem satisfying the initial
conditions $u(x,0)=\phi(x), u_t(x,0)=\psi(x) \;\; 0<x<a(0)$. Moreover
there exists $f\in H_{\mbox{loc}}^1(\mbox{\sr R})\cap L^\infty (\mbox{\sr R})$ such that
\begin{equation}
u(x,t)=f(t+x)-f(t-x)\quad\mbox{a.e. in $Q$},
\end{equation}
and $u\in L^{\infty}(Q)\cap H^1(Q_{\tau})$.
\end{prop}
Our main results are stated as follows:
\begin{thm}[Neumann observability]\label{dirichlet}
Under the assumption \ref{a1}, there exist $T, C^{*}> 0$ such that
for all $u$ solution of the system (\ref{B.Khiar}) with the
Dirichlet boundary condition (\ref{dir}) and initial data
$(\phi,\psi) \in H^1_0(0,a(0)) \times L^2(0,a(0)),$ we have
\begin{equation}\label{Obserneu}
\int_{0}^{T} \left|\ u_{x}(a(t),t)\right|^2 \, dt \geq C^{*} \,
\left(\|\phi\|^2_{H^1_0(0,a(0))} + \|\psi\|^2_{L^2(0,a(0))} \right).
\end{equation}
\end{thm}
\begin{rmk}
We give a similiar result in the appendix, which concern the Dirichlet observability.
\end{rmk}
The exact controllability problem for the system
\begin{equation}\label{cont}
\left\{\begin{array}{lll}
u_{tt}-u_{xx}=0 \;\;\;\;
\mbox{for} \;\;\;\; 0<x<a(t), \, t > 0,\\
u(0,t)=0 \;\;\;\; \mbox{and} \;\;\;\; u(a(t),t)=r(t), \, t > 0,\\
u(x,0)=\phi(x), u_t(x,0)=\psi(x) \;\; 0<x<a(0) \\
\end{array}\right.
\end{equation}
at time $T$ is the following: for each $(\phi, \psi) \in L^2(0,a(0))\times H^{-1}(0,a(0)),$ find
$r \in L^2(0,T)$ such that the corresponding solution to (\ref{cont}) satisfies
$u( .,T) = 0, u_t(.,T) = 0$ in $(0, a(T))$.
\bigskip
Based on the observability estimate mentioned above, we get:
\begin{coro}\label{cor}
Assume that $l_1 > l_2$, then there exist $T>0$ and $r \in L^2(0,T)$ such that the system (\ref{cont}) is exactly controllable at time $T$.
\end{coro}
\bigskip
The paper is organized as follows: In section \ref{sec2} we prove our main
results and in the last section we give further comments on the
quasi periodic case.
\section{Proof of the main results} \label{sec2}
We shall construct a transformation of the time-dependent domain
$[0, a(t)] \times \mbox{\sr R}$ onto $[0, \rho(F) / 2] \times \mathbb{R}$ that
preserves the D'Alembertian form of the wave equations. This
preserving property will reveal very important. Using $H$ given by
(\ref{redF}), we define a domain transformation $\Phi : \mathbb{R}^2
\rightarrow \mathbb{R}^2$ as follows:
\begin{equation}\label{trans}
\left\{\begin{array}{lll}
\xi = (H(x + t) - H(-x + t))/ 2,\\
\tau = (H(x + t) + H(-x + t))/ 2, \\
\end{array}\right.
\end{equation}
for $(x,t) \in \mathbb{R}^2$.
\begin{rmk}
The following propositions can essentially be found in \cite{Ya 5}
(see also the references therein), we reproduce them here for the
reader's convenience and because our presentation is synthetic.
\end{rmk}
\begin{prop}
The transformation $\Phi$ is a bijection of $[0, a(t)] \times
\mathbb{R}$ to $[0, \rho(F) / 2] \times \mbox{\sr R}$ and $\Phi$ maps the
boundaries $x=0$ and $x=a(t)$ onto the boundaries $\xi=0$ and
$\xi=\rho(F) / 2$ (resp).
\end{prop}
\begin{prop}
Let $u(x,t)$ satisfying $(\partial_{t}^{2} - \partial_{x}^{2})
u(x,t)=0$ and $V(\xi,\tau)$ defined by $u (\Phi^{-1}(\xi,\tau))$.
Then the following identity holds
$$
(\partial_{t}^{2} - \partial_{x}^{2}) u(x,t) = K(\xi,\tau)
(\partial_{\tau}^{2} - \partial_{\xi}^{2})V(\xi,\tau)
$$
where $K(\xi,\tau)$ is defined by
$$
4 H' \circ H^{-1}(\xi + \tau) H' \circ H^{-1}(- \xi + \tau) \circ
H^{-1}(\xi + \tau).
$$
\end{prop}
The next lemma will be very useful for the proof of our main
results.
\begin{lemma}\label{second}
Denote by
$$
E_u(t)=\frac{1}{2}\int_{0}^{a(t)}\left[\left|u_t(x,t)\right|^2 +
\left|u_x (x,t)\right|^2 \right] \, dx
$$
the energy of the field $u$, and
$$
E_V(\tau)=\displaystyle\int^{\rho(F) /2}_{0} \left(\left|V_{\xi}
(\xi,\tau)\right|^2 + \left|V_{\tau}(\xi,\tau)\right|^2\right) \,
d\xi,
$$
the energy of the field $V$. There are two positive constants $C_1$
and $C_2$ such that
\begin{equation}
C_1 E_V(\tau)\leq E_u(t)\leq C_2 E_V(\tau).
\end{equation}
\end{lemma}
\begin{proof} We calculate,
$$
\partial_{t}u = \partial_{\xi}V \partial_{t}\xi
+ \partial_{\tau}V \partial_{t}\tau \;\;\; \mbox{and} \;\;\;
\partial_{x}u = \partial_{\xi}V \partial_{x}\xi +
\partial_{\tau}V \partial_{x}\tau
$$
and so,
$$
E_u(t)= \frac{1}{2}\int_{0}^{a(t)} \left[\left|u_t(x,t)\right|^2 +
\left|u_x (x,t)\right|^2 \right] \, dx =
$$
$$
\frac{1}{2}\int_{0}^{a(t)} \left\{[V_{\xi} \xi_{t} + V_{\tau}
\tau_{t}]^2 + \left|V_{\xi} \xi_{x} + V_{\tau} \tau_{x}\right|^2
\right\} \, dx .
$$
Make use of:
$$
\xi_{x} = (\partial_{x}\xi) = (\partial_{t}\tau) = \tau_{t}= [H'(x +
t) + H'(-x + t)] / 2,
$$
$$
\xi_{t} = (\partial_{t}\xi) = (\partial_{x}\tau) = \tau_{x} = [H'(x
+ t) - H'(-x + t)] / 2.
$$
Hence,
$$ \xi_{t}^2 + \xi_{x}^2 = \tau_{t}^2 + \tau_{x}^2
= \frac{1}{2}\left[\left|H'(x + t)\right|^2 + \left|H'(-x +
t)\right|^2 \right],
$$
$$
\xi_{t}\tau_{t} = \xi_{x}\tau_{x} = \frac{1}{4}[(H'(x + t))^2 -
(H'(-x + t))^2].
$$
Back to the energy,
$$
E_u(t)= \int_{0}^{a(t)} \frac{1}{4}\left[\left|V_{\xi}\right|^2 +
\left|V_{\tau}\right|^2 \right] \left[\left|H'(x + t)\right|^2 +
\left|H'(-x + t)\right|^2 \right] \, dx
$$
$$
+ \int_{0}^{a(t)} \frac{1}{2}[V_{\xi} V_{\tau}] \left[\left|H'(x +
t)\right|^2 - \left|H'(-x + t)\right|^2 \right] \, dx.
$$
Also, differentiating $x = ( H^{-1} (\xi + \tau) - H^{-1}(- \xi +
\tau))/ 2$, we obtain
$$dx = 1/2 ((H^{-1})' (\xi +
\tau) + (H^{-1})'(- \xi + \tau)) d\xi.
$$
The inequality $\left|V_{\xi} V_{\tau}\right| \leq 1/2
\left(V_{\xi}^2 + V_{\tau}^2\right)$ yields to:
\begin{equation}\label{inh}
C_1 E_V(\tau)\leq E_u(t)\leq C_2 E_V(\tau),
\end{equation}
for positive constants $C_1, C_2$.
\end{proof}
\begin{rmk}
Applying the transformation $\Phi$, the system (\ref{B.Khiar})-(\ref{dir}) becomes:
\begin{equation}
\label{Beni.Khalled}
\left\{\begin{array}{lll}
\partial_{\tau}^{2}V - \partial_{\xi}^{2} V = 0, \;\;\;\; \mbox{for} \;\;\;\; 0<\xi<\rho(F) /2, \, \tau \in \mathbb{R}, \\
V(0,\tau) = 0, \;\; V(\rho(F) /2,\tau) = 0, \;\; \tau \in \mbox{\sr R}, \\
V(\xi,0) = \phi_1(\xi), \;\; V_{\tau}(\xi,0) = \psi_1(\xi),
\;\; \xi \in (0,\rho(F) /2).
\end{array}
\right.
\end{equation}
\end{rmk} We need the following Lemma.
\begin{lemma}\label{Ob1}
If $T>\rho(F) $, there exists $C(T)>0$ such that for all
$(\phi_1,\psi_1) \in H^1_0(0,\rho(F)/2) \times L^2(0,\rho(F)/2)$ we
have
$$
C(T) \, \int_{0}^{T} |V_{\xi}(\rho(F) /2,\tau)|^2d\tau \geq
\|\phi_1\|^2_{H^1_0(0,\rho(F) /2)} + \|\psi_1\|^2_ {L^2(0,\rho(F)
/2)}.
$$
\end{lemma}
\begin{proof}[\bf{Proof of Theorem~\ref{dirichlet}}]
We consider (\ref{B.Khiar})-(\ref{dir}) and state:
$$
\partial_{x}u = \partial_{\xi}V \partial_{x}\xi
+ \partial_{\tau}V \partial_{x}\tau
$$
Next we have:
$$
\partial_{x}u(a(t),t)= \partial_{\xi}V(\rho(F) /2,\tau)
\partial_{x}\xi(a(t),t)+\partial_{\tau}V(\rho(F) /2,\tau)
\partial_{x}\tau(a(t),t)
$$
Since
$$
\partial_{x}\xi = [H'(x + t)+ H'(-x + t)]/ 2 \;\; \mbox{and}
\;\; \partial_{x}\tau = [H'(x + t)- H'(-x + t)]/ 2,
$$
it follows that:
$$
|\partial_{x}u(a(t),t)|^2= |\partial_{\xi}V(\rho(F) /2,\tau)
\partial_{x}\xi(a(t),t)+\partial_{\tau}V(\rho(F) /2,\tau)
\partial_{x}\tau(a(t),t)|^2
$$
$$
=\frac{1}{4} \left\{|\partial_{\xi}V(\rho(F) /2,\tau)[H'(x + t)+
H'(-x + t)]\right\}^2.
$$
Make use of Young inequalities, (\ref{Obserneu}) is a consequence of
the inequalities (\ref{l1l2}) and (\ref{l2l1}), Lemma \ref{second}
and Lemma \ref{Ob1}.
\end{proof}
\begin{proof}[\bf{Proof of Corollary~\ref{cor}}]
Let us consider
\begin{equation}\label{cont1}
\left\{\begin{array}{lll} \partial_{\tau}^{2}V -
\partial_{\xi}^{2}V = 0, \;\;\;\; \mbox{for} \;\;\;\; 0<\xi<\rho(F) /2, \, \tau \in \mathbb{R}, \\
V(0,\tau) = 0, \;\; V(\rho(F) /2,\tau)= 0, \;\; \tau \in \mbox{\sr R}, \\
V(\xi,0) = V_0(\xi), \;\; V_{\tau}(\xi,0) = V_1(\xi),
\;\; \xi \in (0,\rho(F)/2). \\
\end{array}\right.
\end{equation}
The system (\ref{cont1}) is exactly observable at time $\rho(F)$ that is: there exists $C>0$ such
that for all $\tau \geq \rho(F)$, we have
$$
C(T) \, \int_{0}^{T} |V_{\xi}(\rho(F) /2,\tau)|^2d\tau \geq
\|\phi_1\|^2_{H^1_0(0,\rho(F) /2)} + \|\psi_1\|^2_ {L^2(0,\rho(F)
/2)},
$$
and so the following problem
\begin{equation}\label{cont2}
\left\{\begin{array}{lll} \partial_{\tau}^{2}\tilde{V} -
\partial_{\xi}^{2}\tilde{V} = 0, \;\;\;\; \;\;\;\; 0<\xi<\rho(F) /2, \, \tau \in \mathbb{R}, \\
\tilde{V}(0,\tau) = 0, \;\; \tilde{V}(\rho(F) /2,\tau)= g(\tau), \;\; \tau \in \mbox{\sr R}, \\
\tilde{V}(\xi,0) = \tilde{V}_0(\xi), \;\; \tilde{V}_{\tau}(\xi,0) =
\tilde{V}_1(\xi),
\;\; \xi \in (0,\rho(F)/2)\\
\end{array}\right.
\end{equation}
is exactly controllable at $\rho(F)$ that is for all $(\tilde{V}_0,
\tilde{V}_1) \in L^2(0,\rho(F) /2)\times H^{-1}(0,\rho(F) /2)$,
there exists $g \in L^2(0,\rho(F))$ such that $\tilde{V}(\xi,\tau)=0$ for all $\tau \geq \rho(F)$.\\ Moreover, $g:=V_{\xi}(\rho(F) /2,\tau)
\chi_{(0,\rho(F))}(\tau)$.
\smallskip
So the following transformed system
\begin{equation*}
\left\{\begin{array}{lll} \partial_{t}^{2}u -
\partial_{x}^{2}u = 0, \;\;\;\; \;\;\;\; 0<x<a(t), \, t \in \mathbb{R},\\
u(0,t) = 0, \;\; u(a(t),t)= f(t), \;\; t \in \mbox{\sr R}, \\
u(x,0) = u_0(x), \;\; u_{t}(x,0) =u_1(x),
\;\; x \in (0,a(0))\\
\end{array}\right.
\end{equation*}
is exactly controllable with a time of control $\displaystyle{
T:=|e^{\frac{\rho(F)-h_2}{h_0}}-h_1|}$ and a control $f(t)$ is given by $f(t)= g \left(\frac{H(a(t)+t)+H(-a(t)+t)}{2} \right).$
\end{proof}
\section{Further comments: The quasi periodic case}
One can try to generalize the previous results to the case when $a$
is no longer periodic but has some sort of quasiperiodicity
\footnote{A function $a(t), t \in \mbox{\sr R}$ is called quasiperiodic with
basic frequencies $\omega = (\omega_1 , ..., \omega_m) \in \mbox{\sr R}^{m}$
(briefly $2\pi/\omega-$q.p) if there exists a continuous function
$\hat{g}(\theta), \theta = (\theta_1 , ..., \theta_m) \in \mbox{\sr R}^{m}$
that is $2\pi$-periodic in each $\theta_i , i=1, ..., m$ such that
$a(t) = \hat{a}(\omega t)$ holds. $\hat{g}(\theta)$, is called the
corresponding function and $\displaystyle\frac{2\pi}{\omega} =
(\displaystyle\frac{2\pi}{\omega_1} , ...,
\displaystyle\frac{2\pi}{\omega_m})$ the basic periods of $a$.}.\\
The problem is much more complicated, since there is no rotation
number. However, in \cite{Ya 5} the author uses a weaker notion of
upper (resp. lower) rotation number of $F$ at every point $x$ as
follows:
$$
\overline{\rho}(F) = \displaystyle\limsup_{n\to
+\infty}\frac{F^n(x)-x}{n}
$$
$$ \mbox{ (resp. }
\underline{\rho}(F) =
\displaystyle\liminf_{n\to+\infty}\frac{F^n(x)-x}{n} ).
$$
As a consequence, it is shown that under the same Diophantine
condition \cite{Ca}, \cite{Lan} satisfied by $\overline{\rho}(F)$
(resp. $\underline{\rho}(F)$), the rotation number of $F$ exists and
coincides with the lower (resp. upper) rotation number.
\begin{lemma}\label{reduction}
Assume that $a(t)$ is an $\eta-$q.p function, $\hat{a}(\theta)$ is
real analytic and satisfy $|\hat{a'}(\theta)|<1$ for $\eta, \theta
\in \mbox{\sr R}^m$ and set $\beta = (\displaystyle\frac{2\pi}{\eta_1} ,
...,\displaystyle\frac{2\pi}{\eta_m})$. Assume also that there
exists $C_0>0$ depending on $\beta$ such that $|(k,\beta) + \pi
l/\overline{\rho}(F)|
> \displaystyle\frac{C_0}{|k|^{m+1}}$. Then, there exists a real
analytic function $H(\xi) = \xi+h(\xi)$, where $h(\xi)$ is an
$\eta$-q.p. function, such that
\begin{equation}\label{red1}
H^{-1} \circ F \circ H(\xi) = \xi + \overline{\rho}(F).
\end{equation}
\end{lemma}
\begin{rmk}
Thanks to Lemma \ref{reduction}, Theorem \ref{dirichlet}
is easily extended by similar arguments.
\end{rmk}
\begin{rmk}
Generalizations of the foregoing results may be obtained in a 3D
context, assuming that solutions and given data are functions of
only $r=(x^2+y^2+z^2)^{1/2}$ with respect to the space variables.
Let $\Omega$ be the domain $0<r<a(t)$ and consider,
\begin{equation}
\left\{\begin{array}{lll} u_{tt}-u_{rr}-(2/r)u_r = 0 \;\;\;\;
\mbox{in} \;\;\;\; {\Omega}, \, t > 0,\\
\mbox{with boundary conditions} \;\;\;\; u(0,t)=u(a(t),t)=0, \, t > 0,\\
\mbox{and initial conditions} \;\;\;\; u(r,0)=\phi(r), u_t(r,0)
=\psi(r), \, 0 <r < a(0).
\end{array}
\right.
\end{equation}
Introducing the transformation $u(r,t)=w(r,t)/r$ leads to the
problem:
\begin{equation}
\left\{\begin{array}{lll} w_{tt}=w_{rr} \;\;\;\; 0< r<a(t), \, t > 0,\\
w(0,t)=w(a(t),t)=0, \, t > 0, \\\\
w(r,0)=r\phi(r), \;\; w_t(r,0)= r\psi(r), \, 0 < r < a(0).
\end{array}\right.
\end{equation}
\end{rmk}
\section{Appendix}
In this section, we treat the Dirichlet observability.
\begin{thm}[Dirichlet observability]\label{Neumann}
Under the assumptions \ref{a1} and \ref{a2}, suppose moreover that $l_1<l_2,$ there exist $ T, \;
C^{*}> 0$ such that for all solution $u$ of the system
(\ref{B.Khiar}) with the mixed boundary condition (\ref{neu}) and
initial data $(\phi,\psi) \in H^1_l(0,a(0)) \times L^2(0,a(0)),$ we
have
\begin{equation}\label{Obser}
\int_{0}^{T} \left|u_{t}(a(t),t) \right|^2 \, dt \geq C^{*} \,
\left(\|\phi\|^2_{H^1_l(0,a(0))} + \|\psi\|^2_ {L^2(0,a(0))} \right).
\end{equation}
\end{thm}
\begin{rmk}
Using $\Phi$ given by (\ref{trans}), we transform the system (\ref{B.Khiar})-(\ref{neu})
into:
\begin{equation}\label{Beni.Benou}
\left\{\begin{array}{lll} \partial_{\tau}^{2}V -
\partial_{\xi}^{2}V = 0, \;\;\;\; \mbox{for} \;\;\;\; 0<\xi<\rho(F) /2, \, \tau \in \mathbb{R},\\
V(0,\tau) = 0, \;\; V_{\xi}(\rho(F) /2,\tau)+b(t(\tau))V_{\tau}(\rho(F) /2,\tau) = 0, \;\; \tau \in \mbox{\sr R} \\
V(\xi,0) = \phi_2(\xi), \;\; V_{\tau}(\xi,0) = \psi_2(\xi),
\;\; \xi \in (0,\omega/2).
\end{array}\right.
\end{equation}
\end{rmk}
For the proof of Theorem~\ref{Neumann}, we need the following
lemmas.
\begin{lemma}\label{exv}
Assume that $l_1<l_2$, then there exist positive constants $C$ and
$\omega$ such that
\begin{equation}\label{expv}
E_{V}(\tau)\leq C e^{-\omega \tau} E_V(0).
\end{equation}
\end{lemma}
\begin{proof}
Define the Lyapunov function:
$$E_1(\tau) = \frac{1}{2}\int^{\rho(F)}_0[V_{\xi}^2(\xi,\tau)+ V_{\tau}^2(\xi,\tau)]d\xi+
\delta\int^{\rho(F)}_0 \xi
V_{\xi}(\xi,\tau)V_{\tau}(\xi,\tau)d\xi.$$ We obtain for
$\delta<\frac{1}{\rho(F)},$
\begin{equation}\label{E1E}
0<(1 -\delta\rho(F))E_{V}(\tau) \leq E_1(\tau) \leq (1
+\delta\rho(F))E_{V}(\tau).
\end{equation}
We derive $E_1$ with respect to $\tau$, we get
\begin{eqnarray*}
E_1'(\tau)&=&
[V_{\xi}V_{\tau}]^{\xi=\rho(F)}_{\xi=0}-\frac{\delta}{2}\int^{\rho(F)}_0
[V_{\xi}^2(\xi,\tau)+ V_{\tau}^2(\xi,\tau)]d\xi+\frac{\delta}{2}[\xi
(V_{\xi}^2 +
V_{\tau}^2)]^{\xi=\rho(F)}_{\xi=0}\\
&=&[\frac{\delta}{2}(1+b(t(\tau))^2)-b(t(\tau))]V_{\tau}^2(\rho(F),\tau)-
\frac{\delta}{2}\int^{\rho(F)}_0[V_{\xi}^2(\xi,\tau)+
V_{\tau}^2(\xi,\tau)]d\xi.
\end{eqnarray*}
We choose $\delta$ small enough, taking into account (\ref{btau})
and (\ref{E1E}) we get
$$ E_1'(\tau)\leq -\omega E_1(\tau). $$ The proof is complete.
\end{proof}
\begin{lemma}\label{Obser1bb}
If $T>\rho(F)$, then there exists $C(T)>0$ such that for all
$(\phi_2,\psi_2) \in H_l^1(0,\rho(F)/2) \times L^2(0,\rho(F)/2)$ we
have
\begin{equation}\label{obes1}
C(T) \, \int_{0}^{T} \left|V_{\xi}(\rho(F) /2,\tau)\right|^2 \,
d\tau \geq \|\phi_2\|^2_{H_l^1(0,\rho(F)/2)} +
\|\psi_2\|^2_{L^2(0,\rho(F) /2)},
\end{equation}
and
\begin{equation}\label{obes2}
C(T) \, \int_{0}^{T} \left|V_{\tau}(\rho(F) /2,\tau)\right|^2 \,
d\tau \geq \|\phi_2\|^2_{H_l^1(0,\rho(F)/2)} +
\|\psi_2\|^2_{L^2(0,\rho(F) /2)}.
\end{equation}
\end{lemma}
\begin{proof}
The energy identity for the system (\ref{Beni.Benou}) gives :
$$ E_V(T)-E_V(0)=-\int_{o}^{T}b(t(\tau))|V_{\tau}(\rho(F)/2,\tau)|^2d\tau.$$
Using (\ref{btau}) and (\ref{expv}), we obtain
\begin{eqnarray*}
\int_{0}^{T} \left|V_{\tau}(\rho(F) /2,\tau)\right|^2 \, d\tau &\geq
&
C \int_{0}^{T} b(t(\tau))\left|V_{\tau}(\rho(F) /2,\tau)\right|^2 \, d\tau \\
&\geq & C( E_V(0)- E_V(T))\\
&\geq & CE_V(0)(1- e^{-\omega T}).
\end{eqnarray*}
This permit to conclude the second inequality in Lemma \ref{Obser1bb}. \\
For the first inequality, it suffices to use (\ref{Beni.Benou}) and
(\ref{btau}).
\end{proof}
\begin{proof}[\bf{Proof of Theorem~\ref{Neumann}}]
For the proof of (\ref{Obser}), we state as above:
$$
\partial_{t}u = \partial_{\xi}V \partial_{t}\xi
+ \partial_{\tau}V \partial_{t}\tau.
$$
Next we have:
\begin{eqnarray*}
|\partial_{t}u(a(t),t)|^2
&=&\frac{1}{4} \{|\partial_{\xi}V(\rho(F) /2,\tau)[H'(x + t) - H'(-x +
t)]
\\
&+&\partial_{\tau}V(\rho(F) /2,\tau) [H'(x + t) + H'(-x + t)]\}^2.
\end{eqnarray*}
Make use of Young inequalities, Lemma \ref{second}, (\ref{obes1}),
(\ref{obes2}) and (\ref{l1l2}), we obtain the desired result.
\end{proof}
| {
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A radical proposal (or two)
I got to thinking the other day – yes, I know that can be a dangerous thing – about the 2014 electoral map for Maryland and an intriguing possibility.
Since State Senator E.J. Pipkin resigned a few months back, a sidebar to the story of his succession – as well as that of selecting a replacement for former Delegate Steve Hershey, who was elevated to replace Pipkin – is the fact that Caroline County is the lone county in the state without resident representation. However, with the gerrymandering done by the O'Malley administration to protect Democrats and punish opponents, it's now possible the 2015 session could dawn with four – yes, four – counties unrepresented in that body based on the 2012 lines. Three of those four would be on the Eastern Shore, and would be a combination of two mid-Shore counties and Worcester County, with the fourth being Garrett County at the state's far western end.
Granted, that scenario is highly unlikely and there is probably a better chance all 23 counties and Baltimore City will have at least one resident member of the General Assembly. But what if I had an idea which could eliminate that potential problem while bolstering the hands of the counties representing themselves in Annapolis?
The current composition of the Maryland Senate dates from 1972, a change which occurred in response to a 1964 Supreme Court decision holding that Maryland's system of electing Senators from each county violated the Fourteenth Amendment. Furthermore, Marylanders had directly elected their state Senators long before the Seventeenth Amendment was passed in 1913. Over time, with these changes, the Senate has become just another extension of the House of Delegates, just with only a third of the membership.
So my question is: why not go back to the future and restore our national founders' intent at the same time?
What if Maryland adopted a system where each county and Baltimore City were allotted two Senators, but those Senators weren't selected directly by the voters? Instead, these Senators would be picked by the legislative body of each county or Baltimore City, which would give the state 48 Senators instead of 47. Any tie would be broken by the lieutenant governor similar to the way our national vice-president does now for the United States Senate.
Naturally the Democrats would scream bloody murder because it would eliminate their advantage in the state Senate; based on current county government and assuming each selects two members of their own party the Senate would be Republican-controlled. But that would also encourage more voting on local elections and isn't that what Democrats want? It's probably a better way to boost turnout than the dismal failure of "early and often" voting, which was supposed to cure the so-called ailment of poor participation.
If someone would argue to me that my proposal violates "one man, one vote" then they should stand behind the repeal of the Seventeenth Amendment. How is it fair that I'm one of 2,942,241 people (poorly) represented by Ben Cardin or Barbara Mikulski while 283,206 people in Wyoming are far more capably represented by John Barasso or Mike Enzi? We have counties in Maryland more populous than Wyoming.
No one questions the function or Constitutionality of the U.S. Senate as a body, knowing it was part of a compromise between larger and smaller states in the era of our founding. It's why we have a bicameral legislature which all states save one copied as a model. (Before you ask, Nebraska is the holdout.) What I've done is restored the intent of those who conceived the nation as a Constitutional republic with several balances of power.
But I'm not through yet. If the Senate idea doesn't grab you, another thought I had was to rework the House of Delegates to assure each county has a representative by creating seats for a ratio of one per 20,000 residents. (This essentially equals the population of Maryland's least-populated county, Kent County. Their county could be one single House district.) In future years, the divisor could reflect the population of the county with the least population.
The corollary to this proposal is setting up a system of districts which do not overlap county lines, meaning counties would subdivide themselves to attain one seat per every 20,000 of population, give or take. For my home county of Wicomico, this would translate into five districts and – very conveniently as it turns out – we already have five ready-drawn County Council districts which we could use for legislative districts. Obviously, other counties would have anywhere from 1 to 50 seats in the newly expanded House of Delegates. Even better, because the counties would have the self-contained districts, who better to draw them? They know best which communities have commonality.
Obviously in smaller counties, the task of drawing 2 or 3 districts would be relatively simple and straightforward. It may be a little more difficult in a municipality like Baltimore or a highly-populated area like Montgomery County, but certainly they could come up with tightly-drawn, contiguous districts.
And if you think a body of around 300 seats is unwieldy, consider the state of New Hampshire has 400 members in their lower house. Certainly there would be changes necessary in the physical plant because the number of Delegates and their attendant staff would be far larger, but on the whole this would restore more power to the people and restrict the edicts from on high in Annapolis.
Tonight I was listening to Jackie Wellfonder launch into a brief discussion of whether the Maryland Republican Party should adopt open primaries, an idea she's leaning toward adopting – on the other hand, I think it's nuts. In my estimation, though, these sorts of proposals are nothing more than tinkering around the edges – these ideas I've dropped onto the table like a load of bricks represent real change. I think they should be discussed as sincere proposals to truly make this a more Free State by restoring the balance of power between the people, their local government, and the state government in Annapolis.
Author MichaelPosted on October 29, 2013 October 29, 2013 Categories All politics is local, Delmarva items, Maryland Politics, Politics, State of ConservatismTags Barbara Mikulski, Ben Cardin, E.J. Pipkin, early voting, gerrymandering, John Barasso, Martin O'Malley, Maryland Democratic Party, Maryland GOP, Maryland House of Delegates, Maryland Senate, Mike Enzi, Nebraska, New Hampshire, Steve Hershey, Wicomico County
8 thoughts on "A radical proposal (or two)"
Brian Griffiths says:
Your proposal, while interesting, still doesn't come anywhere close to being Constitutional under the existing precedent of Reynolds v. Sims because it still violates the principle of one man/one vote and still provides for Senators to be selected based on unequal representation. I'm all about thinking outside the box, but your Senate proposal just isn't legal.
Bill Lee says:
Again I like your out of the box thinking concerning the make up of our house in Annapolis. However I am on the side of open primaries. Empirical evidence aside ( which I think supports my opinion) Pure psychology begs the question. How do we prohibit others from voting in our primary, then beg their support in the general? It is a numerical certainty that we cannot win without "outsiders" voting for our candidate. Human nature is much stronger than politics. Slapping one in the face only begets a slap in return, not a kiss of support.
That assumes the Supreme Court is always correct. I think the dissenting judge in Reynolds v. Sims is in the right, as he cited the example of the United States Senate which I'm also using. In my estimation, one key step for restoring original intent would be to repeal the Seventeenth Amendment because the Senate was never intended as a "mini-me" to the House; unfortunately it has turned out that way.
I'd be more inclined to support the concept if the Democrats did it first. They seem to have no problem pandering to unaffiliated voters who can't vote in their primary, despite the fact in Maryland they're not necessary for a majority (although they are in other states and nationally.)
There's no prohibition on any registered voter voting in our primary, so long as they declare themselves Republican. If it's that important to them, come and change your affiliation.
We know that is never going to happen. Imagine just for a minute, there is a really good candidate that happens to be democrat , who is much more in line with your thought process than the republican candidate. Would you change parties just to vote for him? I seriously doubt it, I know I would not.
What is the disadvantage of having open, or at a minimum semi open primaries? The whole they will skew our primary and advance the weaker candidate is not a strong one. If the candidate is so weak that he looses to democratic votes in the primary, how in the world was he going to win the general?
It may not be as apparent in the state, but I look at how our last two presidential nominees were elected and think that's what would happen with an open primary. If several conservatives are in a primary and one squishy moderate panders to the center to win their votes, how much help will that give the GOP brand in the end?
The reason why a lot of these people are unaffiliated is that they've given up on the GOP because it's become Democrat-lite. I don't believe opening our primary to unaffiliated voters is going to send them a message we're going to hew to conservative principles.
Unless I am misunderstanding your explanation,the outcome is based on the premise that enough dems, or unaffiliated will vote for the republicans to skew the outcome. I propose that creates a huge dilemma for the dems themselves. If they defect to vote in our primary, who votes for their best choice? They leave themselves open for the exact same problem you are describing for us.
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Because this tragedy is the reason why the safety regulations we have today exist
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Coffins of Grand Canyon airplane crash victims awaiting burial.
Source A. Y. Owen/The LIFE Images Collection
It began the morning of June 30, 1956. Two flights, the Trans World Airlines Star of the Seine en route to Kansas City and Mainliner Vancouver of United Airlines to Chicago, had been delayed at Los Angeles International Airport — the former for minor maintenance, the latter because of increased holiday traffic before July 4th. Their flight plans were different enough to avoid concern: Both pilots had flown the route countless times. But a few seemingly inconsequential changes were enough to spell disaster. Capt. Jack Gandy of TWA requested clearance to fly "1,000 [feet] on top," over the clouds, which were forming thunderheads. The request was standard, and approved. But because he was flying in what was then "uncontrolled airspace," the maneuver made it his duty to maintain safe separation from other aircraft — a procedure known then as "see and be seen."
Now the policy is "see and avoid," one of the many changes resulting from that ill-fated day. Even after Gandy's shift, his flight course was not destined to intersect with the United flight, a subsequent investigation showed. Crucially, planes flying blind (without outside help) in such conditions were supposed to follow visual flight rules, which means avoiding clouds to try to stay visible. As the weather worsened, both Gandy and the United pilots began to shift to avoid the towering cumulus clouds that were forming.
For years after the event, anybody going down the river … could see a lot of the wreckage from one plane.
Dave Mortenson, vice president of the Grand Canyon Historical Society
And so the two planes collided at 10:30 a.m. after they each passed, investigators believed, the same cloud from opposite sides. The United flight banked to the right at the last moment, a crash analysis found, suggesting the pilots saw Gandy barreling toward them. "It was a see-and-avoid situation for the pilots — and they didn't," says Peter Goelz, former managing director of the National Transportation Safety Board. The 128 aboard all perished, at the time the most egregious plane accident in U.S. history and the only one with triple-digit deaths.
The subsequent stories would bring into focus the primitive standards of air control at the time. Dead spaces, where radars couldn't reach, were more common. Traffic controllers often failed to advise pilots of potential traffic conflicts — in fact, they were prohibited from doing so, an official told the Civil Aeronautics Board during the investigation. "In retrospect, the pilots were flying it the way they had been instructed to fly it," Goelz says. "This accident really galvanized that maybe there was too much peril involved, and there were steps that needed to be taken to minimize it."
Army personnel lay out plastic bags holding remains of some of the 128 victims of the two-plane crash in the Grand Canyon for shipment to the temporary morgue at Flagstaff, Arizona.
Source Bettmann/Getty Images
Congressional hearings in 1957 led to increased funding to hire and train more air traffic controllers and to purchase new radar technology. The next year, the Federal Aviation Agency (renamed Federal Aviation Administration in 1966) was created and given total control over American airspace. Subsequent breakthroughs in safety included collision avoidance programming, TKS de-icing systems and ground-warning proximity devices. Materials that once produced toxic fumes when burned, such as cushions or insulation, were removed from cabins, and airplane seats were made to withstand the force of 16 Gs, which has saved lives in runway crashes.
But there remains room for improved flight safety. Just as much of the Grand Canyon was uncontrolled airspace in the '50s, much of the ocean remains "dead space" today because of the industry's reliance on radar. The FAA is considering a move toward a satellite GPS-based system for tracking aircraft as a result — the type of next-generation development that could help save fuel, lessen environmental impact by having more precise flight plans, and prevent the loss of airplanes, like Malaysia Airlines Flight 370 in 2014.
Two men in the rescue party survey a section of the wrecked fuselage of the TWA plane that crashed in the Grand Canyon.
On April 22, 2014, the TWA-United accident site was declared a National Historic Landmark, the first to commemorate an event that occured in the air. But the exact location of the crash, which has been closed off to park visitors for decades, has remained secret. As a result, it appears to be the only historical site you are actually prohibited from visiting. Dave Mortenson, vice president of the Grand Canyon Historical Society and one of the earliest to raft the Colorado River, says, "For years after the event, anybody going down the river … could see a lot of the wreckage from one plane. It affected you."
Back at the historical marker overlooking the area, visitors read all this, speaking to each other in hushed tones. From miles above, they point to Chuar Butte and Temple Butte, the two rocky outcrops near where each flight fell. "All but three were interred in a mass grave," a woman in a shawl and white skirt reads aloud. The man with her, wearing a brown T-shirt with an American eagle illustrated on it, nods. "Poignant," he says, before walking away to get a closer view of America's most famous canyon.
Nick Fouriezos, OZY Author Follow Nick Fouriezos on Twitter Contact Nick Fouriezos
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\section{Introduction}
\label{sec:intro}
One of the biggest challenges for the cosmological community is the explanation of the current accelerated expansion of the Universe. A theoretical conception, commonly called Dark Energy (DE), is introduced to explain this mysterious phenomenon and whose nature is still unravelled \cite{copeland2006dynamics, amendola2010dark, ruiz2010dark}.
The cosmological constant $\Lambda$, being the
simplest form of the dark energy, together with the Cold Dark Matter (CDM), which is a key component for structure formation in the universe, conform the standard cosmological model or $\Lambda$CDM.
%
This model has had great achievements such as being in excellent agreement with most of the currently available data, for example measurements from the Cosmic Microwave Background radiation \cite{aghanim2020planck}, Supernovae Ia (SNeIa) \cite{betoule2014improved}, Cosmic Chronometers (CC) \cite{stern2010cosmic} and Baryon Acoustic Oscillations (BAO) \cite{alam2017clustering}. Nevertheless, the $\Lambda$CDM model has its own drawbacks: on theoretical grounds the cosmological constant faces several problems, i.e. fine tuning and cosmic coincidence \cite{sahni2002cosmological, peebles2003cosmological}, and from an observational point of view, it also suffers to the so-called Hubble tension -- a measurement disagreement of $H_0$ among datasets \cite{feeney2018clarifying}. The presence of these issues opens the possibility to extensions beyond the standard model by either considering a dynamical DE or modifications to the general theory of relativity \cite{joyce2016dark}.
The search of possible signatures for models beyond the $\Lambda$CDM has led to the creation of an impressive set of high accuracy surveys, already underway or being planned \cite{aghamousa2016desi, tyson2002large, amendola2018cosmology}, to gather a considerable amount of information that constrains the properties of the universe. That is, a viable model that leads to the current accelerating universal expansion is demanded to comply with all the relevant observational data. Extensions to the cosmological constant which allow a redshift-dependent equation-of-state (EoS) $w(z)$ include extra dimensions \cite{brax2003cosmology}, modified gravity \cite{clifton2012modified}, scalar fields \cite{vazquez2020bayesian}, scalar–tensor theories with non-minimal derivative coupling to the Einstein tensor \cite{quiros2018phantom} and graduated dark energy \cite{Akarsu:2019hmw}, just to mention a few.
%
However, in the absence of a fundamental and well-defined theory of dark energy, a time-dependent behaviour can also be investigated by choosing an EoS mathematically appealing or a parameterised form in a simple way, examples of these forms in terms of redshift include a Taylor expansion \cite{chevallier2001accelerating}, polynomial \cite{sendra2012supernova}, logarithmic \cite{odintsov2018cosmological}, oscillatory \cite{Tamayo:2019gqj, liu2009testing} or in terms of cosmic time \cite{Akarsu:2015yea}.
%
Nonetheless, the \textit{a priori} assumption of a specific model may lead to misleading model-dependent results regardless of the dark energy properties, and hence, instead of committing to a particular model the non-parametric inference techniques allow to extract information directly from the observational data to detect features within cosmological functions, for instance $w(z)$. That is, the main aim of a non-parametric approach is to infer (reconstruct) an unknown quantity based mainly on the data and making as few assumptions as possible \cite{wasserman2006all, sahni2002cosmological}.
Several non-parametric techniques are used to reconstruct cosmological functions from the data directly, such as histogram density estimators \cite{sahni2006reconstructing}, Principal Component Analysis (PCA) \cite{sharma2020reconstruction}, smoothed step functions \cite{gerardi2019reconstruction}, gaussian processes \cite{Keeley:2020aym, williams2006gaussian, l2020defying, mukherjee2020revisiting}, Simulation Extrapolation method (SIMEX) \cite{montiel2014nonparametric} and Bayesian nodal free-form methods \cite{hee2017constraining, Vazquez:2012ce}.
After the reconstruction is performed, the function in place can be considered as a new model in order to look for possible deviations from the standard $\Lambda$CDM.
In other words, the result of a non-parametric reconstruction may be used to analyse its similarity with different theoretical models and therefore to select its best description for the data.
%
There are several examples of non-parametric reconstructions of cosmological functions, some of them focus on dark energy features \cite{zhao2017dynamical,sahni2006reconstructing, gerardi2019reconstruction, holsclaw2010nonparametric}, cosmic expansion \cite{montiel2014nonparametric}, deceleration parameter \cite{mukherjee2020revisiting}, growth rate of structure formation \cite{l2020defying}, luminosity distance \cite{wei2017improved, lin2019non} and primordial power spectrum \cite{Handley:2019fll,Vazquez:2012ux}, among many others.
The recent increase in computing power and the vast amount of coming data have allowed the incursion of machine learning methods as analysis tools in observational cosmology \cite{peel2019distinguishing, arjona2020can, wang2020machine, lin2017does, uamneural2021}. In this work we focus on the computational models called Artificial Neural Networks (ANNs). They have been used in a variety of applications, such as image analysis \cite{dieleman2015rotation, ntampaka2019deep}, N-body simulations \cite{rodriguez2018fast, he2019learning} and statistical methods \cite{auld2007fast, alsing2019fast, escamilla2020deep, li2019model}.
%
Artificial Neural Networks have already been applied to non-parametric reconstructions of cosmological functions \cite{wang2020reconstructing} and, unlike other methods such as gaussian processes, ANNs may not assume any statistical distribution of the data. The main aim of this paper is to present a new strategy for non-parametric reconstructions using directly the cosmological observations to feed neural networks and then to generate models based solely on the data without any statistical or cosmological assumptions. We show that our methodology can be applicable to any astronomical datasets and we propose a special treatment for those containing a full covariance matrix, \textit{i.e.} including correlations between measurements.
The rest of the paper has the following structure. In Section \ref{sec:background}, we briefly introduce the cosmological and statistical concepts used throughout this work: cosmological models, functions and observations in section \ref{sec:cosmo}; an overview of neural networks in section \ref{sec:ann} and a short summary of Bayesian inference in section \ref{sec:bayes}. Section \ref{sec:methodoloy} describes the methodology used during the neural network training to generate models based on cosmological data. Section \ref{sec:results} contains the results of the reconstructed functions, namely the Hubble distance $H(z)$, a combination of the growth rate of cosmological perturbations times the matter power spectrum normalisation $f\sigma_8(z)$ and the distance modulus $\mu(z)$ along with its covariance matrix. We use the resulting models and trained neural networks to generate synthetic data, over which we perform a Bayesian inference analysis to test a couple of dark energy models. Finally, in Section \ref{sec:conclusions} we expose our final comments. Furthermore, within the appendices a brief description of variational autocoders is included and how we train them to learn from the covariance matrix with systematic errors.
\section{Cosmological and statistical background}
\label{sec:background}
In this section we introduce some of the cosmological models, functions and datasets used throughout this work.
We also provide a brief overview of the relevant concepts of Bayesian inference and the essentials of Artificial Neural Networks.
Along this paper we use the geometric unit system where $ \hbar = c = 8\pi G = 1$.
\subsection{Cosmological models and datasets}
\label{sec:cosmo}
\subsection*{Models}
The Friedmann equation describing the late-time dynamical evolution for a flat-$\Lambda$CDM model can be written as
%
\begin{equation}
H(z)^2 = H_0^2\left [\Omega_{m,0}(1+z)^3 + (1-\Omega_{m,0})\right],
\label{eq:hzlcdm}
\end{equation}
where $H$ is the Hubble parameter and $\Omega_{m}$ is the matter density parameter; subscript 0 attached to any quantity denotes its present-day $(z= 0)$ value. In this case, the EoS is $w(z) = -1$.
A step further to the standard model is to consider the dark energy being dynamic, where the evolution of its EoS is usually parameterised. A commonly used form of $w(z)$ is to take into account the next contribution of a Taylor expansion in terms of the scale factor $w(a)= w_0 + (1-a)w_a$ or in terms of redshift $w(z) = w_0 + \frac{z}{1+z} w_a$ (CPL model: \cite{chevallier2001accelerating, linder2003exploring}).
The parameters $w_0$ and $w_a$ are real numbers such that at the present epoch $w|_{z=0}=w_0$ and $dw/dz|_{z=0}=-w_a$, and we recover $\Lambda$CDM when $w_0 = -1$ and $w_a=0$.
Hence the Friedmann equation for the CPL parameterisation turns out to be:
%
\begin{equation}
H(z)^2 = H_0^2\left[ \Omega_{m,0}(1+z)^3 + (1-\Omega_{m,0})(1+z)^{3(1+w_0+w_a)} e^{-\frac{3w_a z}{1+z}}\right].
\label{eq:hzcpl}
\end{equation}
As part of some simple models that allow deviations from $\Lambda $CDM we also use the polynomial-CDM model (PolyCDM) \cite{aubourg2015}, that can be thought as a parameterisation of the Hubble function \cite{zhai2017evaluation}. This model has the following Friedmann equation:
%
\begin{equation}
H(z)^2 = H_0^2\left[ \Omega_{m,0}(1+z)^3 + \Omega_{1,0}(1+z)^2+ \Omega_{2,0}(1+z)^1
+ ( 1- \Omega_{m,0} - \Omega_{1,0} - \Omega_{2,0})\right],
\label{eq:hzpolycdm}
\end{equation}
where $\Omega_{1,0}$ and $\Omega_{2,0}$ are two additional parameters, which within the $\Lambda$CDM both of them remain absent. Nevertheless, in \cite{vazquez2018observational} $\Omega_{2,0}$ is interpreted as a 'missing matter' component introduced to allow a symmetry that relates the big bang to the future conformal singularity. We recover $\Lambda$CDM when $\Omega_{1,0} = 0$ and $\Omega_{2,0}=0$.
\subsection*{Datasets}
\textbf{Cosmic chronometers} (CC) are galaxies that evolve slowly and allow direct measurements of the Hubble parameter $H(z)$. These measurements have been collected along several years \cite{jimenez2003constraints, simon2005constraints, stern2010cosmic, moresco2012new, zhang2014four, moresco2015raising, moresco20166, ratsimbazafy2017age}, and now $31$ data points are available within redshifts between $0.09$ and $1.965$,
along with their statistical errors.
%
Given a Friedmann equation from a cosmological model then a theoretical value for $H(z)$ can be obtained and compared directly with these measurements.
The \textbf{growth rate measurement} is usually
referred to the product of $f\sigma_8(a)$ where $ f (a) \equiv d \ln \delta(a)/d\ln a $ is the growth rate of cosmological perturbations given by the density contrast $\delta(a)\equiv \delta \rho/\rho$, being $\rho$ the energy density and
$\sigma_8$ the normalisation of the power spectrum on scales within spheres of $8h^{-1}$Mpc \cite{said2020joint}.
Therefore, the observable quantity $f\sigma_8(a)$ [or equivalently $f\sigma_8(z)$] is obtained by solving numerically
%
\begin{equation}
f\sigma_8 (a) = a\frac{\delta'(a)}{\delta(1)} \sigma_{8,0}.
\end{equation}
%
The $f{\sigma_8}$ data are obtained through the peculiar velocities from Redshift Space Distortions (RSD) measurements \cite{kaiser1987clustering} observed in redshift survey galaxies or by weak lensing \cite{amendola2008measuring}, where the density perturbations of the galaxies are proportional to the perturbations of matter.
%
An extended version of the Gold-2017 compilation is available at \cite{sagredo2018internal}, with $22$ independent measurements of $f\sigma{_8}(z)$ from redshift space distortion measurements from a variety of surveys (see references therein).
\textbf{Supernovae} (SNeIa).
Let us assume a spatially flat universe, for which the relationship between the luminosity distance $d_L$ and the comoving distance $D(z)$ is given by:
%
\begin{equation}
d_L (z) = \frac{1}{H_0}(1+z)D(z), \qquad {\rm with }\qquad D(z) = H_0\int \frac{dz}{H(z)},
\end{equation}
%
Thus, the observable quantity is computed by the distance modulus $\mu(z) = 5 \log d_L(z) + 25$.
\noindent
The SNeIa dataset used in this work corresponds to the Joint Lightcurve Analysis (JLA), a compilation of 740 Type Ia supernovae. It is available in a binned version that consists in 31 data points with a covariance matrix $C_{jla} \in \mathbb{R}^{31 \times 31}$ related to the systematic measurement errors \cite{betoule2014improved}.
\subsection{Artificial neural networks}
\label{sec:ann}
The ANNs consist of several sets of neurons or nodes grouped in layers, and connections between them with an associated number called weight. In general, the learning mechanism of an ANN is as follows:
\begin{itemize}
\item The first layer of neurons reads the features of the dataset. In each connection between neurons is assigned a random number called weight (we use random numbers with normal distribution centred on $0$ with standard deviation of $0.01$). The input data make up a matrix $X_1$ and provide the values for the first layer of nodes. The $X_i$ refers to the values of nodes in the $i$-th layer. The weights make up another matrix $W_i$ and they are the values for the connections between the $i$-th and the $(i+1)$-th layers. The product $Z$ of these two matrices is the following:
%
\begin{equation}
Z_{i+1} = W_i^T X_i,
\end{equation}
%
where $W_i \in \mathbb{R}^{m \times n}$, with $m$ and $n$ the number of nodes in the $i$-th and $(i+1)$-th layers respectively. $X_i$ corresponds to the $i$-th layer, therefore has $m$ dimensions. It is worth to apply the transpose of $W_i$ in order to allow the matrix product.
\item A nonlinear activation (or transfer) function $\phi$ modulates $Z_i$ and assigns values to the next layer of neurons. This process, known as forward propagation, is repeated until the last layer is reached. The values of neurons in subsequent layers are given by:
%
\begin{equation}
X_{i+1} = \phi(Z_{i+1}).
\end{equation}
%
\item The value of the neurons in the last layer must be evaluated by an error function (or loss function) which measures the difference between the value given by the ANN and the expected one. The loss function is minimised by an optimisation algorithm such as \textit{gradient descent} combined with the \textit{backpropagation} algorithm to calculate gradients \cite{rumelhart1986learning, lecun2012efficient}. In this paper we use the mean squared error (MSE) as a loss function which is a usual selection in regression problems:
%
\begin{equation}
{\rm MSE} = \frac{1}{n} \sum_i^n (Y_i - \hat{Y}_i)^2,
\end{equation}
%
where $Y_i$ is a vector with predictions of the ANN, $\hat{Y}_i$ a vector with the expected values and $n$ is the number of predictions (or the length of $Y_i$ and $\hat{Y}_i$). The goal of the training ANN is to minimise the loss function. The number of samples propagated through the network before updating the weights is known as \textit{batch size} and each iteration of the entire data set constitutes an \textit{epoch}.
\item During backward propagation the weights are updated, then forward propagation is performed again. This is repeated until the loss function reaches the desired precision and then the neural network is trained and ready to make predictions.
\end{itemize}
At the beginning of the ANN training, the original dataset is split in two parts: training and validation sets. A common choice is $80\%$ and $20\%$ respectively. The first set is used to train the ANN, while the validation set contains unseen values, therefore it is useful for testing the performance of the ANN and evaluating its ability to produce a good model to the input dataset.
%
On the other hand, the \textit{Universal Approximation Theorem} \cite{hornik1990universal} states that an ANN with at least one hidden layer with a finite number of neurons can approach any continuous function if the activation function is continuous and nonlinear. Therefore an ANN is capable of learning the intrinsic functions inside cosmological datasets.
\\
Two types of artificial neural networks are implemented in this work: FeedForward Neural Networks (FFNN) and AutoEncoders (AE). The FFNN, also called multilayer perceptrons or deep feedforward networks, are the quintessential deep learning models \cite{goodfellow2016deep}. In this type of ANNs the connections between layers and the information flow are straightforward. They are composed of one input layer, at least one hidden layer and an output layer. The input is conformed by the independent variables (or features) of the dataset, while the output contains the dependent variables (or labels).
%
On the other hand, the autoencoders \cite{baldi1989neural} are trained to generate a copy of its input on its output. Autoencoders can be thought as two symmetrical coupled ANNs, where the first (encoder) makes a dimensional reduction for the input and obtains a coded representation (vector embedding or latent space) of the original data. The second part (decoder) takes the coded representation of the data and recovers an instance with the same data type and dimension of the original input. The encoder is a function $f$ that maps the input $x$ with dimension $l$ to an encoded vector $h$ with dimension $m$, with $m < l$:
%
\begin{equation}
f : x \in \mathbb{R}^l \rightarrow h \in \mathbb{R}^m,
\end{equation}
%
where $h_i := f_i(x) = \phi(W^T_i X_i), \; i=1,2,...,m$ with $\phi$ being the activation function. The decoder is the following $g$ function, that maps the encoded representation with dimension $m$ into an output $\hat{x}$ with the same dimension $l$ as the original input $x$:
%
\begin{equation}
g: h \in \mathbb{R}^m \rightarrow \hat{x} \in \mathbb{R}^l.
\end{equation}
%
\noindent
If the activation function, used in the autoencoder, is the identity function, i.e. $\phi(x) = x$, then this type of neural network is analogous to the Principal Component Analysis (PCA) technique. In this work, we use a particular type called variational autoencoder (VAE) \cite{kingma2013auto, rezende2014stochastic} described in the Appendix \ref{sec:appendix1}.
\\
As previously mentioned, the ANNs have the ability to approximate any function, in other words, they can generate computational models that generalise the input data. However, an ANN model has several intrinsic parameters (known as hyperparameters) such as the number of layers, number of nodes, batch size, optimiser algorithm and number of epochs, among others. It is worth to select carefully a good combination of them to guarantee that the ANN model has the capability of generalisation, an incorrect choice of them can produce undesirable models, either underfitted or overfitted with respect to the data.
Although the goal of the neural network training is to minimise the loss function, in our case the MSE, the following relationship should be taken into account:
%
\begin{equation}
{\rm MSE} = {\rm bias}^2 + {\rm variance},
\end{equation}
%
where the bias measures how far away the neural network predictions are from the actual value, while the variance refers to how much the prediction varies at nearby points. As the ANN model gets more complex, the bias can decrease while the variance can increase, this is called the bias-variance dilemma \cite{geman1992neural}. A model with high variance will be overfitted, while a model with high bias will be insufficient to learn the complexity of the data (underfitting). In both cases, the model generated by the neural network have inaccurate predictions. One way to avoid this problem is by monitoring the behaviour of the loss function throughout the training epochs, both in the training set and in the validation set.
%
A common practice to prevent an incorrect fitting in the ANN model is to increase the size of the training set, otherwise we need to calibrate carefully the hyperparameters of our ANN models to achieve acceptable results.
%
There are several approaches to tune the hyperparameters \cite{bardenet2013collaborative, hutter2009paramils, zhang2014four, larochelle2007empirical}. Because our ANNs have relatively simple architectures (between two and five hidden layers and just a few thousands of neurons) we use a common empirical strategy based on a grid of hyperparameters \cite{larochelle2007empirical}.
%
The hyperparameter grid consists of a selection of possible values for all the ANN parameters to be adjusted. The ANN is trained for all combinations included on the grid to find the one that provides the lowest value for the loss function in the validation set. In addition, it is necessary to verify that the loss function of both, the validation set and the training set, have a convergent behaviour to ensure that the ANN model is well trained (neither overfitting nor underfitting).
%
Whenever underfitting occurs it is mainly because the behaviour of the loss function in the test and training sets shows that the model has a high error in its predictions and, therefore, also a high bias. On the other hand, overfitting occurs when the loss function evaluated at the validation shows an increasing trend, or because there is a considerable gap with the loss function of the training set, thus, it has a high variance. We use the difference between the predictions of the last two epochs ($\Delta {\rm MSE}_{val}$) to get an idea about the variance of the ANN model, the smaller this error, the smaller the variance.
Another important concept is the dropout (DO), a regularisation technique \cite{srivastava2014dropout} that allows smaller values to be achieved in the loss function and prevents overfitting. It consists in randomly turning off neurons during training, so the neurons that operate at each epoch are different. The associated hyperparameter is a scalar value that indicates the probability of turning off a neuron in each epoch. Due to its random nature, the dropout can be used as a Monte Carlo simulation \cite{gal2015dropout}. When an ANN is trained, the dropout can be implemented in such a way that each prediction is different because the active neurons are different at each epoch. Therefore, it is possible to make several predictions, and thus obtain the average and standard deviations. Using this formalism, dubbed Monte Carlo dropout (MC-DO) \cite{gal2015dropout} we can obtain a statistical uncertainty of a trained ANN model. We apply the dropout method to the FFNNs implemented in this work and compare the results with those solely with FFNNs.
\subsection{Bayesian inference}
\label{sec:bayes}
Given a set of observational data and a mathematical expression for a cosmological model, a conditional probability density function can be constructed in terms of the model parameters and the observables.
There are many ways to infer the combination of parameters that best fit to the data. In cosmology, Bayesian inference algorithms have been used prominently \cite{trotta2008bayes, feroz2009, leclercq2018bayesian}; however, methods such as the Laplace approximation \cite{taylor2010analytic}, genetic algorithms \cite{nesseris2012new, arjona2020can}, simulated annealing \cite{hannestad1999stochastic} or particle swarm optimisation \cite{prasad2012cosmological} have also been explored.
Here, we use the Bayesian inference to test the quality of our synthetic data generated by the trained ANNs in comparison with the real data. Bayesian statistics is a paradigm in which probabilities are computed given the prior knowledge of the data \cite{padilla2019, medel2021}. It is based on Bayes' Theorem:
%
\begin{equation}
P(\theta|D)= \frac{P(D|\theta)P(\theta)}{P(D)},
\end{equation}
%
where $D$ represents the observational dataset and $\theta$ the set of free parameters in the theoretical model. $P(\theta)$ is the prior probability density function and represents the previous knowledge of the parameters $\theta$. $P(D|\theta)$ is the likelihood function and indicates the conditional probability density function of the data given the model. Finally, $P(D)$ is a normalisation constant, that is, the likelihood marginalisation and is called the Bayesian evidence and it is very useful in model comparison, for example it has been used in several papers to compare dark energy models through the Bayes factor and the Jeffrey's scale \cite{vazquez2020bayesian, Vazquez:2012ce}.
\section{Methodology}
\label{sec:methodoloy}
\begin{figure}[t!]
\captionsetup{justification=raggedright,singlelinecheck=false,font=footnotesize}
\centering
\makebox[12cm][c]{
\includegraphics[trim= 40mm 40mm 40mm 50mm, clip, width=5.5cm, height=4cm]{img/ffnnHZ.png}
\includegraphics[trim= 40mm 40mm 40mm 50mm, clip, width=5.5cm, height=4cm]{img/ffnnFS8.png}
\includegraphics[trim= 50mm 40mm 40mm 40mm, clip, width=5.5cm, height=4cm]{img/ffnnjla.png}
}
\caption{Neural network architectures chosen for cosmic chronometers (CC), $f{\sigma_8}$ measurements and distance modulus in JLA respectively; the batch size found for each case was: $16$, $1$ and $1$. In the last architecture, there is only one node in the output layer because the errors are computed with a variational autoencoder (described in the Appendix \ref{sec:appendix1}) given the original covariance matrix of the systematic errors. Blue numbers indicate the nodes in each layer.}
\label{fig:arch}
\end{figure}
Throughout this work we use FFNNs for datasets with diagonal covariance matrices and autoencoders when
correlations between the measurements are present, for instance within the JLA dataset.
In general, for the three types of cosmological observations (CC, $f{\sigma_8}$ and SNeIa) we have followed the next steps to find out a suitable neural network model for the corresponding data:
\begin{itemize}
\item We train several neural network configurations to gain insights about the complexity that their architecture require to model the data. According to the results of the loss function, we choose a number of layers.
%
\item Several values are suggested for each hyperparameter of the neural network, based on the intuition achieved in the first step, a grid is formed that must be traversed to find the combination that provides the minimum value of the loss function. Among the hyperparameters it is the batch size, the number of nodes per layer and, in some cases, the dropout.
%
\item The best ANN architectures found for each case are shown in Figure \ref{fig:arch}. The first two correspond to the CC and $f{\sigma_8}$ datasets respectively, for which $320$ combinations were tested up to three hidden layers: number of nodes in $\{50, 100, 150, 200\}$ and the batch size in $\{1,4,8,16,32\}$. We found that for the compressed JLA dataset a one-layer neural network works best, so we refined the third architecture among 20 combinations, varying the number of nodes in $\{30, 50, 100, 150, 200\}$ and the batch size in $\{1, 2, 4, 8\}$.
%
\item We train the neural network with the combination of hyperparameters chosen in the previous step with a correct number of epochs. We verify the behaviour of the loss function in the training and validation sets to check that our model is neither underfitted nor overfitted. The effect of the epochs in the learning process using the first two ANN architectures of Figure \ref{fig:arch} is shown in Figure \ref{fig:epochs}.
%
\item Once the neural network is trained, we can generate synthetic data points with absent redshift in the original datasets.
\item By making several predictions with the neural network, the reconstruction of the data can be appreciated and compared with the original data. If the statistical behaviour of the synthetic data is not consistent, the neural networks must be retrained.
%
\item We store the output, for a certain number of predicted data points as well as the new covariance matrix in order to be able to do the Bayesian inference of the cosmological models with these artificial data.
%
\item We compare the parameter estimation of the synthetic data with the original set to verify they are statistically consistent and to analyse their differences. For this purpose, we use the \texttt{SimpleMC}\footnote{www.github.com/ja-vazquez/SimpleMC} package \cite{simplemc}, initially released at \cite{aubourg2015}, along with a modified version of the \textit{dynesty} nested sampling library \cite{speagle2020dynesty}, which allows to do the parameter estimation and Bayesian evidence calculation.
\end{itemize}
We have developed a package called \texttt{CRANN}\footnote{Cosmological Reconstructions with Artificial Neural Networks (CRANN). The link will be available upon acceptance.} that contains the ANNs models already trained to produce synthetic cosmological data given a set of arbitrary redshifts. All the ANNs used in this work and their hyperparameter tuning were based on \texttt{Tensorflow}\footnote{www.tensorflow.org} and \texttt{Keras}\footnote{www.keras.io} Python libraries.
In the case of cosmic chronometers and $f{\sigma_8} $ measurements we use FFNNs. These types of data have a diagonal covariance matrix and hence it can be arranged into a single column of the same length as the number of measurements. Therefore, these two datasets have three features (columns): the redshift $z$, the function $f(z)$, and the related error (taken from the diagonal of the covariance matrix). Once the FFNN is trained, for a given of set of points (redshifts) we can output the cosmological function together with its simulated statistical error. It should be noted that the neural networks learn to generate the cosmological function and the error, where the latter is the result of modelling the original statistical errors from the observational data set.
\begin{figure}[t!]
\centering
\captionsetup{justification=raggedright,singlelinecheck=false,font=footnotesize}
\makebox[12cm][c]{
\includegraphics[trim= 10mm 20mm 20mm 15mm, clip, width=4.5cm, height=3.cm]{img/epHD20.png}
\includegraphics[trim= 31mm 20mm 20mm 15mm, clip, width=4.3cm, height=3.cm]{img/epHD100.png}
\includegraphics[trim= 31mm 20mm 20mm 15mm, clip, width=4.3cm, height=3.cm]{img/epHD500.png}
\includegraphics[trim= 31mm 20mm 20mm 15mm, clip, width=4.3cm, height=3.cm]{img/epHDmil.png}
}
\makebox[12cm][c]{
\includegraphics[trim= 10mm 0mm 20mm 15mm, clip, width=4.5cm, height=3.cm]{img/epFS820.png}
\includegraphics[trim= 31mm 0mm 20mm 15mm, clip, width=4.3cm, height=3.cm]{img/epFS8100.png}
\includegraphics[trim= 31mm 0mm 20mm 15mm, clip, width=4.3cm, height=3.cm]{img/epFS8500.png}
\includegraphics[trim= 31mm 0mm 20mm 15mm, clip, width=4.3cm, height=3.cm]{img/epFS8mil.png}
}
\caption{Effect of the number of epochs in the training with the CC dataset (top) and with the $f{\sigma_8}(z)$ measurements (bottom). The first case ($20$ epochs) shows underfitting, while considering $1000$ epochs shows overfitting. In the $f{\sigma_8}$ dataset, the cases for $500$ and $1000$ epochs present overfitting. In both cases, we choose $100$ epochs due to the lowest value of MSE and $\Delta {\rm MSE}$ in the validation set. Green points display real data-points with error bars, and in purple synthetic data along with red error bars.}
\label{fig:epochs}
\end{figure}
%
On the other hand, through the analysis of the JLA SNeIa compilation, we also use a FFNN to learn the behaviour of distance modulus $\mu(z)$ in a similar fashion we did for the CC and $f{\sigma_8}$. However, in order to handle the full covariance matrix we use a VAE as described in the Appendix \ref{sec:appendix1}; using this type of neural network allows us to map the distribution of the distance modulus to the distribution of the coded representation of the autoencoder to generate new covariance matrices.
%
One restriction of this method to bear in mind is that the new matrix has to have the same dimension as the original one. However, we can generate a matrix given any combination of new redshifts, provided that this set has the same length as the original measurements.
In addition to the above procedure, we slightly modify the FFNNs to implement an epistemic calculation of their uncertainties using the dropout \cite{gal2015dropout}. In this way, we add dropout between the layers of the FFNNs and run the Monte Carlo dropout several times to obtain average values and uncertainties for each prediction (as described in Section \ref{sec:ann}). We combine our FFNN designs with the implementation of MC-DO layers from \texttt{astronn}\footnote{https://github.com/henrysky/astroNN}\cite{leung2019deep}, and compare the results of this method with the previous ANNs implementations.
Because dropout is a regularisation technique, the number of epochs is irrelevant for a large enough set.
%
The error predictions and the uncertainties are independent, therefore the total standard deviation is:
\begin{equation}
\sigma = \sqrt{er_{p}+\sum_i u_i^2 },
\label{eq:sigma}
\end{equation}
\noindent
where $u_i$ is the epistemic uncertainty involved with the FFNN used and $er_{p}$ is the error prediction.
As mentioned above, once the non-parametric reconstruction is obtained we can generate synthetic data and use them to perform a Bayesian inference procedure. For these purposes, we use the following flat priors: for the matter density parameter today $\Omega_m \in [0.05, \; 0.5]$, for the physical baryon density parameter $\Omega_b h^2 \in [0.02, \; 0.025]$, for the reduced Hubble constant $h \in [0.4 , \; 0.9]$, and for the amplitude of the (linear) power spectrum $\sigma_8 \in [0.6 , \; 1.0]$. When assuming the CPL parameterisation, we use $w_0 \in [-2.0 , \; 0.0]$ and $w_a \in [-2.0 , \; 2.0]$; and for the PolyCDM model, we use $\Omega_1 \in [-1.0 ,\; 1]$ and $\Omega_2 \in [-1.0 ,\; 1]$. The $h$ parameter refers to the dimensionless reduced Hubble parameter today ${H}/{100}$kms$^{-1}$Mpc$^{-1}$.
\section{Results}
\label{sec:results}
In order to perform the reconstructions of the Hubble parameter $H(z)$, the growth rate measurement $f{\sigma_8}(z)$ and the distance modulus $\mu(z)$ we apply two different methods implemented with the feedforward neural networks shown in Figure \ref{fig:arch}: i) using the trained neural network (FFNN) and ii) along with the FFNN, by considering uncertainties with the Monte Carlo dropout (FFNN+MC-DO). To test the quality of our ANNs predictions we perform the Bayesian inference procedure for the CPL and PolyCDM models with the original data and with the data generated by the neural networks. Then, and to improve the constraints on the free parameters, in some cases we also include a compressed version of Planck-15 information (treated as a BAO experiment located at redshift $z= 1090$ \cite{aubourg2015}).
\subsection*{Reconstruction of $H(z)$}
Once the chosen FFNN is trained with the CC dataset (green points in Figure \ref{fig:recHZ}), we input new redshift values ($z$) and let the network to predict the corresponding values for $H(z)$ and their errors. In the left-panel of Figure \ref{fig:recHZ}, we generate $1000$ synthetic data (magenta points) with their respective error bars (red lines) with the FFNN. These errors are the result of the ANN modelling the errors contained in the original dataset.
Besides the intrinsic error associated with the datasets, we consider an uncertainty related with the FFNN by adding a Monte Carlo dropout between each layer of the chosen FNNN architecture, under the method described in \cite{gal2015dropout}. Among several tests to dropout values between $[0, 0.5]$, we found that a good value for the dropout is $0.3$ and we trained the FFNN with MC-DO along $1000$ epochs.
%
After training the new FFNN with dropout between each layer, we can make the predictions to $1000$ unseen redshifts and, with $100$ executions of MC-DO for each prediction. We obtain the right-panel of Figure \ref{fig:recHZ}, that contains the total standard deviation considering the uncertainties of the neural network. In this case, the result has variations caused by the probabilistic nature of the MC-DO and, now, the error bars contain information about the statistical uncertainty of the FFNN trained model; indeed the error is larger than in the FFNN alone case.
\begin{figure}[t!]
\centering
\captionsetup{justification=raggedright,singlelinecheck=false,font=footnotesize}
\makebox[11cm][c]{
\includegraphics[trim= 0mm 0mm 0mm 0mm, clip, width=8.5cm, height=5.5cm]{img/recHZl.png}
\includegraphics[trim= 0mm 0mm 0mm 0mm, clip, width=8.5cm, height=5.5cm]{img/recHZDOl.png}
}
\caption{$H(z)$ reconstructions produced with $1000$ synthetic data points generated with FFNNs. \textit{Left}: Purple line represents the FFNN predictions for $H(z)$ along with their error bars in red colour. \textit{Right}: Similarly to FFNN but adding MC-DO, we executed $100$ times the Monte Carlo dropout to compute the uncertainties of the predictions, therefore the purple points are the average predictions of the MC-DO executions and the red error bars are the uncertainties of the FFNN plus the error predictions (see Equation \ref{eq:sigma}). In both cases, we compare the non-parametric reconstruction with the original cosmic chronometers (green bars) and $H(z)$ from $\Lambda$CDM, as shown in the labels. The small panels displayed the receptive behaviour of the loss function (MSE) in the training (red) and validation (green) sets.}
\label{fig:recHZ}
\end{figure}
An interesting feature shown in both panels of Figure \ref{fig:recHZ} is that, despite the original dataset does not contain a measurement for $H(z=0)$, the FFNN prediction is $H_{pred}(z=0) = 75.09 \pm 15.49 \;$ km s$^{-1}$ Mpc$^{-1}$ and the prediction of the FFNN with Monte Carlo dropout is $H_{pred}(z=0) = 77.07 \pm 15.91~ $km s$^{-1}$ Mpc$^{-1}$. Both results have a better matching to the one measured with Cepheid variables \cite{riess20162}: $H_0 = 73.24 \;$km s$^{-1}$ Mpc$^{-1}$. In addition, it can be appreciated that as the redshift increases, the model generated by the FFNNs approaches $\Lambda$CDM with $H_0 = 67.40 \;$ km s$^{-1}$ Mpc$^{-1}$ measured by Planck mission \cite{aghanim2020planck}. It is worth to remember that the ANN models do not have any prior cosmological or statistical assumption so they have been built solely from the data.
\subsection*{Reconstruction of $f{\sigma_8}(z)$}
Similarly to the Hubble parameter case, we apply the same methodology but now with measurements of the $f\sigma_8$ function \cite{sagredo2018internal}. In order to generate the reconstruction plot shown in the left panel of Figure \ref{fig:recFS8}, once the FFNN is trained, we generate $1000$ synthetic data points (red points). Besides, we put the evaluation of $f{\sigma_{8}}(z)$ from CPL model in three different scenarios of $w_0$ and $w_a$.
We added Monte Carlo dropout to the FFNN, shown in the second ANN architecture of Figure \ref{fig:arch}, to be able to calculate the uncertainties of the ANN. We train this new FFNN along 2000 epochs. In this case, among several tests to dropout values between $[0, 0.5]$, we choice a dropout of $0.1$ because it had the best performance. Then we obtain, with 1000 synthetic $f{\sigma_{8}}$ data points, the reconstruction of the right-panel of Figure \ref{fig:recFS8}, where the purple line is the average obtained by MC-DO predictions and the error bars contain an error conformed by the standard deviations (uncertainties) of MC-DO for each prediction plus the error predictions.
We can notice, that in both cases, the models plotted are within the reconstruction and hence this dataset by itself may provide loose constraints on the CPL parameters. However, the values $w_0 = -0.8$ and $w_a = -0.4$ (brown line) seem to have a better agreement with the reconstruction, as we shall see below.
\begin{figure}[t!]
\centering
\captionsetup{justification=raggedright,singlelinecheck=false,font=footnotesize}
\makebox[11cm][c]{
\includegraphics[width=8.5cm, height=5.2cm]{img/recFS8l.png}
\includegraphics[width=8.5cm, height=5.2cm]{img/recFS8dol.png}
}
\caption{Reconstruction of $f\sigma_8(z)$ with $1000$ synthetic data points (red dots) and their respective errors (red bars) learned by ANNs. \textit{Left}: FFNN alone, red line is conformed by the predictions of $f{\sigma_{8}}$. \textit{Right}: FFNN using Monte Carlo dropout, the averages of $100$ executions of MC-DO are indicated with the red line and their standard deviations are added to the error predictions. In both cases the small panels displayed the receptive behaviour of the loss function (MSE) in the training (red curve) and validation (green curve) sets.}
\label{fig:recFS8}
\end{figure}
\subsection*{Distance modulus $\mu (z)$ reconstructions}
%
Regarding to the distance modulus $\mu(z)$, we train a FFNN (last ANN in Figure \ref{fig:arch}) for a given redshift. We assume a gaussian distribution for the predictions of the distance modulus and using a trained VAE we produce a new covariance matrix (see Appendix \ref{sec:appendix1}). Once the FFNN is trained, we can generate synthetic data points for unseen redshifts and reconstruct the $\mu(z)$ function as it can be appreciate in the left panel of Figure \ref{fig:recJLA}. In particular, we use 31 log-uniformly distributed redshifts over the interval $z \in [0.01, 1.3]$. To apply MC-DO, we did several tests to dropout values in the interval $[0, 0.5]$ and we found that a dropout with $0.01$ value has a good performance. We executed $100$ times MC-DO to obtain the right panel of Figure \ref{fig:recJLA}.
%
\begin{figure}[t!]
\centering
\makebox[11cm][c]{
\includegraphics[trim=4mm 10mm 20mm 10mm, clip, width=8.8cm, height=8.5cm]{img/ffnn_jla.png}
\includegraphics[trim= 28.5mm 10mm 20mm 10mm, clip, width=8cm, height=8.5cm]{img/ffnn_jla_do.png}
}
%
\captionsetup{justification=raggedright,singlelinecheck=false,font=footnotesize}
\caption{\textit{Left}: 31 new data points (red dots) generated with FFNN. \textit{Right}: 31 new data points (red dots) generated with FFNN+MC-DO. In both cases it is shown their receptive behaviour of the loss function (MSE) in the training (red curve) and validation (green curve) sets along the chosen number of epochs for each case (300 and 1800). The 31 green dots are the original points from the binned version of JLA.}
\label{fig:recJLA}
\end{figure}
An important point to bear in mind is that when using a full covariance matrix we need to restrict to $31$ synthetic data points in order to generate the covariance matrix with the VAE for mapping to the new points in the latent space. The synthetic covariance matrix (Figure \ref{fig:covmatrix}) was generated by the VAE, with the distance modulus predictions from the FFNNs (refer to the appendix for a detail description).
\begin{figure}[t!]
\centering
\captionsetup{justification=raggedright,singlelinecheck=false,font=footnotesize}
\makebox[11cm][c]{
\includegraphics[trim= 30mm 0mm 15mm 0mm, clip, width=4.2cm, height=4.1cm]{img/originalcov_.png}
\includegraphics[trim= 30mm 0mm 15mm 0mm, clip, width=4.2cm, height=4.1cm]{img/vae_cov_ffnn_.png}
\includegraphics[trim=28mm 0mm 15mm 0mm, clip, width=4.2cm, height=4.1cm]{img/vae_cov_ffnn_do_.png}
}
\caption{\textit{Left}: Original covariance matrix with systematic errors from JLA compilation (binned version). Covariance matrices predicted by the VAE (\textit{Middle}) and VAE with MC-DO (\textit{Right}).}
\label{fig:covmatrix}
\end{figure}
%
\subsection*{Parameter estimation with synthetic data}
Neural networks allow us to produce data models with several parameters (neural network weights) which are difficult to interpret, however with the use of synthetic data generated by these models we can analyse them with Bayesian inference and compare their results with those obtained from the original data. Thus we performed the Bayesian inference analysis to estimate the best fit parameters of the CPL and PolyCDM models. The aim of this procedure is to look for possible deviations of the $\Lambda$CDM model with the neural networks approach.
In addition to the three original datasets (cosmic chronometers, $f{\sigma_8}$ measurements and binned JLA compilation), we have created two datasets for each type of observation from the trained FFNNs with and without MC-DO. As a proof of the concept, the new datasets for CC and $f{\sigma_8}$ consisted of 50 random uniformly distributed points in redshift, while for SNeIa they were 31 log-uniformly distributed in redshift (same size as the original dataset). For the SNeIa case we also generated its respective covariance matrix with the decoder part of the trained VAE.
We have used the data from CC, $f\sigma_8$ measurements and JLA separately, and also some combinations of them. The most representative results are in Figure \ref{fig:posteriorall} along with Table \ref{tab:results}, which contains mean values and standard deviations, and they have been sorted according to the datasets used as a source (original, FFNN, and FFNN+MC-DO), and to the models involved ($\Lambda$CDM, CPL, and PolyCDM). It is indicated when the Planck point has been added to the data sets. Results are displayed for the reduced Hubble parameter $h$, $\sigma_8$, $w_0$ and $w_a$ measurements for the CPL model, and $\Omega_1$ with $\Omega_2$ when the model is PolyCDM. In addition, the last column of the Table \ref{tab:results} contains the $-2\log\mathcal{L}_{max}$ of the Bayesian inference process for each case.
Before analysing each scenario separately, it is worth mentioning that there are some generalities in the results. In general, it can be noted that when using a single source separately (no Planck information added) the constraints are consistent among each other, that is, they all have a similar best-fit (maximum likelihood) and are in agreement with the $\Lambda$CDM model.
In the case of parameter estimation using exclusively the CC dataset, the first two panels of Figure \ref{fig:posteriorall} (and first block in Table \ref{tab:results}) show that the best-fit values are mutually contained within their $1\sigma$ standard deviations and are in agreement with the $\Lambda$CDM values. However, we can notice that when Planck information is added
the reduced Hubble parameter value slightly increases for the ANNs.
For instance, for the CPL model with original data the constraints are $h=0.673 \pm 0.046$ whereas for the synthetic data it increases to $0.713 \pm 0.059$ for the FFNN and $0.726 \pm 0.063$ for the FFNN+MC-DO; in fact these values obtained by the synthetic data are closer to the Hubble parameter value of the Cepheid variables than to the Planck mission value. This issue, as a supplement to Figure \ref{fig:recHZ}, shows that the neural network models generated by cosmic chronometers are sensing the Hubble tension, although considering the size of the standard deviation values, all the results of the parameter estimation are still statistically consistent with each other.
Something similar happens when added Planck information to JLA SNeIa and assuming the CPL model. With the original data the constraints are $h=0.695 \pm 0.021$ whereas for the synthetic data they increase to $0.704 \pm 0.025$ for the FFNN and $0.712 \pm 0.026$ for the FFNN+MC-DO. However both datasets are still statistically consistent, within $1\sigma$, with the $\Lambda$CDM parameters.
If taken into account the JLA+CC combination with the Planck information, the increment of the reduced Hubble parameter is still present but also a small deviation of $\Lambda$CDM (about $1\sigma$) for FFNN+MC-DO, with constraints of $w_0=-0.957 \pm 0.141$ and $ w_a=-0.563 \pm 0.669$.
On the other hand, considering only measurements of $f\sigma_8$+Planck for the synthetic data, the $w_0$ and $w_a$ constraints suggest a slight deviation from $\Lambda$CDM. With the FFNN data the values are $w_0 = -0.657 \pm 0.172$ and $w_a = -0.493 \pm 0.265$, and for FFNN+MC-DO we have $w_0 =-0.673 \pm 0.183$ and $w_a =-0.364 \pm 0.221$. In fact, it can be seen in the Figure \ref{fig:posteriorall} that the cosmological constant is right on the limits of the $2\sigma$ contours.
By using all datasets combined CC+$f\sigma_8$+JLA we have performed a Bayesian inference to the three models $\Lambda$CDM, CPL and PolyCDM. With the original datasets we found consistency throughout the models with the $\Lambda$CDM parameters and slight shift when using the synthetic data, for instance higher values for the reduced Hubble parameter, lower for the $\sigma_8$ parameter and for $w_0$: $-0.916\pm 0.065$ (FFNN) and $-0.925 \pm 0.068$ (FFNN+MC-DO).
Deviations of the standard values are enhanced once we use synthetic data along with Planck information. This can be seen on the constraints of the PolyCDM model for the FFNN source: $\Omega_1=0.272 \pm 0.194$, $\Omega_2=-0.092 \pm 0.058$. Also, based on the improvement in the fit alone ($\sqrt{2\Delta \ln \mathcal{L}_{max} }$), and using the same source, we found a preference to the data for the CPL model of $1.5\sigma$ and $1.7\sigma$ for the PolyCDM. That is, the Artificial Neural Network by itself is finding deviations from the standard cosmological model.
The above discussion suggests that, if the models generated by the neural networks are correct, hypothetical new observations within the range of the existing ones would tend to move away from $\Lambda$CDM. Therefore, parameter estimation in the CPL and PolyCDM models in conjunction to the models generated by the neural networks suggest that $\Lambda$CDM does not have the best match to the data. In all cases, the addition of the Planck point increases the tension and the need for a model beyond $\Lambda$CDM.
To reinforce the idea that models generated by the neural networks depart from $\Lambda$CDM, from the posterior distribution samples for CPL, we obtained the posterior distribution of its corresponding EoS, as shown in Figure \ref{fig:EoSrec} (using \texttt{fgivenx} Python library \cite{fgivenx}). From these plots it can be seen that $w=-1$ lies in the most probable region within $1\sigma$ with the original data; however, in the case of the synthetic data the cosmological constant moves away from the most probable region, still within $1\sigma$ without considering the Planck point, and outside $1\sigma$ when it is taken into account.
Finally, as part of the Bayesian analysis, we perform a model comparison with Bayesian evidences $Z$ through the Bayes' factor $B$ and the Jeffrey's scale \cite{vazquez2020bayesian}. Table \ref{tab:bayevidence} shows the log-Bayes' factor of $\Lambda$CDM compared to CPL and PolyCDM models using the different sources of data. It can be seen that with the synthetic data the penalisation of having extra parameters decreases from strong advantage to an inconclusive advantage due to the improvement of the fit in both models. However, it is worth noting that $\Lambda$CDM stays with a slight advantage.
%
\setcounter{table}{1}
\begin{table}[hbt!]
\scriptsize
\centering
\captionsetup{justification=raggedright, singlelinecheck=false, font=footnotesize}
\begin{tabular}{p{2.2cm} p{2.2cm}p{2.2cm}p{2.2cm}}
\hline
\hline
Source & CPL & PolyCDM \\
\hline
Original & 3.651 & 2.837 \\
FFNN & 1.823 & 0.687 \\
FFNN+MC-DO & 2.464 & 1.159 \\
\hline
\\
\end{tabular}
\caption{Log-Bayes' factor $\ln(B) = \ln (Z_{\Lambda CDM}) - \ln (Z)$ of $\Lambda$CDM with respect the other models using the same data source for each case. The combined dataset used in this table is JLA+CC+$f_{\sigma 8}$+Planck}.
\label{tab:bayevidence}
\end{table}
%
\begin{figure}[t!]
\centering
\captionsetup{justification=raggedright,singlelinecheck=false,font=footnotesize}
\makebox[11cm][c]{
\includegraphics[trim=3mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postHDwowa.png}
\includegraphics[trim=3mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postHDplanckwowa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postFS8plancks8wa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postFS8wowa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postFS8planckwowa.png}
}
\makebox[11cm][c]{
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postJLAwowa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postJLAplanckwowa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postHDJLAwowa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postHDJLAplanckwowa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postALLwoh.png}
}
\makebox[11cm][c]{
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postALLplanckwoh.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postALLwowa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postALLplanckwowa.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postpolycdm1.png}
\includegraphics[trim=3.5mm 0mm 3mm 0mm, clip, width=3.8cm, height=3.7cm]{img/postpolycdm2.png}
}
\caption{2D marginalised posterior distributions from different combinations of datasets: original data, synthetic datasets from FFNN and FFNN+MC-DO. The green dashed lines ($w_0=-1$, $w_a=0$) and ($\Omega_1 = 0$, $\Omega_2 = 0$) correspond to the $\Lambda$CDM model. The constraints are plotted with $1\sigma$ and $2\sigma$ confidence contours.}
\label{fig:posteriorall}
\end{figure}
\begin{figure}[t!]
\centering
\captionsetup{justification=raggedright,singlelinecheck=false,font=footnotesize}
\makebox[11cm][c]{
\includegraphics[trim=4mm 4mm 11mm 4mm, clip , width=5.5cm, height=3.5cm]{img/cplEoS.png}
\includegraphics[trim=4mm 4mm 11mm 4mm, clip , width=5.5cm, height=3.5cm]{img/cplEoS_ann.png}
\includegraphics[trim=4mm 4mm 11mm 4mm, clip , width=5.5cm, height=3.5cm]{img/cplEoS_ann_do.png}
}
\makebox[11cm][c]{
\includegraphics[trim=4mm 4mm 11mm 4mm, clip , width=5.5cm, height=3.5cm]{img/cplEoS_planck.png}
\includegraphics[trim=4mm 4mm 11mm 4mm, clip , width=5.5cm, height=3.5cm]{img/cplEoS_ann_planck.png}
\includegraphics[trim=4mm 4mm 11mm 4mm, clip , width=5.5cm, height=3.5cm]{img/cplEoS_ann_do_planck.png}
}
\caption{Posterior probability distribution functions of the Dark Energy EoS considering CPL parameterisation, by using original data, FFNN and FFNN+MC-DO respectively.
Planck-15 point information is additionally included in the lower panels. $1-3\sigma$ confidence intervals are plotted as black lines.}
\label{fig:EoSrec}
\end{figure}
\setcounter{table}{0}
\begin{table}[hbt!]
\scriptsize
\centering
\captionsetup{justification=raggedright, singlelinecheck=false, font=footnotesize}
\begin{tabular}{p{2.0cm} p{1.2cm}p{1.2cm}p{2.0cm}p{2.2cm}p{2.0cm}p{2.0cm}p{1.4cm}}
\hline
\hline
Source & Model & & Datasets: & \textbf{CC} & & & $ -2 \ln \mathcal{L}_{max}$ \\
& & ~ & $h$ & $w_0$ & $w_a$ & $\;$ & \\
\hline
Original & $\Lambda$CDM & - & $0.678 \pm 0.039$ & $--$ & $--$ & $\;$ & $14.502$ \\
& CPL & - & $0.703 \pm 0.064$ & $-1.223 \pm 0.447$ & $-0.061 \pm 1.075$ & $\;$ & $14.290$ \\
& CPL & + Planck & $0.673 \pm 0.046$ & $-0.867 \pm 0.326$ & $-0.325 \pm 0.824$ & $\;$ & $14.638$ \vspace{0.1cm} \\
FFNN & $\Lambda$CDM & - & $0.698 \pm 0.057$ & $--$ & $--$ & $\;$ & $0.176$ \\
& CPL & - & $0.703 \pm 0.071$ & $-1.072 \pm 0.431$ & $-0.179 \pm 1.025$ & $\;$ & $0.042$ \\
& CPL & + Planck & $0.713 \pm 0.059$ & $-0.962 \pm 0.337$ & $-0.485 \pm 0.890$ & $\;$ & $0.120$ \vspace{0.1cm} \\
FFNN+MC-DO & $\Lambda$CDM & - & $0.699 \pm 0.063$ & $--$ & $--$ & $\;$ & $0.346$ \\
& CPL & - & $0.689 \pm 0.078$ & $-1.014 \pm 0.450$ & $-0.227 \pm 1.003$ & $\;$ & $0.284$ \\
& CPL & + Planck & $0.726 \pm 0.063$ & $-1.029 \pm 0.355$ & $-0.377 \pm 0.897$ & $\;$ & $0.808$ \\
\hline
\hline
& & & Datasets: & ${f\sigma_8}$ & & &$ -2 \ln \mathcal{L}_{max}$ \\
& & ~ & $h$ & $w_0$ & $w_a$ & $\sigma_8$ & \\
\hline
Original & $\Lambda$CDM & - & $0.648 \pm 0.147$ & $--$ & $--$ & $0.787 \pm 0.115$ & $11.932$ \\
& CPL & - & $0.638 \pm 0.135$ & $-0.742 \pm 0.264$ & $-0.144 \pm 0.468$ & $0.777 \pm 0.111$ & $11.908$ \\
& CPL & + Planck & $0.648 \pm 0.062$ & $-0.801 \pm 0.229$ & $-0.225 \pm 0.254$ & $0.771 \pm 0.109$ & $11.944$ \vspace{0.1cm} \\
FFNN & $\Lambda$CDM & - & $0.650 \pm 0.144$ & -- & $--$ & $0.694 \pm 0.172$ & $0.292$ \\
& CPL & - & $0.648 \pm 0.142$ & $-0.701 \pm 0.271$ & $-0.290 \pm 0.540$ & $0.777 \pm 0.111$ & $0.284$ \\
& CPL & + Planck & $0.628 \pm 0.046$ & $-0.657 \pm 0.172$ & $-0.493 \pm 0.265$ & $0.756 \pm 0.109$ & $0.374$ \vspace{0.1cm} \\
FFNN+MC-DO & $\Lambda$CDM & - & $0.651 \pm 0.147$ & $--$ & $--$ & $0.652 \pm 0.170$ & $0.984$ \\
& CPL & - & $0.632 \pm 0.140$ & $-0.674 \pm 0.270$ & $-0.156 \pm 0.489$ & $0.775 \pm 0.110$ & $0.960$ \\
& CPL & + Planck & $0.622 \pm 0.046$ & $-0.673 \pm 0.183$ & $-0.364 \pm 0.221$ & $0.756 \pm 0.103$ & $1.038$ \\
\hline
\hline
& & & Datasets: & \textbf{JLA} & & & $ -2 \ln \mathcal{L}_{max}$ \\
& & ~ & $h$ & $w_0$ & $w_a$ & $\;$ & \\
\hline
Original & $\Lambda$CDM & - & $0.638 \pm 0.146$ & $--$ & $--$ & $\;$ & $33.214$ \\
& CPL & - & $0.652 \pm 0.141$ & $-0.901 \pm 0.238$ & $-0.216 \pm 0.899$ & $\;$ & $32.354$ \\
& CPL & + Planck & $0.695 \pm 0.021$ & $-0.880 \pm 0.140$ & $-0.606 \pm 0.696$ & $\;$ & $30.528$ \vspace{0.1cm} \\
FFNN & $\Lambda$CDM & - & $0.645 \pm 0.144$ & $--$ & $--$ & $\;$ & $14.670$ \\
& CPL & - & $0.640 \pm 0.137$ & $-1.092 \pm 0.277$ & $0.287 \pm 0.957$ & $\;$ & $13.888$ \\
& CPL & + Planck & $0.704 \pm 0.025$ & $-1.061 \pm 0.178$ & $-0.018 \pm 0.811$ & $\;$ & $14.808$ \vspace{0.1cm} \\
FFNN+MC-DO & $\Lambda$CDM & - & $0.643 \pm 0.142$ & $--$ & $--$ & $\;$ & $16.446$ \\
& CPL & - & $0.641 \pm 0.135$ & $-1.037 \pm 0.248$ & $-0.245 \pm 0.996$ & $\;$ & $16.274$ \\
& CPL & + Planck & $0.712 \pm 0.026$ & $-0.994 \pm 0.165$ & $-0.395 \pm 0.802$ & $\;$ & $16.504$ \\
\hline
\hline
& & & Datasets: & \textbf{CC+JLA} & & &$ -2 \ln \mathcal{L}_{max}$ \\
& & ~ & $h$ & $w_0$ & $w_a$ & $\;$ & \\
\hline
Original & $\Lambda$CDM & - & $0.690 \pm 0.030$ & $--$ & $--$ & $\;$ & $47.822$ \\
& CPL & - & $0.687 \pm 0.030$ & $-0.980 \pm 0.173$ & $-0.156 \pm 0.939$ & $\;$ & $47.830$ \\
& CPL & + Planck & $0.687 \pm 0.018$ & $-0.946 \pm 0.137$ & $-0.232 \pm 0.592$ & $\;$ & $47.918$ \vspace{0.1cm} \\
FFNN & $\Lambda$CDM & - & $0.705 \pm 0.037$ & $--$ & $--$ & $\;$ & $14.846$ \\
& CPL & - & $0.695 \pm 0.037$ & $-1.010 \pm 0.165$ & $0.315 \pm 0.715$ & $\;$ & $14.096$ \\
& CPL & + Planck & $0.708 \pm 0.021$ & $-1.031 \pm 0.150$ & $-0.167 \pm 0.688$ & $\;$ & $15.478$ \vspace{0.1cm} \\
FFNN+MC-DO & $\Lambda$CDM & - & $0.703 \pm 0.038$ & $--$ & $--$ & $\;$ & $16.808$ \\
& CPL & - & $0.698 \pm 0.039$ & $-0.968 \pm 0.155$ & $-0.046 \pm 0.859$ & $\;$ & $16.688$ \\
& CPL & + Planck & $0.717 \pm 0.019$ & $-0.957 \pm 0.141$ & $-0.563 \pm 0.669$ & $\;$ & $17.252$ \\
\hline
\hline
& & & Datasets: & \textbf{CC+JLA+} ${f\sigma_8}$ & & & $ -2 \ln \mathcal{L}_{max}$ \\
& & ~ & $h$ & $w_0$ & $w_a$ & $\sigma_8$ & \\
\hline
Original & $\Lambda$CDM & - & $0.695 \pm 0.032$ & $--$ & $--$ & $0.795 \pm 0.115$ & $60.244$ \\
& $\Lambda$CDM & +Planck & $0.690 \pm 0.013$ & $--$ & $--$ & $0.790 \pm 0.109$ & $60.33$ \\
& CPL & - & $0.692 \pm 0.029$ & $-0.933 \pm 0.086$ & $0.009 \pm 0.476$ & $0.763 \pm 0.111$ & $59.840$ \\
& CPL & + Planck & $0.685 \pm 0.015$ & $-0.961 \pm 0.057$ & $-0.122 \pm 0.194$ & $0.763 \pm 0.112$ & $59.832$ \vspace{0.1cm} \\
FFNN & $\Lambda$CDM & - & $0.721 \pm 0.034$ & $--$ & $--$ & $0.786 \pm 0.117$ & $16.278$ \\
& $\Lambda$CDM & +Planck & $0.704 \pm 0.013$ & $--$ & $--$ & $0.7500 \pm 0.105$ & $19.191$\\
& CPL & - & $0.712 \pm 0.032$ & $-0.916 \pm 0.065$ & $0.150 \pm 0.432$ & $0.786 \pm 0.114$ & $15.076$ \\
& CPL & + Planck & $0.712 \pm 0.015$ & $-0.941 \pm 0.057$ & $-0.417 \pm 0.246$ & $0.733 \pm 0.100$ & $17.044$ \vspace{0.1cm} \\
FFNN+MC-DO & $\Lambda$CDM & - & $0.713 \pm 0.035$ & $--$ & $--$ & $0.775 \pm 0.116$ & $18.096$ \\
& $\Lambda$CDM & +Planck & $0.702 \pm 0.012$ & $--$ & $--$ & $0.753 \pm 0.100$ & $20.842$ \\
& CPL & - & $0.706 \pm 0.037$ & $-0.925 \pm 0.068$ & $0.222 \pm 0.443$ & $0.763 \pm 0.105$ & $17.734$ \\
& CPL & + Planck & $0.711 \pm 0.017$ & $-0.970 \pm 0.055$ & $-0.318 \pm 0.247$ & $0.723 \pm 0.097$ & $19.034$ \\
\hline
& & & & $\Omega_1$ & $\Omega_2$ & & $ -2 \ln \mathcal{L}_{max}$ \\
\hline
Original & PolyCDM & - & $0.693 \pm 0.029$ & $0.089 \pm 0.416$ & $0.034 \pm 0.302$ & $0.788 \pm 0.108$ & $59.832$ \\
& PolyCDM & +Planck & $0.696 \pm 0.017$ & $0.147 \pm 0.238$ & $-0.020 \pm 0.071$ & $0.779 \pm 0.107$ & $59.972$ \vspace{0.1cm} \\
FFNN & PolyCDM & - & $0.712 \pm 0.035$ & $0.110 \pm 0.444$ & $0.048 \pm 0.302$ & $0.776 \pm 0.109$ & $15.186$ \\
& PolyCDM & +Planck & $0.736 \pm 0.021$ & $0.272 \pm 0.194$ & $-0.092 \pm 0.058$ & $0.781 \pm 0.105$ & $16.422$ \vspace{0.1cm} \\
FFNN+MC-DO & PolyCDM & - & $0.707 \pm 0.036$ & $-0.054 \pm 0.435$ & $0.134 \pm 0.302$ & $0.775 \pm 0.114$ & $17.730$ \\
& PolyCDM & +Planck & $0.732 \pm 0.019$ & $0.173 \pm 0.195$ & $-0.062 \pm 0.057$ & $0.762 \pm 0.105$ & $18.936$ \\
\hline
\\
\end{tabular}
\caption{Parameter estimation using Bayesian inference with datasets from different sources: original, FFNN alone and FFNN using Monte Carlo dropout.}
%
\label{tab:results}
\end{table}
\section{Conclusions}
\label{sec:conclusions}
Throughout this work, we have shown that well-calibrated artificial neural networks have the capacity to produce non-parametric reconstructions from which synthetic cosmological data, statistically consistent with the originals, can be generated even when the datasets are small.
We have explored the generation of synthetic covariance matrices through VAE, and the results have allowed us to carry out Bayesian inference without drawbacks. However, for larger datasets, we believe that it will be convenient to use convolutional layers in the autoencoder and a slightly different approach to dealing with the computing demand.
Using the Monte Carlo dropout method allows to have more information about the predictions of neural networks through their epistemic estimation of uncertainty. The results obtained have also contributed to both the methodological and the cosmological analysis and to validate the outputs from the ANNs without this method as they are very similar.
The models generated by the neural networks were produced exclusively from the data, therefore they offer the possibility of reconstructing cosmological functions without assumptions about the data distribution and without assuming any cosmological model as a starting point. From the non-parametric reconstructions produced with the neural networks, we were able to observe how the Hubble parameter changes as the cosmic chronometers are at higher redshifts, as suggested by the current Hubble tension. We could also note that SNeIa are observations very much in accordance with $\Lambda$CDM and, in contrast, that $\Lambda$CDM is not the best model to describe $f\sigma_8$ measurements. Overall, using Bayesian inference on the CPL and PolyCDM models with the synthetic neural network data, we have observed that the $\Lambda$CDM model does not perfectly match these data and loses some of the advantage given by the original observations.
It is worth mentioning that the cosmological results obtained in this work are limited to current cosmological observations and have been sufficient to show some interesting cosmological features from the data alone. We have shown that our method can be a good complement to the traditional Bayesian analysis and, moreover, could be applied to other types of cosmological observations and models, as well as to perform some forecasts. In this way, we can see that the use of neural networks, from their models created for the data and the generation of synthetic data, can complement the analysis of cosmological models and improve the interpretations of their behaviours. We plan to apply similar techniques to other types of cosmological data, including a complete set of covariance matrices.
\section*{Acknowledgements}
This work was partially supported by CONACyT-Mexico scholarship, BEIFI-IPN. RG-S acknowledges the support provided by SIP20200666-IPN and SIP20210500-IPN grants and FORDECYT-PRONACES-CONACYT CF-MG-2558591. JAV acknowledges the support provided by FOSEC SEP-CONACYT Investigaci\'on B\'asica A1-S-21925, FORDECYT-PRONACES-CONACYT 304001 and UNAM-DGAPA-PAPIIT IA104221.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,059 |
Investing in real estate and property is a pursuit that will create streams of passive income for you in the future, potentially. Most people value passive income as one of the best ways to make money for your future. Active income streams mean that you are getting paid directly for the time you put in. If you run out of time, you run out of money. There are many reasons why getting your name on property deeds and land ownership titles will be ideal for your future success.
The prices of land continue to skyrocket. That's possibly to keep up with inflation, but partly to do with the notion that everyone wants private property. Investing in land is lucrative, but it can be done now with minimal risk. If you find land on Zillow or other property sites, you will find you can get a few acres for as low at $10 000 in the States. This means that you will soon be on your way to developing that land with housing, or reselling it in a few years' time for triple the amount. And even after you pay capital gains on the sale, you will still come out ahead.
Some people make empires out of running rental property. That's why investing in land can lead you to a career in property management. If there ever was a good reason as to why invest in land, this is it. You will maximize your investment. You can build housing on the unit. And then you will be able to rent the property out to the vast market out there.
If you believe that you shouldn't have to always trade time for money in a direct correlation, then this might be the avenue for you to look into. When you work on the property up front, later the checks will keep rolling in. Sure you might have to visit the site for maintenance, but you will find that you will be able to squirrel away funds to pay for what you invested. And after that, anything you gain above maintenance fees is just pure profit. There are not too many business opportunities that generate this same type of passive revenue. Retail stores might be another example, and in this instance too, you can build a commercial property on your land if you get the proper permits. The sky is the limit when it comes to people enjoying bright horizons after they initially invest in land.
The above list should introduce you to the many benefits of land ownership. There is nothing that beats the feeling of having something in your own name. And the deed cannot be taken away from you. Anything leased, such as a car, can be repossessed. But a bit of land will stay in your name after you purchase it, provided that you stay out of debt. There are numerous reasons to join the ranks of land owning individuals. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,686 |
Q: grep of executable commands with variable Have a file with thousands of lines like so :
echo " device " ; login 'command' ip > device_command" $(date +"%Y_%m_%d_%I_%M_%p")".txt
I run my script like so:
./script.sh | grep "device"
..in order to target specific device. Each line within the script is executable. Question is how do grep it from an other script and EXECUTE the line, not just echo it out.
#!/bin/bash
nameFind=$1
grep "$nameFind" script.sh | while read line
do
"result of grep "
done
A: One way would be to simply pipe it to a shell:
./script.sh | grep "device" | sh
Simple example:
$ cat input
echo one
echo two
echo three
$ grep t input | sh
two
three
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 850 |
Paolo Mosetti (29 January 1939 – 17 February 2009) was an Italian Olympic rower.
References
External links
1939 births
2009 deaths
Italian male rowers
Rowers at the 1960 Summer Olympics
Olympic rowers of Italy
Sportspeople from Trieste
European Rowing Championships medalists
Mediterranean Games gold medalists for Italy
Mediterranean Games medalists in rowing
Competitors at the 1963 Mediterranean Games | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,140 |
{"url":"https:\/\/www.bayesianspectacles.org\/the-problem-of-old-evidence\/","text":"# The Problem of Old Evidence\n\nTo my shame and regret, I only recently found the opportunity to read the book \u201cBayesian philosophy of science\u201d (BPS), by Jan Sprenger and Stephan Hartmann. It turned out to be a wonderful book, both in appearance, typesetting, and in contents. The book confirmed many of my prior beliefs ;- ) but it also made me think about the more philosophical topics that I usually avoid thinking about. One example is \u201cthe problem of old evidence\u201d (due to Glymour, 1980). Entire forests have been felled in order for philosophers to be able to debate the details of this problem.\n\nBelow I will provide my current perspective on this problem, considered by Sprenger and Hartmann to present \u201cone of the most troubling and persistent challenges for Bayesian Confirmation Theory\u201d (p. 132). It is highly likely that my argument is old, or even beside the point. I am not an expert on this particular problem. But first let\u2019s outline the problem.\n\n### Old Evidence\n\nIn order to provide the correct context I will quote liberally from PBS:\n\nThe textbook example from the history of science is the precession of the perihelion of the planet Mercury (\u2026). For a long time, Newtonian mechanics failed to account for this phenomenon; and postulated auxiliary hypotheses (e.g., the existence of another planet within the orbit of Mercury) failed to be confirmed. Einstein realized in the 1910s that his General Theory of Relativity (GTR) accounted for the perihelion shift. According to most physicists, explaining this \u201cold evidence\u201d (in the sense of data known previously) conferred a substantial degree of confirmation on GTR, perhaps even more than some pieces of novel evidence, such as Eddington\u2019s 1919 solar eclipse observations. (\u2026)\n\nWe can extract a general scheme (\u2026): A phenomenon $E$ is unexplained by the available scientific theories. At some point, it is discovered that theory $T$ accounts for $E$. The observation $E$ is \u201cold evidence\u201d: at the time when the relationship between $T$ and $E$ is developed, the scientist is already certain, or close to certain, that the phenomenon $E$ is real. Indeed, in the GTR example, astronomers had been collecting data on the Mercury perihelion shift for many decades.\u201d\u00a0 (PBS, pp. 131-132)\n\nPBS then presents the problem in the form of Bayes rule:\n\n$$p(T | E) = p(T) \\frac{p(E | T)}{p(E)},$$\n\nand argue: \u201cWhen $E$ is old evidence and already known to the scientist, her degree of belief in $E$ is maximal: $p(E) = 1$. Because $T$ predicts $E$, also $p(E | T) = 1$.\u201d It follows that $E$ cannot confirm $T$.\n\n### Issue 1: Old Evidence or Unrecognized Evidence?\n\nFrom my perspective, the problem of old evidence does not highlight a limitation of Bayesian confirmation theory, but a limitation of the human intellect. To make this clear, let\u2019s change the scenario such that $T$ comes first, and $E$ comes later. For instance, assume that Einstein first developed GRT and that the astronomical data on the perihelion shift were collected five years later. Crucially, however, assume that when these astronomical data first became available, nobody thought that they would be relevant to GRT. Epistemically we are then in exactly the same situation as the one that the field was in immediately after Einstein first proposed GRT: in both scenarios there is a theory $T$, there are data $E$, but these data are (falsely!) judged to be non-diagnostic or predictively irrelevant. That is, scientists mistakenly judged that\n\n$$\\frac{p(E | T)}{p(E | \\text{not-}T)} = 1,$$\n\nthat is, theory $T$ neither helps nor hurts in explaining, accounting for, or predicting $E$.\n\nThus, it does not matter whether $E$ came before or after $T$. What matters is that $E$ was incorrectly judged to be irrelevant. At some point after $E$ has been observed and $T$ has been developed (regardless of the temporal order of these two events), scientists think about the problem more deeply and then discover that they have made an error in judgment, and that $E$ is in fact diagnostic and predictively relevant:\n\n$$\\frac{p(E | T)}{p(E | \\text{not-}T)} >> 1.$$\n\nThe fault therefore appears to lie not with Bayesian confirmation theory, but with researchers not being omniscient.\n\n### Issue 2: p(E) = 1?\n\nOne may object that the equation immediately above is incorrect, as $E$ is known to have occurred and hence $p(E | \\text{not-}T)$ is simply 1. Perhaps the crux of the problem is meant to be that predictive performance can be assessed only for events whose outcome is still uncertain. I must admit that I find this whole business of arguing that $p(E) = 1$ rather strange. Suppose I tell you that I played a dice game yesterday and my opponent, who is notorious for cheating, rolled four sixes in seven throws. It appears completely legitimate (to me) to assess the probability of this having happened under a fair-die hypothesis versus a loaded-die hypothesis. I view $p(E | \\text{not-}T)$ as a prediction that follows from the model $\\text{not-}T$, and the *model* does not know whether or not $E$ has occurred. Another example: suppose we wish to assess the predictive adequacy of a weatherperson W. We provide W with abundant information about the weather, and W then issues a probabilistic prediction about the amount of precipitation for the next day. It does not matter whether this information happens to refer to data from a previous year, from a previous day, or from today (with the amount of precipitation still hidden in the future). We wish to assess the predictive adequacy of W, so all that matters is that W itself is agnostic about the outcome. Epistemically, for W the probability of the outcome is definitely not 1. The same holds, I would argue, when we wish to assess the predictive performance of statistical models. When the models do not have access to the outcome, it does not matter whether or not that outcome has already manifested itself to the scientist \u2013 we are evaluating the models, not the scientists.\n\n#### References\n\nGlymour, C. (1980). Theory and evidence. Princeton, NJ: Princeton University Press.\nSprenger, J., & Hartmann, S. (2019). Bayesian philosophy of science. Oxford: Oxford University Press. https:\/\/academic.oup.com\/book\/36527","date":"2023-03-28 03:25:16","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\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 33, \"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\": 3, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6503176093101501, \"perplexity\": 775.353309043145}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296948756.99\/warc\/CC-MAIN-20230328011555-20230328041555-00747.warc.gz\"}"} | null | null |
#ifndef HKP_LINEAR_PARAMETRIC_CURVE_H
#define HKP_LINEAR_PARAMETRIC_CURVE_H
#include <Physics/Constraint/Data/PointToPath/hkpParametricCurve.h>
extern HK_EXPORT_PHYSICS const hkClass hkpLinearParametricCurveClass;
/// Piecewise linear curve class
class HK_EXPORT_PHYSICS hkpLinearParametricCurve : public hkpParametricCurve
{
public:
HK_DECLARE_CLASS_ALLOCATOR(HK_MEMORY_CLASS_BASE);
HK_DECLARE_REFLECTION();
/// Default constructor.
///
/// Default smoothing parameter is 0.01f.
/// Loop is "non-closed" by default.
/// To create a closed loop, the last segment should overlap the first segment exactly.
/// This is easily accomplished by adding the first TWO points again at the end of the path.
/// Or, if your last point is coincident to the first point, just add the second point of your
/// path again as the last point. After this, setting the m_closedLoop flag to true will give
/// you the desired behavior.
hkpLinearParametricCurve();
/// Given a parametric value, map to a point on the curve.
virtual void getPoint( hkReal t, hkVector4& pointOnCurve ) const;
/// Map a parametric value and a point to the nearest point on the curve. Returns the parametric value.
virtual hkReal getNearestPoint( hkReal t, const hkVector4& nearPoint, hkVector4& pointOnPath ) const;
/// Return the normalized tangent at the point on the curve specified by the parametric value.
virtual void getTangent( hkReal t, hkVector4& tangent ) const;
/// Return the smallest parametric value that is on the curve.
virtual hkReal getStart() const;
/// Return the largest parametric value that is on the curve.
virtual hkReal getEnd() const;
/// Return physical length along the curve from the start of the curve to this parametric point.
virtual hkReal getLengthFromStart( hkReal t ) const;
/// Return the normalized vector that defines "up" for a path.
/// WARNING: For this curve, we calculate a binormal as being normal to the tangent
/// but not necessarily continuous.
/// A far better method would be to have a binormal specified at each point and interpolate between them.
virtual void getBinormal( hkReal t, hkVector4& up ) const;
/// Returns true if the path is closed. i.e., the beginning is the end and the end is the beginning.
virtual hkBool isClosedLoop() const;
// hkpLinearParametricCurve interface
/// Add a point to the path.
void addPoint( const hkVector4& p );
/// Gets the smoothing factor.
/// Smoothing factor determines how much to smooth out seams between line segments.
hkReal getSmoothingFactor() const;
/// Sets the smoothing factor.
/// Smoothing factor determines how much to smooth out seams between line segments.
/// Smoothing factor is the % from the end points of each line segment to begin smoothing.
/// A value of 0.025 allows for fairly sharp angles. Set it to 0 if you want the curve to be
/// discontinuous in the first derivative.
void setSmoothingFactor( hkReal smooth );
/// Get an array of points to draw.
virtual void getPointsToDraw(hkArray<hkVector4>& pathPoints) const;
/// Closes or uncloses the loop.
/// A closed loop needs to have the last segment overlap the first segment exactly.
void setClosedLoop( hkBool closeLoop );
/// Transform all the points in the curve
virtual void transformPoints( const hkTransform& transformation );
///Create an exact copy of the curve
virtual hkpParametricCurve* clone();
public:
hkReal m_smoothingFactor;
hkBool m_closedLoop;
hkVector4 m_dirNotParallelToTangentAlongWholePath;
hkArray<hkVector4> m_points;
hkArray<hkReal> m_distance;
public:
hkpLinearParametricCurve(hkFinishLoadedObjectFlag f) : hkpParametricCurve(f), m_points(f), m_distance(f) {}
};
#endif // HKP_LINEAR_PARAMETRIC_CURVE_H
/*
* Havok SDK - NO SOURCE PC DOWNLOAD, BUILD(#20140907)
*
* Confidential Information of Havok. (C) Copyright 1999-2014
* Telekinesys Research Limited t/a Havok. All Rights Reserved. The Havok
* Logo, and the Havok buzzsaw logo are trademarks of Havok. Title, ownership
* rights, and intellectual property rights in the Havok software remain in
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* acceptance of the End User licence Agreement for this product. A copy of
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,070 |
\section{Introduction}
Monte Carlo methods are the most fundamental pillar for pricing models in quantitative finance,
and a considerable amount effort has been made to refine their properties,
from variance reduction techniques~\cite[Chapter 4]{Glasserman}
(importance sampling, antithetic variables or control variates)
to discretisation of stochastic differential equations~\cite{Kloeden}
and their multilevel extensions~\cite{Giles1, Giles2, Giles3}.
Indeed, most financial instruments do not admit closed-form pricing formulae,
and any numerical technique is thus at the mercy of instabilities and approximation errors.
This of course generated an appetite for variance reduction techniques,
whereby more stability can be achieved with similar computation time and level of accuracy.
There has recently also been a lot of interest in leveraging the power of machine learning tools,
mainly on the use of neural networks to solve pricing~\cite{Madan, Ferguson, Ghee, Hahn, Hutchinson, Kondratyev}
and calibration~\cite{Bayer, Horvath, Stone} problems.
We consider here an alternative approach, which involves the application of simple regression techniques to improve Monte Carlo estimates. We borrow the idea from Alonso, Tracey and Wolpert~\cite{Tracey, Tracey2}
who were first motivated by applications in aeronautics.
The main idea is to learn appropriate control variates for option pricing problems,
in order to achieve high levels of variance reduction.
This task faces two challenges; the first is to obtain unbiased estimates of the pricing payoff function,
as the Stacked Monte Carlo technique proposed in~\cite{Tracey,Tracey2} did not apply to derivatives pricing.
The second challenge is to select the appropriate fitting functions for our pricing problems,
such that the model closely approximates the solution whilst being sufficiently simple to minimise computation times.
We start below by presenting the stacked Monte Carlo in its simple form, namely to numerically compute
integrals, before applying it pricing options in the Black-Scholes and in the Heston model.
We shall consider European options as well as Asian options, showing how to adapt the method to
multivariate problems.
We then carry out numerical tests highlighting both the simplicity and the efficiency of the method.
\section{Stacked Monte Carlo}
The Stacked Monte Carlo (StackMC) method~\cite{Tracey, Tracey2}, is a post-processing technique for reducing the error in Monte Carlo estimates.
The main idea is to learn a control variate from the Monte Carlo draws,
such that the distribution of the learnt function approximates that of the original problem.
While doing so, it is crucial to avoid bias and overfitting, as this may worsen the final result.
We rewrite the solution to a Monte Carlo problem~$f(x)$ in terms a learnt function~$g(x)$
\begin{equation}\label{eq:StackMCInt}
\widehat{f} =\int{f(x)p(x)\mathrm{d} x}
= \int{\alpha g(x)p(x)\mathrm{d} x} + \int{\left[f(x)-\alpha g(x)\right] p(x)}
= \alpha \widehat{g} + \int{\left[f(x)-\alpha g(x)\right] p(x)},
\end{equation}
where~$\alpha$ is some constant and $\widehat{g} := \int{g(x)p(x)\mathrm{d} x}$.
If~$g$ is a reasonable fit to~$f$,
the difference $f - \alpha g$ for some appropriately chosen~$\alpha$ will have lower variance
than the original problem.
Over the observations $(x^{(i)})_{i=1,\ldots,N}$, the estimate
\begin{equation}\label{eq:methods:f_stack}
\widetilde{f} := \alpha\widehat{g} + \frac{1}{N}\sum_{i=1}^N \left[f\left(x^{(i)}\right) - \alpha g\left(x^{(i)}\right)\right]
\end{equation}
has variance $\sigma^2_{\widetilde{f}} = \sigma^2_{f} + \alpha^2 \sigma^2_{g} - 2\alpha \sigma_{f,g}$
which is minimised as soon as
\begin{equation}\label{eq:methods:alpha}
\alpha = \frac{\sigma_{f}}{\sigma_{g}}\rho_{f,g},
\end{equation}
which implies
$\sigma^2_{\widetilde{f}} = (1-\rho_{f,g}^2)\sigma^2_{f}$,
and the condition $\rho_{f,g}\neq 0$ is enough to guarantee variance reduction.
Intuitively, if~$g$ is a good fit then~$\rho$ should be high and the estimate~$\widehat{g}$ is trusted,
otherwise it is ignored.
It remains to compute the function~$g$ and to estimate~$\alpha$,
both of which are achieved by fitting the samples of~$f$.
\subsection{Model choice for the control variate}
In order to achieve significant reduction while controlling the computational cost,
it is essential to choose a fitting function~$g$ for which the integral $\widehat{g}$ in~\eqref{eq:methods:f_stack}
is available in closed form, or at least is costless to compute.
A convenient choice is a simple polynomial of order~$L$:
\begin{equation}
g(x) = \sum_{n=0}^{L}\widehat{c}_n x^n,
\end{equation}
where the coefficients $\widehat{\boldsymbol{c}} = (\widehat{c}_n)_{i=0, \ldots, L}$
is defined as the least-square minimiser
$$
\widehat{\boldsymbol{c}} := \argmin_{\boldsymbol{c}} \left\| f(x) - \sum_{n=0}^L c_n x^n \right\|^2.
$$
\begin{remark}\label{rem:Multig}
\begin{enumerate}[(i)]\
\item If the problem is multidimensional (in the case of basket options),
with $\boldsymbol{x} = (x_i)_{i=1,\ldots,m}$, we may then consider a multivariate polynomial
$g(\boldsymbol{x}) = \sum_{|\boldsymbol{\alpha}|\leq L} c_{\boldsymbol{\alpha}}\boldsymbol{x}^{\boldsymbol{\alpha}}$,
using multi-index notations, where $\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_m) \in \mathbb{N}^m$.
\item Since Call option payoffs are discontinuous,
it may be convenient to choose piecewise-polynomial functions,
with zero value in part of the domain, for example,
for some truncation plane~$\boldsymbol{a}$,
\begin{equation}
g(\boldsymbol{x}) = \left\{
\begin{array}{ll}
0, & \text{if } \displaystyle \boldsymbol{a}^T \boldsymbol{x} + \boldsymbol{a}_0 < 0\\
\displaystyle \sum_{|\boldsymbol{\alpha}|\leq L} c_{\boldsymbol{\alpha}}\boldsymbol{x}^{\boldsymbol{\alpha}}, & \text{if }\boldsymbol{a}^T \boldsymbol{x} + \boldsymbol{a}_0 \geq 0.
\end{array}\right.
\label{eq:methods:pl_g}
\end{equation}
\end{enumerate}
\end{remark}
\subsection{Fitting the control variate}
Tracey et al.~\cite{Tracey} proposed using K-folds cross-validation to achieve an unbiased function estimate. This technique involves dividing the data into several non-overlapping training sets (or folds),
and using the left out points to test the fit.
We split the~$N$ samples of the variable~$\boldsymbol{x}$ into~$K$ folds of equal size
to create the sets $\boldsymbol{x}^{(k),\text{train}}$ and $\boldsymbol{x}^{(k),\text{test}}$,
for each $k=1,\ldots, K$.
Each individual data sample is included in a test set once, and in a training set $K-1$ times.
A different function $g_k(\boldsymbol{x})$ is estimated using the~$N-N_k$ samples in each training set,
and tested against the $N_k$ left-out samples from the $k$-th fold.
We thus obtain an estimate of the StackMC solution for each set from~\eqref{eq:methods:f_stack},
\begin{equation}\label{eq:methods:smc_k}
\widetilde{f}_k := \alpha\widehat{g}_k + \frac{1}{N_k}\sum_{i=1}^{N_k}
\left[f\left(x_i^{(k),\text{test}}\right) - \alpha g_k\left(x_i^{(k),\text{test}}\right)\right],
\qquad\text{for }k=1,\ldots, K.
\end{equation}
And the final `stacked' solution is the average over the folds
\begin{equation}\label{eq:SMC}
\widetilde{f}_{\text{SMC}} := \frac{1}{K} \sum_{k=1}^K \widetilde{f}_k.
\end{equation}
\subsection{Estimating the control variate parameter} \label{sec:methods:alpha}
The parameter~$\alpha$ is estimated from the out-of-sample points
using~\eqref{eq:methods:alpha} with the classical unbiased empirical estimates
$\widehat{\sigma}_{f,g}$, $\widehat{\sigma}^2_{f}$ and~$\widehat{\sigma}^2_{g}$ defined as
\begin{equation*}
\begin{array}{rlrl}
\widehat{\mu}_g & := \displaystyle \frac{1}{N}\sum_{k=1}^K \sum_{i=1}^{N_k} g_k\left(x_i^{(k),\text{test}}\right), &
\widehat{\mu}_f & := \displaystyle \frac{1}{N}\sum_{i=1}^N f(x_i),\\
\widehat{\sigma}^2_{g} & := \displaystyle \frac{1}{N-1}\sum_{k=1}^K \sum_{i=1}^{N_k} \left[g_k\left(x_i^{(k),\text{test}}\right)-\mu_g\right]^2,&
\widehat{\sigma}^2_{f} & := \displaystyle \frac{1}{N-1} \sum_{i=1}^N (f(x_i) - \mu_f)^2, \\
\widehat{\sigma}_{f,g} & := \displaystyle \frac{1}{N-1} \sum_{k=1}^K\sum_{i=1}^{N_k}\left[f\left(x_i^{(k),\text{test}}\right) - \mu_f\right]\left[g\left(x_i^{(k),\text{test}}\right) - \mu_g\right].
\end{array}
\end{equation*}
\subsection{Integrating the control variate}
We consider the computation of the analytical integral $\widehat{g}$ of the control variate function in the case where~$g$ is a polynomial as in Remark~\ref{rem:Multig}(i):
$$
\widehat{g}(\boldsymbol{x}) = \int\sum_{|\boldsymbol{\alpha}|\leq L} c_{\boldsymbol{\alpha}}\boldsymbol{x}^{\boldsymbol{\alpha}}p(\boldsymbol{x})\mathrm{d} \boldsymbol{x}
= \sum_{|\boldsymbol{\alpha}|\leq L} c_{\boldsymbol{\alpha}}\int\boldsymbol{x}^{\boldsymbol{\alpha}}p(\boldsymbol{x})\mathrm{d} \boldsymbol{x}
= \sum_{|\boldsymbol{\alpha}|\leq L} c_{\boldsymbol{\alpha}}\EE{\boldsymbol{x}^{\boldsymbol{\alpha}}}.
$$
In this case, the problem reduces to the computation of moments.
If the random variable is drawn from a one-dimensional zero-mean Normal distribution with variance~$\sigma^2$,
as in the standard Black-Scholes model, we may use the property:
\begin{equation}
\EE{X^n} = \left\{
\begin{array}{ll}
0, & \text{if }n\text{ is odd},\\
\displaystyle \frac{n! \sigma^n}{ 2 ^ {n / 2} (n/2)!}, & \text{if }n\text{ is even},
\end{array}
\right.
\end{equation}
In the multidimensional (centered) Gaussian case with variance-covariance matrix~$\Sigma$
(as will be useful later for Asian options),
the moments can be derived from the moment generating function
$\EE{\mathrm{e}^{\boldsymbol{u} \boldsymbol{X}}} = \exp\left(\frac{1}{2}\boldsymbol{u}^T\Sigma \boldsymbol{u}\right)$.
In the case where the control variate has a piecewise linear form,
\begin{equation}
g(\boldsymbol{x}) = \left\{
\begin{array}{ll}
\displaystyle c_0 + \sum_j c_j x_j, & \text{if } \boldsymbol{x} \in \Omega_{+},\\
0, & \text{otherwise},
\end{array}\right.
\end{equation}
where $\Omega_{+}:= \left\{\boldsymbol{x}: \boldsymbol{a}^T \boldsymbol{x} + a_0 \geq 0 \right\}$,
so that the computation of~$\widehat{g}$ boils down to computing the zeroth and first moments of the truncated Gaussian distribution
$$
\int_{\Omega_+}{n(\boldsymbol{x})\mathrm{d} \boldsymbol{x}}
\qquad\text{and}\qquad
\int_{\Omega_+}{x_j n(\boldsymbol{x})\mathrm{d} \boldsymbol{x}},
\qquad\text{for each }j = 1,\ldots, M,
$$
where $n(\cdot)$ denotes the multivariate Gaussian density.
\begin{remark}\label{rem:LinearCaseComput}
The integral above on a truncated domain can in fact be computed in closed form when the integrand
(without the Gaussian density) is linear, as here.
Following~\cite{Sharples}, since a linear combination of a multivariate Gaussian is Gaussian, we can write
$$
\int_{\Omega_+}n(\boldsymbol{x})\mathrm{d} \boldsymbol{x}
= \int_{-a_0}^{\infty} \frac{1}{\|\boldsymbol{a}\|\sqrt{2\pi}}\exp\left\{-\frac{x^2}{2\|\boldsymbol{a}\|^2}\right\}\mathrm{d} x
= 1 - \mathcal{N}\left(-\frac{a_{0}}{\|\boldsymbol{a}\|}\right).
$$
For the first moment, we can use~\cite[Theorem 5]{Sharples} to write
$$
\int_{\Omega_+}x_j n(\boldsymbol{x})\mathrm{d} \boldsymbol{x}
= \frac{a_j}{\left\|\boldsymbol{a}\right\|\sqrt{2\pi}}\exp\left(-\frac{a_{0}^2}{2\|\boldsymbol{a}\|^2}\right),
\qquad\text{for any }j=1,\ldots, M.
$$
\end{remark}
\section{Pricing options in the Black-Scholes model}
We now test the Stacked Monte Carlo method presented above on the pricing of options\footnote{All tests were conducted on a desktop with 2 Intel Xeon 2.00 GHz processors and 16 GB RAM,
running Windows 7 Enterprise. The code was written in \texttt{Python~3.6}, using \texttt{Numpy~1.15} and \texttt{Scikit-learn~0.20}. For comparison purposes, all calculations were computed on a single thread.}.
We use the \texttt{random} method from \texttt{NumPy},
which employs a Mersenne-Twister generator, to generate all Gaussian samples.
All numerical results reported here are obtained by running the Monte Carlo solver ten times
and by taking the average of the parameter of interest (solution, confidence interval, relative improvement) over the runs.
This avoids any bias with respect to the seed--determined by the system time from the pseudo-random number generator--especially for low numbers of paths.
\subsection{Stacked Monte Carlo for European options}\label{sec:results_european}
We first consider the price of a European Call option with payoff $(S_T-K)_+$,
for some strike~$K$, when the underlying stock price~$S$ follows the Black-Scholes model
\begin{equation}\label{eq:BS}
\frac{\mathrm{d} S_t}{S_t} = r \mathrm{d} t + \sigma \mathrm{d} W_t,
\end{equation}
starting from $S_0>0$ for some one-dimensional Brownian motion~$W$.
To apply the StackMC algorithm, we note that
$\log(S_T)$ is a Gaussian random variable, and hence the price of the Call option is given by
$$
\mathrm{e}^{-rT}\EE{S_T - K}_+ = \mathrm{e}^{-rT}\int_{\mathbb{R}}\left(S_0\mathrm{e}^{\left(r-\frac{\sigma^2}{2}\right)T + \sigma\sqrt{T}x} - K\right)_{+}n(x)\mathrm{d} x,
$$
where~$n(\cdot)$ denotes the Gaussian density, which is exactly of the form~\eqref{eq:StackMCInt}.
We start by fitting a polynomial function $g(\cdot)$ (of order $L=4$),
and follow the StackMC methodology above with
$K=2$ folds and $N=10^5$ Gaussian samples.
We then estimate~$\alpha$ as in Section~\ref{sec:methods:alpha}.
With the parameters:
$$
(S_0, K, r, \sigma, T) = (100, 100, 5\%, 20\%, 1),
$$
for which the exact price is $C_{\mathrm{BS}} = 10.4506$, the results are shown in Table~\ref{tab:OptionPrices}.
\begin{table}[h]
\begin{tabular}{rrr|rrr|r|rr}
\multicolumn{ 3}{c|}{MC} & \multicolumn{ 3}{c|}{Stacked Monte Carlo} & \multicolumn{ 1}{c|} {Total} & \multicolumn{ 2}{c} {Improvement} \\
\hline
Price & $\text{CI}_\text{MC}$ & Time & Price & $\text{CI}_\text{SMC}$ & Time & Time & Absolute & Ratio \\
\hline
10.4395 & 0.0913 & 0.57 & 10.4507 & 0.0061 & 0.30 & 0.87 & 0.0851 & 14.90 \\
\hline
\\
\end{tabular}
\caption{StackMC vs standard Monte Carlo for a European Call option}
\label{tab:OptionPrices}
\end{table}
Here, CI denotes the half-width of the confidence interval defined as
$$
\left[\mu - u_{l/2}\frac{\sigma_N}{\sqrt{N}},\ \mu + u_{l/2} \frac{\sigma_N}{\sqrt{N}} \right],
$$
where~$\mu$ is the mean, $\sigma_N$ the sample standard deviation, and $u_{l/2}$ the Gaussian quantile ($u_{l/2} = 1.96$ for $l=95\%$).
The absolute improvement is defined as $\text{CI}_\text{MC}-\text{CI}_\text{SMC}$,
and the improvement ratio is $\text{CI}_\text{MC}/\text{CI}_\text{SMC}$.
All indicated times are in seconds.
To standardise the reporting of run times and render these independent of PC performance,
we consider Table~\ref{tab:OptionPrices} as time unit, namely the time taken to price a European Call option with~$10^5$ Monte Carlo draws ($0.57$ second).
We expect run times to scale (approximately) linearly with the number of simulations.
Results below are reported in these units unless otherwise specified.
We also quote an equivalent Monte Carlo time, defined as an estimate of the time
it would have taken to achieve the confidence interval of StackMC using MC alone.
This is the MC time multiplied by the square of the improvement ratio,
and must be compared to the total runtime, i.e. the time taken to perform both MC and StackMC.
Under such measure, the previous results read
\begin{equation*}
\begin{array}{cccc}
\text{MC} & \text{StackMC} & \text{Total} & \text{Equivalent} \\
1 & 0.52504 & 1.525 & 228.08\\
\end{array}
\end{equation*}
On average, the StackMC procedure achieved a nearly 15-fold improvement in the size of the confidence interval with respect to simple MC, at the cost of approximately $0.3$ seconds runtime. Considering that the Monte Carlo solution converges at a rate of~$\mathcal{O}\left(N^{-1/2}\right)$, to achieve a similar variance reduction using Monte Carlo alone,
we must increase the number of simulations by a the improvement ratio squared (about~$228$).
Given that runtime scales linearly with the number of simulations,
this would have taken significantly longer than the additional time taken by the stacking procedure, which is only about half of the MC runtime.
The code we are using here is not fully optimised, and significant speed improvements
can be made for both the MC and StackMC implementations by exploiting parallelisation,
minimising data loops, and increasing algorithm efficiency.
Some speed-up gains may be greater for Monte Carlo, given the high potential for parallelisation~\cite{Joshi}.
\subsection{Dependency on model parameters}
We now investigate how the number of paths (Table~\ref{tab:NbPaths} and Figure~\ref{fig:results:mcvsmc_nsteps}), cross-validation technique,
and fitting model affect the result of Section~\ref{sec:results_european}, holding all else equal.
For the cross-validation method, we increase the number of folds used in the K-folds method,
and compare the results with those obtained via simple random sub-sampling (without folds).
We vary the proportion of data in the training sample, and use the remainder to estimate the control variate parameter~$\alpha$ (Figure~\ref{fig:results:EURCrossVal}).
\begin{table}[h!]
\begin{tabular}{|r|rr|rr|rr|rr|}
\hline
& \multicolumn{2}{|c|}{MC} & \multicolumn{2}{c|}{StackMC} & \multicolumn{2}{c|}{Improvement} & \multicolumn{2}{c|}{Time} \\
\hline
Paths & CI & Time & CI & Runtime & Absolute & Ratio & Total & Equivalent \\
\hline
$10^3$ & 0.9059 & 0.01 & 0.0601 & 0.01 & 0.08 & 15.18 & 0.02 & 2.27 \\
$10^4$ & 0.2895 & 0.10 & 0.0194 & 0.06 & 0.27 & 14.92 & 0.16 & 22.57 \\
$10^5$ & 0.0913 & 1.00 & 0.0061 & 0.53 & 0.09 & 14.90 & 1.53 & 222.08 \\
$10^6$ & 0.0288 & 10.15 & 0.0019 & 5.38 & 0.03 & 14.96 & 15.53 & 2271.72 \\
\hline
\multicolumn{9}{c}{}
\end{tabular}
\caption{Effect of the number of simulations on the variance reduction.}
\label{tab:NbPaths}
\end{table}
\begin{figure}[h!]
\subfigure[Size of the confidence interval]{\includegraphics[scale=0.4]{MCvsSMC.png}}
\subfigure[Absolute confidence interval reduction]{\includegraphics[scale=0.4]{MCvsSMC_improvement.png}}
\caption{Variance reduction for a European Call option, between MC and StackMC.}
\label{fig:results:mcvsmc_nsteps}
\end{figure}
The Monte Carlo variance is reduced as expected at a rate of $\mathcal{O}\left(N^{-1/2}\right)$, and the StackMC variance is consistently lower (Figure~\ref{fig:results:mcvsmc_nsteps}).
The improvement brought by the stacking procedure decreases as the number of simulations increases, and is much more stable.
In Figure~\ref{fig:results:EURCrossVal},
we observe a very modest improvement when increasing the number of folds.
The computational expense increased with the number of folds,
due in most part to the time spent fitting all the functions~$g_k$.
The random sub-sampling performs worse than K-folds in all cases, and performance depends of the balance between data used to estimate the model parameters (in-sample data), and to calculate~$\alpha$ (out-of-sample data). Runtime is lower than that of K-folds, and decreases as the proportion of training data increased,
largely due to the shorter time spent estimating~$\alpha$ on a smaller test set.
\begin{figure}[h!]
\subfigure[Confidence interval reduction ratio]{\includegraphics[scale=0.4]{kfolds_random_improvement.png}}
\subfigure[Computation time of the stacking procedure]{\includegraphics[scale=0.4]{kfolds_random_runtime.png}}
\caption{Variance reduction by cross-validation between $K$-folds and random sub-sampling.}
\label{fig:results:EURCrossVal}
\end{figure}
\subsubsection{Choice of fit model} \label{sec:results_fit_model}
To investigate the effect of the fit model, we perform the stacking procedure using polynomial functions~$g(\cdot)$
of increasing order, and compare these results with those obtained for a piecewise linear fitting function (Figure~\ref{fig:results:mcvsmc_poly}).
Rather than performing a non-linear fit to determine the inflection point,
the piecewise function is obtained by filtering the zero-valued training data,
and by estimating a linear fit to the remaining points.
Values to the left of the intercept with the horizontal axis are then set to zero.
\begin{figure}[!htbp]
\subfigure[Size of the confidence interval]{\includegraphics[scale=0.4]{MCvsSMC_poly.png}}
\subfigure[Confidence interval reduction ratio]{\includegraphics[scale=0.4]{MCvsSMC_poly_improvement.png}}
\caption{Variance reduction between MC and StackMC by fit model for a European Call option.}
\label{fig:results:mcvsmc_poly}
\end{figure}
We observe a large improvement between orders~$1$ and~$2$,
and small improvements thereafter up to order~$8$, after which the results worsen
(Figure~\ref{fig:results:mcvsmc_poly}).
On average, the performance of the piecewise linear fitter is comparable to that of the polynomial functions (Figure~\ref{fig:results:mcvsmc_poly}).
It is worth noting that our method for finding the piecewise linear function described above
was chosen for computational speed.
For comparison purposes, we also applied a non-linear method (using \texttt{Scikit-learn}) to fit a generic piecewise-linear function with one inflection point,
and found the gains in variance reduction to be almost identical,
albeit with a much higher computational cost.
\subsection{Pricing Asian options}
We are now interested in testing the Stacked Monte Carlo procedure on some path-dependent option,
and we consider the case of an arithmetic Asian option, whose payoff is given by
$$
\Phi_T := \left(\frac{1}{M}\sum_{i=1}^{M}S_{t_i} - K\right)_+,
$$
for some strike~$K$ and some monitoring dates $0<t_1<\ldots<t_{M}$.
Under no-arbitrage arguments, the price of the Asian option reads, at time zero,
$\mathrm{e}^{-rT}\EE{\Phi_T}$.
In the Black-Scholes model~\eqref{eq:BS}, by independence of the Gaussian increments, we can write
(with $t_0=0$)
$$
\sum_{i=1}^{M}S_{t_i}
= S_{t_0}\sum_{i=0}^{M-1}\prod_{j=0}^{i}\frac{S_{t_{j+1}}}{S_{t_j}}
= S_0\sum_{i=0}^{M-1}\prod_{j=0}^{i}\exp\left\{\left(r-\frac{\sigma^2}{2}\right)(t_{j+1}-t_{j}) + \sigma\sqrt{t_{j+1}-t_{j}}X_j\right\}
=: f(X_1,\ldots, X_M),
$$
where $(X_1,\ldots, X_M)$ is a centered Gaussian vector with identity covariance matrix.
The fitting problem is thus of dimension~$M$, where the variables~$X_1, \ldots, X_M$
correspond to the Brownian increments.
With the same parameters as in Section~\ref{sec:results_european},
Figure~\ref{fig:results:asians_toy} shows the results of the StackMC
procedure using polynomial surfaces of degrees~$2$ and~$4$,
and a piecewise linear surface as in Section~\ref{sec:results_fit_model}.
\begin{figure}[!htbp]
\subfigure[Polynomial of order 2.]{
\includegraphics[scale=0.32]{poly2.png}}
\subfigure[Polynomial of order 4.]{
\includegraphics[scale=0.32]{poly4.png} }
\subfigure[Piecewise linear]{
\includegraphics[scale=0.32]{piecewise_linear.png}}
\caption{StackMC variance reduction on an Asian option with $M=2$.}
\label{fig:results:asians_toy}
\end{figure}
Tables~\ref{tab:AsianComparre2}, \ref{tab:AsianComparre50} and~\ref{tab:AsianComparre100} illustrate the method with
respectively~$M=2$, $M=50$ and~$M=100$, in order to to illustrate the effect of the dimensionality.
In this example, the advantage of using a piecewise linear fitter is apparent:
not only does this scheme deliver a greater improvement in the variance,
but the computational effort required is such that it enables scaling of the problem to high dimensions.
This property is essential to the application of the StackMC methodology to Asian options,
for which each path is defined by an $M$-dimensional vector of Brownian increments.
The runtime and memory requirements increase greatly when the number of time steps increases,
and the performance of the second-order polynomial fit is consistently worse than that of the piecewise linear function.
\begin{table}[!htbp]
\begin{tabular}{rrrrr|r|rr|}
\cline{6-8}
\multicolumn{ 1}{c}{} & & & & & Improvement & \multicolumn{ 2}{c|}{Time} \\
\cline{2-8}
\multicolumn{ 1}{c|}{} & Fit model & Price & CI & Runtime & Ratio & Total & Equivalent \\
\hline
\multicolumn{ 1}{|r|}{MC} & & 8.1024 & 0.0698 & 21.36 & & & \\
\hline
\multicolumn{ 1}{|c|}{} & Polynomial 2 & 8.1108 & 0.0097 & 1.23 & 7.18 & 22.59 & 1100.43 \\
\multicolumn{ 1}{|r|}{StackMC} & Polynomial 4 & 8.1112 & 0.0064 & 1.54 & 10.97 & 22.90 & 2570.85 \\
\multicolumn{ 1}{|c|}{} & Piecewise Linear & 8.1117 & 0.0042 & 1.27 & 16.71 & 22.63 & 5963.16 \\
\hline
\\
\end{tabular}
\caption{Comparison of MC and StackMC for the Asian option with $M=2$.}
\label{tab:AsianComparre2}
\end{table}
\begin{table}[!htbp]
\begin{tabular}{rrrrr|r|rr|}
\cline{6-8}
& & & & & Improvement & \multicolumn{ 2}{c|}{Time} \\
\cline{2-8}
\multicolumn{ 1}{c|}{\multirow{1}{*}{}} & Fit model & Price & CI & Runtime & Ratio & Total & Equivalent \\
\hline
\multicolumn{ 1}{|r|}{MC} & & 5.8532 & 0.0502 & 22.24 & & & \\
\hline
\multicolumn{ 1}{|r|}{\multirow{2}{*}{StackMC}} & Polynomial 2 & 5.8582 & 0.0073 & 102.64 & 6.87 & 124.88 & 1049.08 \\
\multicolumn{ 1}{|c|}{} & Piecewise Linear & 5.8567 & 0.0502 & 3.10 & 19.88 & 25.34 & 8785.44 \\
\hline
\\
\end{tabular}
\caption{Comparison of MC and StackMC for Asian option pricing with $M=50$
(the memory requirements for the 4th-order surface fit is prohibitive, and hence omitted).}
\label{tab:AsianComparre50}
\end{table}
\begin{table}[!htbp]
\begin{tabular}{rrrrr|r|rr|}
\cline{6-8}
\multicolumn{ 1}{c}{} & & & & & Improvement & \multicolumn{ 2}{c|}{Time} \\
\cline{2-8}
\multicolumn{ 1}{c|}{} & Fit model & Price & CI & Runtime & Ratio & Total & Equivalent \\
\hline
\multicolumn{ 1}{|r|}{MC} & & 5.8209 & 0.0499 & 23.10 & & & \\
\hline
\multicolumn{ 1}{|r|}{\multirow{2}{*}{StackMC}} & Polynomial 2 & 5.8100 & 0.0076 & 1149.01 & 6.60 & 1172.12 & 1005.32 \\
\multicolumn{ 1}{|c|}{} & Piecewise Linear & 5.8102 & 0.0025 & 5.59 & 19.84 & 28.69 & 9092.11 \\
\hline
\\
\end{tabular}
\caption{Comparison of MC and StackMC for Asian option pricing with $M=100$.}
\label{tab:AsianComparre100}
\end{table}
\vspace{1cm}
We add one final numerical example (Table~\ref{tab:AsianComparre365}) for the Asian case,
in line with market considerations,
namely with a piecewise linear model and daily time intervals ($M=365$ and $T=1$ year).
\begin{table}[!htbp]
\begin{tabular}{|r|rrr|rrr|rr|rr|}
\hline
& \multicolumn{ 3}{|c}{Monte Carlo} & \multicolumn{ 3}{|c}{Stacked Monte Carlo} & \multicolumn{ 2}{|c}{Improvement} & \multicolumn{ 2}{|c|}{Time} \\
\hline
Paths & Price & CI & Time & Price & CI & Time & Abs & Ratio & Total & Equivalent \\
\hline
1E4 & 5.7867 & 0.1570 & 2.97 & 5.7759 & 0.0084 & 2.10 & 0.0149 & 18.77 & 5.07 & 1045.88 \\
1E5 & 5.7727 & 0.0496 & 29.56 & 5.7762 & 0.0025 & 18.12 & 0.0471 & 19.84 & 47.68 & 11635.73 \\
1E6 & 5.7755 & 0.0157 & 299.46 & 5.7759 & 0.0008 & 199.37 & 0.0149 & 19.92 & 498.83 & 118783.86 \\
\hline
\multicolumn{11}{c}{}
\end{tabular}
\caption{Comparison of MC and StackMC for Asian option pricing with $M=365$.}
\label{tab:AsianComparre365}
\end{table}
For $N=10^5$ Monte Carlo simulations,
the additional time spent on the stacking procedure is about~$18$ in our time units
(equivalent to about ten seconds),
which yields a nearly 20-fold improvement in the variance.
\section{Stacked Monte Carlo for stochastic volatility models}
\label{sec:results_stochvol}
We now adapt the Stacked Monte Carlo method to European and Asian options in local stochastic volatility models of the form
\begin{equation}\label{eq:SDEStochVol}
\left\{
\begin{array}{rll}
\displaystyle \frac{\mathrm{d} S_t}{S_t} & = \displaystyle r \mathrm{d} t + \sigma_{\mathrm{loc}}(S_t)\xi(t, V_t)\mathrm{d} W_t, & S_0>0,\\
\mathrm{d} V_t & = \displaystyle b(t, V_t)\mathrm{d} t + a(t, V_t)\mathrm{d} B_t, & V_0>0,
\end{array}
\right.
\end{equation}
where~$r$ denotes the risk-free interest rate, and~$W, B$ are two Brownian motions with correlation~$\varrho\in [-1, 1]$.
The coefficients~$b(\cdot)$, $a(\cdot)$, $\sigma_{\mathrm{loc}}(\cdot)$ and $\xi(\cdot)$ are left undefined,
and are such that a unique solution to the system exists.
Sets of sufficient conditions are classical, and can be found in~\cite[Chapter 5]{Karatzas} for example.
\subsection{Methodology}
We start by discretising the system~\eqref{eq:SDEStochVol} following some Euler scheme.
For the purpose of our methodology, any convergent schemes suffices,
and an overview of such discretisations can be found in~\cite{Kloeden}.
In particular, we start by drawing two matrices of standard Gaussian samples~$\boldsymbol{W}\in\mathcal{M}_{N,M}$
and~$\boldsymbol{B}\in\mathcal{M}_{N,M}$
(corresponding to the standardised Brownian increments of the stock price and the variance respectively),
such that the correlation between~$\boldsymbol{W}_{ij}$ and ~$\boldsymbol{B}_{ij}$
is equal to~$\rho$, for any $i=1,\ldots, N$ and $j=1,\ldots, M$,
where~$N$ and~$M$ respectively denote the number of paths and the number of time steps.
For each path, we can then compute the price of the option under consideration;
we shall denote by $\boldsymbol{\Pi} = (\Pi_1, \ldots, \Pi_N)$ the corresponding vector of prices.
Dividing the sample into $K$ folds of sizes $N_1, \ldots, N_K$,
we denote by $\boldsymbol{\Pi}^k \in \mathbb{R}^{N_k}$ the part of the vector~$\boldsymbol{\Pi}$ corresponding to the $k$-th fold and by $\boldsymbol{\breve{\Pi}}^k \in \mathbb{R}^{N-N_k}$ the vector~$\boldsymbol{\Pi}$ without~$\boldsymbol{\Pi}^k$
($k=1, \ldots, K$), and similarly for~$\boldsymbol{W}^k \in\mathcal{M}_{N_k,M}$ and $\breve{\boldsymbol{W}}^k\in\mathcal{M}_{N-N_k,M}$ (column-wise).
Up to reordering, we assume that the elements of~$\boldsymbol{\Pi}$ and the rows of~$\boldsymbol{W}$ are ordered,
so that the first~$N_1$ of them correspond to the first fold, and so on.
For any $k=1, \ldots, K$, a predictor~$\overline{\boldsymbol{\Pi}}^k$ of~$\boldsymbol{\Pi}^k$ is then obtained as
$$
\overline{\boldsymbol{\Pi}}^k\left(\boldsymbol{W}^k\right) := \mathcal{P}\left(\boldsymbol{W}^k\right),
$$
where~$\mathcal{P}$ denotes a polynomial of any order, the coefficients of which are calibrated
through the fit of~$\breve{\boldsymbol{\Pi}}^k$ on the cloud of points~$\breve{\boldsymbol{W}}^k$.
Note again that we do not use the $k$-th fold for the regression, but only for prediction
(the in-sample data).
This fit is essentially a supervised machine learning problem that can be solved
either by least-square regressions or other techniques such as neural networks or random forests~\cite{Hastie}.
In order to set up the control variate, for each fold~$k$, we compute the integral
\begin{equation}\label{eq:ControlVarHeston}
\widehat{\boldsymbol{\Pi}}^k := \int \overline{\boldsymbol{\Pi}}^k\left(\boldsymbol{W}^k\right) n(\boldsymbol{W}^k)\mathrm{d}\boldsymbol{W}^k
= \int \mathcal{P}\left(\boldsymbol{W}^k\right)n(\boldsymbol{W}^k)\mathrm{d}\boldsymbol{W}^k,
\end{equation}
and the version of the control variate~\eqref{eq:methods:smc_k} in the present context reads
$$
\widetilde{\boldsymbol{\Pi}}^k := \frac{1}{N_k}\sum_{j=1}^{N_k}\boldsymbol{\Pi}^k_j
+ \alpha\left(\widehat{\boldsymbol{\Pi}}^k - \frac{1}{N_k}\sum_{j=1}^{N_k}\overline{\boldsymbol{\Pi}}^k\left(\boldsymbol{W}^k_j\right)\right)
$$
With our notations, we have $\boldsymbol{\Pi}^k_j = \boldsymbol{\Pi}_{N_1 + \cdots+N_{k-1} + j}$;
the final stacked estimator corresponding to~\eqref{eq:SMC} is then
$$
\widetilde{\boldsymbol{\Pi}}_{\text{SMC}} := \frac{1}{K} \sum_{k=1}^K \widetilde{\boldsymbol{\Pi}}^k.
$$
On the numerical side, the computation of~\eqref{eq:ControlVarHeston}
may not be that straightforward.
However, in the spirit of Remark~\ref{rem:LinearCaseComput},
we restrict the integration domain to
$\left\{\boldsymbol{W}^k: \mathcal{P}\left(\boldsymbol{W}^k\right)\geq 0\right\}$, which is natural as the payoff should remain positive
for usual type of options such as Calls and Puts
(other constraints can be considered should one be interested in other types of payoff functions).
The linearity of~$\mathcal{P}$ as well as this truncation domain thus yield the closed-form expressions
from Remark~\ref{rem:LinearCaseComput} for the integral~\eqref{eq:ControlVarHeston}.
\begin{remark}
In~\eqref{eq:SDEStochVol}, we could replace the Brownian motion~$B$ by
a more general continuous Gaussian process~$G$ such as a fractional Brownian motion
or a Gaussian Volterra process, in the spirit of the recent rough volatility wave~\cite{RoughVol1, RoughVol2, RoughVol3, RoughVol4, RoughVol5}.
Since any continuous Gaussian Volterra process~$G$
has a representation of the form $G_t = \int_{0}^{t}K(s,t)\mathrm{d} B_s$, for some Brownian motion~$B$
defined on the same filtration as~$G$ and some kernel~$K(\cdot)$,
then the knowledge of the increments~$\boldsymbol{B}$ of~$B$ provides the increments of~$G$,
and the Stacked Monte Carlo methodology above still applies.
\end{remark}
\subsection{Application to the Heston model}
In order to motivate our results numerically,
we specialise~\eqref{eq:SDEStochVol} to the Heston~\cite{Heston} model,
under which the stock price satisfies the system
\begin{equation}\label{eq:Heston}
\left\{
\begin{array}{rll}
\displaystyle \frac{\mathrm{d} S_t}{S_t} & = \displaystyle r \mathrm{d} t + \sqrt{V_t}\mathrm{d} W_t, & S_0>0,\\
\mathrm{d} V_t & = \displaystyle \kappa(\theta-V_t)\mathrm{d} t + \xi\sqrt{V_t}\mathrm{d} B_t, & V_0>0,
\end{array}
\right.
\end{equation}
for some parameters $ \kappa, \theta, \xi>0$, where~$W$ and~$B$
are two Brownian motions with correlation~$\varrho\in [-1, 1]$.
Several Euler schemes exist in the literature for the Heston model,
in particular keeping track of the necessary positivity of the variance process, and we refer the interested reader to~\cite{Alfonsi, Alfonsi1, Alfonsi2} for an overview.
We consider the following set of parameters:
$$
(S, K, r, T, V_0, \kappa, \theta, \xi, \varrho) =
(100, 100, 3.19, 1, 1.02, 6.21, 1.9, 0.61, -0.7),
$$
for which the reference price for of the European Call option, as reported in~\cite{Broadie}, is equal to $6.8061$.
Table~\ref{tab:HestonStackEuro} and Table~\ref{tab:HestonStackAsian} show the results of the procedure with $M=365$ time steps.
The results are not as clear as before, and only lead to a modest improvement in the variance.
\begin{table}[h!]
\begin{tabular}{|r|rrr|rrr|rr|rr|}
\hline
& \multicolumn{ 3}{|c|}{Monte Carlo} & \multicolumn{ 3}{|c}{Stacked Monte Carlo} & \multicolumn{ 2}{|c}{Improvement} & \multicolumn{ 2}{|c|}{Time} \\
\hline
Nb Simulations & Price & CI & Time & Price & CI & Time & Abs & Ratio & Total & Equivalent \\
\hline
1E4 & 6.7930 & 0.1454 & 58.22 & 6.8158 & 0.0905 & 2.13 & 0.0549 & 1.61 & 60.35 & 150.20 \\
5E4 & 6.8197 & 0.0652 & 290.58 & 6.8113 & 0.0388 & 8.70 & 0.0263 & 1.68 & 299.29 & 818.58 \\
1E6 & 6.8118 & 0.0460 & 600.01 & 6.8088 & 0.0272 & 18.33 & 0.0188 & 1.69 & 618.34 & 1712.07 \\
\hline
\multicolumn{11}{c}{}
\\
\end{tabular}
\caption{StackMC for a European option in the Heston model.}
\label{tab:HestonStackEuro}
\end{table}
\begin{table}[h!]
\begin{tabular}{|r|rrr|rrr|rr|rr|}
\hline
& \multicolumn{ 3}{c|}{Monte Carlo} & \multicolumn{ 3}{c|}{Stacked Monte Carlo} & \multicolumn{ 2}{c|}{Improvement} & \multicolumn{ 2}{c|}{Time} \\
\hline
Nb Simulations & Price & CI & Time & Price & CI & Time & Abs & Ratio & Total & Equivalent \\
\hline
1E4 & 3.6115 & 0.0762 & 59.28 & 3.6150 & 0.0469 & 2.09 & 0.0293 & 1.62 & 61.37 & 156.46 \\
5E4 & 3.6222 & 0.0342 & 304.00 & 3.6182 & 0.0200 & 7.93 & 0.0142 & 1.71 & 311.92 & 887.88 \\
1E5 & 3.6159 & 0.0241 & 611.52 & 3.6173 & 0.0140 & 18.63 & 0.0101 & 1.72 & 630.14 & 1809.95 \\
\hline
\multicolumn{11}{c}{}
\\
\end{tabular}
\caption{StackMC for an Asian option in the Heston model.}
\label{tab:HestonStackAsian}
\end{table}
An example of the agreement between the payoff function and the control variate is displayed in Figure~\ref{fig:results:stochvol}.
We note that the correlation is much higher in the case of constant volatility.
Further, when a piecewise linear fit is achieved by filtering the non-zero data and fitting a linear function to the remainder,
the fit appears biased toward smaller estimates (Figure~\ref{fig:results:stochvol}(b)).
This is because in the presence of the variance-produced `noise',
there are data points drawn for negative~$\boldsymbol{x}$-values with a (small) positive payoff,
which are not being filtered, biasing the fit to the left.
This effect could be mitigated by fitting a piecewise linear function to the full dataset
using a non-linear method (Figure~\ref{fig:results:stochvol}(c)).
Similarly, a larger variance reduction might be obtained by fitting a generic function,
such as an $n^\text{th}$ order polynomial.
However we are restricted by computational constraints to considering simple linear functions.
\begin{figure}[h!]
\subfigure[Constant volatility.]{
\includegraphics[scale=0.3]{correlation_asian.png}}
\subfigure[Stochastic volatility.]{
\includegraphics[scale=0.3]{correlation_asian_stochvol.png} }\label{fig:results:stochvolb}
\subfigure[Stochastic volatility.]{
\includegraphics[scale=0.3]{correlation_asian_stochvol_nonlinear.png}}
\caption{Correlation between the data~$f$ and the control variate~$g$ for Asian options;
(a) and~(b) are obtained by filtering non-zero data and performing a linear fit,
while~(c) is determined by fitting a piecewise linear function to all the data using a non linear method.}
\label{fig:results:stochvol}
\end{figure}
\subsection{Comparison with other variance reduction methods}
We finally compare the results achieved using StackMC with those obtained with other commonly employed variance reduction methods.
We concentrate on European and Asian Call options in the Black-Scholes model~\eqref{eq:BS}
as in Section~\ref{sec:results_european}
In all cases we draw~$100,000$ simulations, apply K-fold cross validation with $K=2$,
and choose a piecewise linear control variate function (fitted with a linear method to the positive data).
\subsubsection{Antithetic updates}
We compare StackMC to a standard Monte Carlo method with antithetic updates~\cite{Glasserman}.
With the same values as in Section~\ref{sec:results_european},
From the results in Table~\ref{tab:StacKMCAnti},
we see that StackMC achieves much greater variance reduction, for both European and Asian options.
\begin{table}[h!]
\begin{tabular}{r|rr|rr|rr|rr|}
& \multicolumn{ 4}{c|}{European option} & \multicolumn{4}{c|}{Asian option} \\
\cline{2-9}
& \multicolumn{ 2}{c|}{Result} & \multicolumn{ 2}{c|}{Improvement} & \multicolumn{ 2}{|c|}{Result} & \multicolumn{ 2}{|c|}{Improvement}\\
\cline{2-9}
& Price & CI & Abs & Ratio & Price & CI & Abs & Ratio\\
\hline
MC & 10.450 & 0.0914 & & & & & & \\
\hline
Antithetic MC & 10.452 & 0.0646 & 0.0268 & 1.41 & 5.7803 & 0.0351 & 0.0145 & 1.41\\
{}
StackMC & 10.452 & 0.0061 & 0.0852 & 14.90 & 5.7763 & 0.0025 & 0.0471 & 19.75\\
\hline
\multicolumn{9}{c}{}
\\
\end{tabular}
\caption{StackMC with antithetic updates.}
\label{tab:StacKMCAnti}
\end{table}
\subsubsection{Geometric mean control variate}
For an arithmetic Asian option in the Black-Scholes models, it is common practice to use
the geometric version as a control variate~\cite{Joshi, Kemna},
so that the original pricing problem is replaced with
\begin{equation}\label{eq:results:geom_cv}
\widehat{C}_{\text{A}} := \left(C_{\text{A}} - C_{\text{G}}\right) + \widehat{C}_{\text{G}},
\end{equation}
with~$C_{\text{A}}$ and~$C_{\text{G}}$ the arithmetic and Geometric Call option prices.
The difference $ (C_{\text{A}} - C_{\text{G}})$ is computed using the Monte Carlo simulations.
Note that $C_{\text{G}}$ is available in closed form, and is equal to~\cite{Kemna}
$$
C_{\text{G}} = \mathrm{e}^{-rT}\left[S_0\mathrm{e}^{aT}\mathcal{N}(d_+) - K\mathcal{N}(d_-)\right],
$$
with
$$
d_{\pm} := \frac{\log(S_0/K) + \left(a\pm \frac{\sigma^2}{3}\right)T}{\sigma\sqrt{T}/3}
\qquad\text{and}\qquad
a := \frac{1}{2}\left(r-\frac{\sigma^2}{2}\right).
$$
The comparison between StackMC and the geometric mean as control variate for an Asian option with parameters as in Section~\ref{sec:results_european} and~$M=365$ is shown in
Table~\ref{tab:CampreStackGeom}.
\begin{table}[h!]
\begin{tabular}{r|rr|rr|}
& \multicolumn{ 2}{c|}{Result} & \multicolumn{ 2}{c|}{Improvement} \\
\cline{2-5}
& Price & CI & Abs & Ratio \\
\hline
MC & 5.7811 & 0.0050 & & \\
\hline
CV MC & 5.7795 & 0.0023 & 0.0473 & 21.71 \\
StackMC & 5.7758 & 0.0025 & 0.0470 & 19.82 \\
\hline
\multicolumn{5}{}\\
\end{tabular}
\caption{Comparison of StackMC and MC with Geometric control variate.}
\label{tab:CampreStackGeom}
\end{table}
The variance reduction achieved with the geometric mean is slightly larger than that of StackMC. However, we may also apply Stacked MC to the new control variate
problem~\eqref{eq:results:geom_cv} and compound the improvements.
The results of the StackMC applied to the modified problem~\eqref{eq:results:geom_cv} are summarised in Table~\ref{tab:GeomMeanStackMCControl}.
\begin{table}[h!]
\begin{tabular}{r|rr|rr|}
& \multicolumn{ 2}{c|}{Result} & \multicolumn{ 2}{c|}{Improvement} \\
\cline{2-5}
& Price & CI & Abs & Ratio \\
\hline
\multicolumn{1}{|c|}{StackMC+CV} & 5.7796 & 0.0011 & 0.0485 & 46.12 \\
\hline
\multicolumn{5}{}
\\
\end{tabular}
\caption{StackMC on the modified problem.}
\label{tab:GeomMeanStackMCControl}
\end{table}
The advantage of the stacking method is that whilst using the geometric mean is only successful in the particular case of the arithmetic Asian option, StackMC has general validity and can be applied (in theory) to any problem.
As such, it applies to the control variate modified problem~\eqref{eq:results:geom_cv},
which delivers an additional compounded variance reduction (Figure~\ref{fig:results:varreduction}).
\begin{figure}[h!]
\includegraphics[scale=0.5]{cv_correlation.png}
\caption{Correlation between the data and the fit function for the modified Asian option pricing problem using the Geometric mean as control variate.}
\label{fig:results:varreduction}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,604 |
Q: CMake Warning, cannot generate a safe runtime search path for target I am trying to build Pangolin v0.5 by my own. However when executing cmake I get the following warning for several builds:
here for example with VideoViewer
Cmake Warning at tools/VideoViewer/CMakeLists.txt:5 (add_executable):
Cannot generate a safe runtime search path for target VideoViewer becaause
files in some directories may conflict with libraries in implicit
directorys:
runtime library[libpython2.7.so.1.0] in /usr/lib/x86_64-linux-gnu may be
hidden by files in: /opt/conda/lib
However, the compiliation works, and after this I do make install
This results in a later error when I try to build another project that relies on Pangolin. There he cannot find certain classes etc., which resuls in a build error.
I am on Ubuntu 16.04 on a Docker Image, I have Anaconda installed for Python.
My LD_LIBRARY_PATH looks like the following:
/usr/local/nvidia/lib:/usr/local/nvidia/lib64/:opt/conda/lib
Removing the /opt/conda/lib part from the LD_LIBRARY_PATH did not help.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 563 |
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